From 758800f92fd84ee036e9b10e91a7facd54d3712c Mon Sep 17 00:00:00 2001 From: LightChainr Date: Sun, 13 Sep 2026 01:46:30 +0800 Subject: [PATCH 01/17] Consolidate geometric balance manuscript and prove arbitrary-period full-law criterion Add orientation-uniform staircase crossings with explicit periodic closure and finite-group translation packing. This proves log N / ell -> 0 is necessary and sufficient for the whole birth law on arbitrary honest integer-period tori, while consolidating #735's weaker geometry for balance-root consistency. Five additive files; five local tests and 135168 finite configurations checked. No full repository CI or publication novelty claim. No existing research asset or navigation file modified. Refs #613 #735 #736 #650. --- docs/manuscripts/geometric-balance/README.md | 29 + .../geometric-balance/manuscript.md | 558 ++++++++++++++++++ .../oblique-corridor-controls.json | 296 ++++++++++ scripts/oblique_winding_corridor.py | 327 ++++++++++ tests/test_oblique_winding_corridor.py | 54 ++ 5 files changed, 1264 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/README.md create mode 100644 docs/manuscripts/geometric-balance/manuscript.md create mode 100644 results/research-control-20260913/oblique-corridor-controls.json create mode 100644 scripts/oblique_winding_corridor.py create mode 100644 tests/test_oblique_winding_corridor.py diff --git a/docs/manuscripts/geometric-balance/README.md b/docs/manuscripts/geometric-balance/README.md new file mode 100644 index 00000000..96a3ccd8 --- /dev/null +++ b/docs/manuscripts/geometric-balance/README.md @@ -0,0 +1,29 @@ +# Geometric balance: one manuscript, one new lemma + +Read [manuscript.md](manuscript.md). It combines the root theorem of #735 with +#736's full-law question, and supplies the missing arbitrary-direction +staircase corridor. The resulting all-period full-law criterion is +`log N / ell -> 0`; balance-root consistency only needs `ell -> infinity`. + +The new proof uses axis-aligned NN rectangles and disjoint translates in the +finite quotient group. It does not rotate the physical interaction, need an +ambient-primitive shortest vector, or import a fixed-width continuum limit. +No unrelated source/Jordan calculation is a dependency. + +From the repository root: + +```sh +python -m unittest discover -s tests -p 'test_oblique_winding_corridor.py' -v +python scripts/oblique_winding_corridor.py --output /tmp/oblique-corridor-new.json +``` + +The script refuses to overwrite an existing result. The committed result is +`results/research-control-20260913/oblique-corridor-controls.json`. +Python standard library only. Five local tests and 135,168 tiny configurations +were executed; full repository CI was not run. No Monte Carlo was performed. + +The manuscript states the external RSW, site-sharpness and matching inputs, +contains the entire root and concentration arguments, and gives a bounded +closest-source comparison. It is an author-supplied proof; no independent +publication acceptance or originality certification is claimed. Existing +proofs, data, frozen designs and research branches are retained unchanged. diff --git a/docs/manuscripts/geometric-balance/manuscript.md b/docs/manuscripts/geometric-balance/manuscript.md new file mode 100644 index 00000000..431961fe --- /dev/null +++ b/docs/manuscripts/geometric-balance/manuscript.md @@ -0,0 +1,558 @@ +# Balance without concentration: two geometric scales for square-site percolation on integer-period tori + +**Working mathematical manuscript — 2026-09-13.** This text consolidates the +probability argument in #735 and the axial full-law result in #736. Its new +step is an orientation-uniform staircase corridor and translation-packing +argument, which proves the full-law criterion for arbitrary integer periods. +It is an author-supplied proof under the explicit standard inputs below, not +an independently accepted publication or a claim of literature priority. + +## Abstract + +Let independent nearest-neighbour site percolation be defined on +\(\mathbb Z^2/\Lambda\), where \(\Lambda\) is a rank-two integer lattice, with +\(N=[\mathbb Z^2:\Lambda]\) vertices and shortest nonzero Euclidean period +\(\ell\). Write \(r\in\{0,1,2\}\) for the rank of the ambient homology image, +and \(P_j(p)=\Pr_p(r=j)\). The topological balance root solves \(P_2=P_0\). +Using site sharpness and matching duality, its convergence to the infinite +square-site critical probability is uniform over all period lattices as +\(\ell\to\infty\), with no area, aspect, or shear constraint. In contrast, +for any sequence of honest tori with \(N\to\infty\), the entire two-birth +mixture converges to a point mass at that critical probability **if and only +if** \(\log N/\ell\to0\). Necessity uses critical square-site box crossing: +an integer staircase of axis-aligned crossing rectangles closes around the +actual shortest period, at arbitrarily small exponential cost below +criticality. A finite-group packing lemma produces sufficiently many disjoint +translates without rotating the interaction or requiring an ambient-primitive +period. We also give an elementary, all-period endpoint-splitting corollary +when \(\log N/\ell\to\infty\). No critical exponent, conformal-invariance +assumption, numerical critical probability, or growing transfer matrix enters. + +## 1. Model, inputs, and statements + +### 1.1 The finite quantities + +The physical interaction is the nearest-neighbour (NN) square lattice with +unit edges. Quotient by a full-rank subgroup \(\Lambda\le\mathbb Z^2\), while +retaining lifted edge displacements. Assume each periodic unit square is an +embedded cell with four distinct corners; this is automatic when the shortest +period is sufficiently large. These are the **honest tori** throughout. A +change of period basis is not a rotation of the NN interaction. + +For an occupied vertex set \(\omega\), let \(G_\omega\) be its induced NN +graph in \(T_\Lambda=\mathbb R^2/\Lambda\), and put +\[ + r(\omega)=\dim_{\mathbb Q}\operatorname{im} + [H_1(G_\omega;\mathbb Q)\longrightarrow H_1(T_\Lambda;\mathbb Q)]. +\] +The source variable is \(X=r-1\), and +\[ + M_\Lambda(p)=\mathbb E_pX=P_2^\Lambda(p)-P_0^\Lambda(p),\qquad + F_\Lambda(p)=\tfrac12\mathbb E_pr=\tfrac12(1+M_\Lambda(p)). \tag{1} +\] +No directional wrapping marginal is substituted for rank. In particular a +rank-one spiral has nonzero projections on two coordinates but still rank one. + +Give each vertex an independent uniform label in \([0,1]\), and occupy labels +at most \(p\). Write \(T_1,T_2\) for the first times at which rank is at least +one and two; a simultaneous jump is allowed. Then +\[ + F_\Lambda(p)=\tfrac12\Pr(T_1\le p)+\tfrac12\Pr(T_2\le p). \tag{2} +\] +Thus \(F\) is a CDF, that of an independent fair choice between the two birth +times. It is not the rank law at a fixed parameter. Denote its inverse by +\(Q_\Lambda\), and its median, equivalently the balance root, by +\(p_\Lambda=Q_\Lambda(1/2)\). The conditional rank odds +\[ + H_\Lambda(p)=\frac{P_2^\Lambda(p)}{P_0^\Lambda(p)+P_2^\Lambda(p)} \tag{3} +\] +are a different function. + +### 1.2 Imported inputs and deterministic duality + +Only the following infinite-volume inputs are used. + +**S: site sharpness.** For NN and for its matching graph NN+NNN, each fixed +subcritical parameter has a bound +\[ + \Pr_p(0\leftrightarrow \partial B_R)\le C(p)e^{-c(p)R},\quad c(p)>0. + \tag{4} +\] +Distances can be Euclidean, with a change in constants because both step sets +are finite. Duminil-Copin--Tassion [S], Theorem 1.1(3), is printed for bonds; +its section 1.2 explicitly discusses the site adaptation. We use the adapted +site conclusion, not the value or threshold theorem for square bond percolation. + +**D: the matching critical relation.** With \(p_c=p_c^{\rm site}(\mathrm{NN})\), +\[ + p_c^{\rm site}(\mathrm{NN+NNN})=1-p_c. \tag{5} +\] +A direct source is Grimmett--Li [D], Theorem 5.5 in the amenable section. +Their Theorem 1.1(a) gives the more general relation with \(p_u\). We use the +amenable square-lattice instance, not a claim about every planar matching pair. + +**R: critical square-site box crossing.** There is \(c_0>0\) such that for +every integer \(s\ge1\), the probability at \(p_c\) of an occupied horizontal +crossing of a \(3s\)-by-\(2s\) axis-aligned rectangle is at least \(c_0\). +Zeng [R], Theorem 1.1, states the requisite square-site RSW result. It is used +as a preprint theorem; no publication status beyond the cited text is assumed. +The full-law necessity uses R. Balance-root consistency and full-law +sufficiency use S and D but not R. + +The finite identity +\[ + r_{\rm NN}(\omega)+r_{\rm NN+NNN}(\omega^c)=2 \tag{6} +\] +is the deterministic digital-Alexander lemma used in the earlier repository +proofs. For completeness, its topological content is as follows. Take a closed +regular neighbourhood \(U\) of the black NN graph and its complementary +subsurface \(V\). The images \(A,C\) of their first homology in that of the +torus satisfy \(C=A^\perp\) for the intersection form. Indeed, the annihilator +of \(A\) identifies with the kernel of restriction \(H^1(T)\to H^1(U)\); +the relative exact sequence, excision, and Poincare--Lefschetz duality identify +that kernel with the image from \(H_1(V)\). Their dimensions sum to two. +The white matching graph has the same ambient homology image as \(V\): in a +face retain the diagonal only for an opposite-white-pair pattern; every other +active white diagonal has a white boundary replacement in that contractible +face. The retained local graph is an embedded spine up to filling local faces, +which does not change the ambient image. This is the precise 4/8 adjacency +convention. The existing `notes/digital-alexander-duality-proof.md` supplies +its facewise implementation. It is not inferred from the numerical controls +in this paper. Short-period cell degeneracies are outside this lemma. + +### 1.3 Main conclusions + +**Theorem A (balance consistency, consolidated from #735).** The balance +root is unique on every honest torus, and +\[ + \lim_{L\to\infty}\ \sup_{\Lambda:\ell(\Lambda)\ge L} + |p_\Lambda-p_c|=0. \tag{7} +\] +More precisely, for each fixed \(p0\), +independent of area and period shape, such that +\[ + \frac{P_2^\Lambda(p)}{P_0^\Lambda(p)} + \le \exp[-\kappa(p)N/\ell],\qquad \ell\ge L(p). \tag{8} +\] +Above \(p_c\), the inverse ratio has the same type of estimate. Every fixed +interior quantile of \(H\) in (3) also converges uniformly under \(\ell\to\infty\). +This is a sufficient geometry for balance; necessity for balance is not asserted. + +**Theorem B (sharp full-law geometry).** For any sequence of honest integer- +period tori with \(N_n\to\infty\), the following are equivalent: + +1. \(\log N_n/\ell_n\to0\). +2. For every fixed \(pp_c\), \(F_n(p)\to1\). +3. Each of \(T_{1,n},T_{2,n}\) converges in probability to \(p_c\). +4. The birth-time mixture in (2) converges weakly to \(\delta_{p_c}\). +5. \(Q_n(u)\to p_c\) for every fixed \(u\in(0,1)\). +6. The convergence in 5 is uniform on every compact subinterval of \((0,1)\). + +The extension of necessity from axial periods (#736) to **all** integer +periods is proved in sections 4--6. It uses only axis-aligned NN rectangles, +not a rotated-lattice RSW assertion. + +**Corollary C (extreme elongation, all period shapes).** If instead +\(\log N_n/\ell_n\to\infty\), then for each \(p\in(0,1)\), +\[ + P_0^n(p),P_2^n(p)\to0,\qquad F_n(p)\to\tfrac12, +\] +and the mixture converges to \(\tfrac12\delta_0+\tfrac12\delta_1\). +When also \(\ell_n\to\infty\), its finite median still converges to \(p_c\) +by Theorem A. This includes genuinely oblique examples, not only rectangles. + +## 2. Finite strict monotonicity + +Adding sites cannot decrease the ambient homology image. Both events +\(r\ge1\) and \(r\ge2\) are increasing and nonconstant. Along an empty-to- +full occupation chain each changes value somewhere. Such a pivotal assignment +of the other sites has strictly positive product probability at every +\(00. +\] +The endpoint values are \(-1,+1\), proving the unique root and invertibility +of \(F\). Also \(P_2'>0\) and \(P_0'<0\), so \(H'>0\) on the interior. +A simultaneous rank jump does not invalidate either argument. + +## 3. Why balance needs no area restriction + +This section puts the arbitrary-period argument of #735 into the same +manuscript; it is not a second independent proof certificate for that PR. + +### 3.1 Shortest period and transverse height + +Choose a shortest period \(u\), so \(|u|=\ell\). It is primitive in +\(\Lambda\), since a proper lattice multiple would not be shortest. Complete +it to a basis \((u,v)\) with determinant \(N>0\), and subtract an integer +multiple of \(u\) from \(v\) so that \(|u\cdot v|\le\ell^2/2\). Since +\(|v|\ge\ell\), the transverse height is +\[ + h=N/\ell\ge\sqrt3\ell/2. \tag{9} +\] +Neither vector must be primitive in ambient \(\mathbb Z^2\). The circle +coordinate \(\theta(x)=\det(u,x)/\ell\pmod h\) is well defined. On vertices, +\(\det(u,x)\pmod N\) implements it exactly. A rank-two image contains a +closed walk with nonzero transverse winding. Every physical step changes +\(\theta\) by at most \(\sqrt2\), on either adjacency. + +### 3.2 Overlapping local balls: a lower bound for no winding + +Assume \(\ell\ge64\), put \(R=\ell/64\), and let \(a_R(p)\) be the one-arm +probability including the occupied origin. Stopping a path on its first exit +makes this an event on vertices at Euclidean distance at most \(R+\sqrt2\). +That ball injects into the torus because its diameter is less than \(\ell\). +The corresponding local event at each of the \(N\) vertices has exactly the +infinite-lattice probability \(a_R\). + +A nonzero winding walk has a lift that escapes this radius. Consequently, +absence of every local arm implies rank zero. The arm-absence events overlap +but are decreasing. Harris positive association, not independence, gives +\[ + P_0\ge(1-a_R)^N. \tag{10} +\] +Harris association for a product measure follows by induction on the sites: +condition on one Bernoulli variable, apply induction to the conditional +covariances, and use that the two conditional means are monotone in the same +direction for the remaining covariance. + +### 3.3 Disjoint transverse bands: an upper bound for rank two + +Use \(k=\lfloor8N/\ell^2\rfloor\) transverse bands of physical width +\(\ell/8\), leaving the residual strip unused. Shift boundaries off vertices. +In each band require an occupied path using only its vertices, joining the +lower and upper boundary layers of thickness \(\sqrt2\). + +A lift with nonzero transverse winding traverses every band. For a chosen +lifted band, take its first exit through the upper side after a visit below +the lower side, and the last entrance through the lower side preceding that +exit. The intervening path lies inside the band. This last-entry construction +allows arbitrary backtracking. The endpoint separation is at least +\(\ell/8-2\sqrt2>R\), and hence witnesses a local arm from an entry vertex. + +The entry layer contains at most \(B=4\lceil\ell\rceil\) vertices. To see +this uniformly in tilt, center unit squares at the lattice vertices; they tile +the quotient with area one each. Squares centered in a layer of width +\(\sqrt2\) lie in a layer of width \(2\sqrt2\). The latter has area +\(2\sqrt2\ell\), since transverse fibres have length \(\ell\). +Thus a band crossing has probability at most \(Ba_R\). The bands have disjoint +site supports, so their events **are** independent. Therefore +\[ + P_2\le(Ba_R)^k,\qquad k\ge4N/\ell^2. \tag{11} +\] +This remains valid for matching diagonals: edge lengths are at most +\(\sqrt2\), and edges between unused bands are not part of the events. + +### 3.4 Rate comparison + +At a fixed subcritical parameter, S gives \(a_R\le C e^{-c\ell/64}\). +For sufficiently large \(\ell\), independently of \(N\), +\[ + \log(Ba_R)\le-c\ell/128,\quad a_R\le1/2,\quad + 2a_R\le c/(64\ell). +\] +Equations (10)--(11) yield +\[ + \log(P_2/P_0) + \le -cN/(32\ell)+2Na_R + \le -cN/(64\ell). \tag{12} +\] +For \(p>p_c\), apply this to the matching graph at \(1-p\) and use (5)--(6). +Two fixed parameters \(p_c\pm\varepsilon\) trap the unique finite root +uniformly over all lattices with large \(\ell\). This proves A, including +the conditional-odds assertion. No value of the probabilities at \(p_c\) +and no quantitative control of \(c(p)\) near \(p_c\) is needed. + +## 4. An occupied staircase ring around an arbitrary integer period + +We now construct the new necessity mechanism. Constants are deliberately +conservative; their optimization is irrelevant to the criterion. + +**Lemma 4.1 (deterministic staircase).** Given \(s\in\mathbb N\) and a +nonzero integer vector \(u\), one can specify at most \(8\ell/s\) horizontal +or vertical rectangle-crossing events, for \(\ell=|u|\ge64s\), such that: + +- every rectangle has side lengths between \(2s\) and \(3s\); +- their projected intersection forces a closed occupied walk with lift + displacement exactly \(u\), including the periodic seam; +- the lifted vertex support \(\widetilde S\) lies within distance \(4s\) + of the segment \([0,u]\). + +When \(u\) is a shortest period of \(\Lambda\), every individual rectangle +injects into \(T_\Lambda\), since its diameter is at most \(\sqrt{13}s<\ell\). + +**Construction and proof.** A lattice reflection or quarter-turn is an exact +symmetry, so for notation arrange \(u=(a,b)\), \(a\ge b\ge0\). No arbitrary +angle rotation is performed. Set \(k=\lceil a/s\rceil\), and +\[ + z_j=(\lfloor ja/k\rfloor,\lfloor jb/k\rfloor),\quad 0\le j\le k. + \tag{13} +\] +Between \(z_j\) and \(z_{j+1}\), move horizontally to +\((z_{j+1,x},z_{j,y})\), then vertically to \(z_{j+1}\). Omit zero moves. +The resulting centers \(c_0,\ldots,c_J\) satisfy +\(c_0=0,c_J=u\), \(J\le2k\), and each axis step has length at most \(s\). + +At every \(c_i\), \(0\le i0\) +there are a fixed integer \(s=s_\eta\) and a fixed parameter +\(p_\eta\in(0,p_c)\), such that for every shortest integer period +\(u\) with \(\ell\ge64s\), the event \(\mathcal G_{u,s}\) of Lemma 4.1 +satisfies +\[ + \Pr_{p_\eta}(\mathcal G_{u,s})\ge e^{-\eta\ell}. \tag{14} +\] + +**Proof.** Choose \(00\). + +First suppose a further subsequence has \(\ell_n\to\infty\). Choose +\(\eta=d/2\) in (16). Its exponent in absolute value is at least +\[ + \frac{e^{(d/2)\ell_n}}{64s_\eta\ell_n}\longrightarrow\infty. +\] +Thus \(P_0^n(p_\eta)\to0\) at a **fixed** \(p_\etap_c\), apply (18) to the matching complement using D and (6). +Thus \(P_2^n(p)\to1\), proving assertion 2. This sufficient argument is +already present in #613; it is not new necessity disguised as a union bound. + +Monotonicity of \(F_n\) traps all \(Q_n(u)\), \(u\in[\delta,1-\delta]\), +between \(p_c-\varepsilon\) and \(p_c+\varepsilon\) for large \(n\). +This proves 2 => 6 => 5. Assertion 4 is equivalent to 2 by the CDF criterion +for weak convergence to a point mass. Finally, because \(T_1\le T_2\), +\[ + \Pr(T_1\le p)\le2F_n(p),\qquad + \Pr(T_2>p)\le2[1-F_n(p)], +\] +so 2 implies 3. Conversely 3 and (2) imply 4. All assertions in B follow. + +### 6.3 Extreme elongation + +Inequality (17) does not require RSW. It is valid for every honest period +lattice. Apply it also to the matching graph, which contains the same forced +NN path, at \(1-p\). Duality gives +\[ + P_2^\Lambda(p)\le + \exp\left[-\frac{N}{2\ell^2}(1-p)^{\sqrt2\ell}\right]. \tag{19} +\] +If \(\log N/\ell\to\infty\), both exponents in (17) and (19) diverge for +every fixed \(01\). With \(k_n=\lceil e^{n^2}\rceil\), +the birth mixture splits to the endpoints while the balance root converges +to \(p_c\). With \(k_n=\lceil e^{\sqrt n}\rceil\), all fixed quantiles +converge to \(p_c\). With \(k_n=\lceil e^{dn}\rceil\), \(d>0\), Theorem B +rules out full-law concentration but does not specify its limiting nonmedian +quantiles. No thin-continuum scaling law is used for these lattice sequences. + +## 7. Prior work, claim boundaries, and what remains + +The ambient homology observable, matching-function root, positive association, +RSW gluing, and translation packing are not presented as inventions of this +manuscript. The following comparison separates source scope from our deductions. + +| Source | Scope inspected in the primary text | Relation to this manuscript | +|---|---|---| +| Mertens--Ziff [MZ], equations (20)--(21) and the subsequent root paragraph | Finite matching lattices, wrapping contrasts, and convergence of roots on square sequences | Establishes the existing observable/root context; not an arbitrary-period full-law necessity statement | +| Duncan--Kahle--Schweinhart [DKS], section 1.1, Theorems 1--4 | Cubical plaquettes on uniform \(N\mathbb Z^d\) quotients and scaled permutohedral systems; in 2D the basic examples are bond square/site triangular | Same ambient-image question, but different finite sequence/model hypotheses; no automatic extension to arbitrary square-site period lattices | +| Zeng [R], Theorem 1.1 | Uniform critical square-site rectangle crossings | Imported box lower bound; the staircase and its endpoint/seam argument are supplied here | +| Duminil-Copin--Tassion [S], Theorem 1.1(3), section 1.2 | Subcritical sharpness, with explicit site adaptation | Imported one-arm decay, not a computed near-critical modulus | +| Grimmett--Li [D], Theorem 5.5; Theorem 1.1 and Remark 1.4 | Site matching critical relation in the amenable case and its wider uniqueness-threshold form | Imported infinite-graph relation; finite rank duality is a separate topological statement | +| Repository #735 and #736 | Arbitrary-period root argument; axial full-law necessity | Author-level predecessors consolidated here; sections 4--6 remove the axial restriction | + +This bounded source comparison does not certify novelty. It did not include +systematic citation-graph traversal, books, theses, or all strip-percolation +literature. In particular, earlier results on elongated strips may package +related rare-opportunity arguments in different language. No claim of absence +from print follows from the searches made for this delivery. + +The manuscript does not solve the original-U candidate-map problem, identify +an irrelevant field, prove an \(L^{-4}\) shift, furnish a new numerical bound +for \(p_c\), establish fixed-width-to-continuum interchange, or predict the +nonmedian limiting law at finite positive \(\log N/\ell\). Theorem A does +not assert its systole condition is necessary for the median. Theorem B's +necessity is for the entire law, not a failure of every selected quantile. + +The real next mathematical comparison is the consolidated theorem versus its +closest existing strip and homological statements. An independent reader can +challenge the seam construction, the finite-group support packing, or the +imported site inputs directly in this one text. No further width table, +source-jet hierarchy, or unrelated Jordan example is a dependency. + +## 8. Executable control and integration boundary + +`scripts/oblique_winding_corridor.py` implements only the new finite geometry: +integer quotient coordinates, staircase rectangles, planar rectangle-crossing +BFS, and a separate spanning-forest winding detector on the physical NN graph. +The tiny exhaustive controls use injecting rectangles, not the asymptotic +\(\ell\ge64s\) constants. They test the deterministic gluing rather than RSW. +The larger full-support cases check actual reduced oblique bases, including +ambient-nonprimitive shortest vectors; rational distances verify the tube +bound. The packing check uses independent finite cyclic-group examples. + +Execution details are in +`results/research-control-20260913/oblique-corridor-controls.json`. +Five local mathematical tests passed. The three tiny tori comprise 135,168 +configurations with no ring-without-winding failure. These checks do not prove +the all-size probability inputs or supply independent publication review. +No Monte Carlo, paid compute, GPU, or full Matching-One repository CI was run. +The code depends only on the standard library, not on the unmerged width-four +certificate stack. Old proofs and frozen results are not overwritten. + +## References + +[S] H. Duminil-Copin and V. Tassion, *A new proof of the sharpness of the phase +transition for Bernoulli percolation and the Ising model*. arXiv:1502.03050v3, +Theorem 1.1 and section 1.2. Primary HTML read: +https://arxiv.org/html/1502.03050v3 + +[D] G. Grimmett and Z. Li, *Hyperbolic site percolation*. arXiv:2203.00981, +Theorem 5.5 (amenable matching pairs), also Theorem 1.1 and Remark 1.4. +Primary theorem text read: https://arxiv.org/html/2203.00981 +Their *Percolation critical probabilities of matching lattice-pairs*, +arXiv:2205.02734v3, introduction (1.3), gives the companion context: +https://arxiv.org/html/2205.02734v3 + +[R] X. Zeng, *A Russo Seymour Welsh Theorem for critical site percolation on +\(\mathbb Z^2\)*, arXiv:1309.2273v1, Theorem 1.1. Primary theorem and setup +read: https://arxiv.org/html/1309.2273 +Kohler-Schindler--Tassion, *Crossing probabilities for planar percolation*, +arXiv:2011.04618, is general RSW background, not a replacement for checking +the critical square-site input: https://arxiv.org/html/2011.04618 + +[MZ] S. Mertens and R. M. Ziff, *Percolation in Finite Matching Lattices*, +arXiv:1603.07289v2. Primary HTML definitions and root discussion read: +https://arxiv.org/html/1603.07289v2 + +[DKS] P. Duncan, M. Kahle and B. Schweinhart, *Homological percolation on a +torus: plaquettes and permutohedra*, arXiv:2011.11903v4. Primary section 1.1 +and theorem statements read: https://arxiv.org/html/2011.11903v4 + +Repository predecessors (not independent sources of validation): +#735 at `9d29d014df28af7c635e6859d98a95ffe2b34d06`; +#736 at `64d809b4404f80ff3f9adf9713337cc76008e92d`; +the finite digital-Alexander note on main. Navigation reset #738 was read +on main together with its subsequent focus clarification. diff --git a/results/research-control-20260913/oblique-corridor-controls.json b/results/research-control-20260913/oblique-corridor-controls.json new file mode 100644 index 00000000..79b61c4c --- /dev/null +++ b/results/research-control-20260913/oblique-corridor-controls.json @@ -0,0 +1,296 @@ +{ + "schema": "matching-one.oblique-corridor.v1", + "date": "2026-09-13", + "scope": "deterministic NN staircase gluing and finite-group packing; not an RSW computation", + "tiny_exhaustive": [ + { + "u": [ + 4, + 0 + ], + "v": [ + 1, + 3 + ], + "scale": 1, + "N": 12, + "rectangles": 12, + "all_rectangles_inject": true, + "configurations": 4096, + "ring_configurations": 279, + "ring_counts_by_occupation": [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 76, + 124, + 66, + 12, + 1 + ], + "implication_failures": [] + }, + { + "u": [ + 4, + 1 + ], + "v": [ + 0, + 4 + ], + "scale": 1, + "N": 16, + "rectangles": 15, + "all_rectangles_inject": true, + "configurations": 65536, + "ring_configurations": 3282, + "ring_counts_by_occupation": [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 31, + 210, + 619, + 974, + 870, + 440, + 120, + 16, + 1 + ], + "implication_failures": [] + }, + { + "u": [ + 4, + 2 + ], + "v": [ + 0, + 4 + ], + "scale": 1, + "N": 16, + "rectangles": 18, + "all_rectangles_inject": true, + "configurations": 65536, + "ring_configurations": 2365, + "ring_counts_by_occupation": [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 10, + 90, + 332, + 656, + 724, + 416, + 120, + 16, + 1 + ], + "implication_failures": [] + } + ], + "geometry": [ + { + "u": [ + 64, + 0 + ], + "v": [ + 17, + 83 + ], + "scale": 1, + "ell_squared": 4096, + "N": 5312, + "ambient_gcd_u": 64, + "events": 192, + "lifted_support_size": 201, + "max_distance_squared_to_segment": "2", + "rank_of_full_corridor_support": 1 + }, + { + "u": [ + 63, + 16 + ], + "v": [ + -33, + 127 + ], + "scale": 1, + "ell_squared": 4225, + "N": 8529, + "ambient_gcd_u": 1, + "events": 237, + "lifted_support_size": 246, + "max_distance_squared_to_segment": "24649/4225", + "rank_of_full_corridor_support": 1 + }, + { + "u": [ + 32, + 57 + ], + "v": [ + -130, + 85 + ], + "scale": 1, + "ell_squared": 4273, + "N": 10130, + "ambient_gcd_u": 1, + "events": 267, + "lifted_support_size": 276, + "max_distance_squared_to_segment": "21025/4273", + "rank_of_full_corridor_support": 1 + }, + { + "u": [ + 48, + 48 + ], + "v": [ + -80, + 112 + ], + "scale": 1, + "ell_squared": 4608, + "N": 9216, + "ambient_gcd_u": 48, + "events": 288, + "lifted_support_size": 297, + "max_distance_squared_to_segment": "9/2", + "rank_of_full_corridor_support": 1 + }, + { + "u": [ + 128, + 33 + ], + "v": [ + -50, + 263 + ], + "scale": 2, + "ell_squared": 17473, + "N": 35314, + "ambient_gcd_u": 1, + "events": 291, + "lifted_support_size": 830, + "max_distance_squared_to_segment": "264196/17473", + "rank_of_full_corridor_support": 1 + }, + { + "u": [ + -65, + 17 + ], + "v": [ + -35, + -188 + ], + "scale": 1, + "ell_squared": 4514, + "N": 12815, + "ambient_gcd_u": 1, + "events": 246, + "lifted_support_size": 255, + "max_distance_squared_to_segment": "26569/4514", + "rank_of_full_corridor_support": 1 + } + ], + "packing": [ + { + "group_order": 23, + "pattern": [ + 0, + 1, + 4 + ], + "difference_size": 7, + "centers": [ + 0, + 2, + 7, + 9, + 14, + 16 + ], + "cover_inequality": true + }, + { + "group_order": 64, + "pattern": [ + 0, + 2, + 5, + 11 + ], + "difference_size": 13, + "centers": [ + 0, + 1, + 8, + 15, + 16, + 23, + 30, + 31, + 38, + 45, + 46 + ], + "cover_inequality": true + }, + { + "group_order": 101, + "pattern": [ + 0, + 1, + 2, + 8, + 9 + ], + "difference_size": 13, + "centers": [ + 0, + 3, + 13, + 16, + 26, + 29, + 39, + 42, + 52, + 55, + 65, + 68, + 78, + 81, + 91 + ], + "cover_inequality": true + } + ], + "total_exhaustive_configurations": 135168, + "all_implications_hold": true +} diff --git a/scripts/oblique_winding_corridor.py b/scripts/oblique_winding_corridor.py new file mode 100644 index 00000000..e5698dd0 --- /dev/null +++ b/scripts/oblique_winding_corridor.py @@ -0,0 +1,327 @@ +#!/usr/bin/env python3 +"""Finite controls for a staircase of NN crossings around an integer period. + +The probability theorem is in docs/manuscripts/geometric-balance/manuscript.md. +This script does not estimate p_c or prove the imported RSW/sharpness inputs. +All geometry and graph calculations below use integers / Fraction. +""" +from __future__ import annotations + +import argparse +import json +from collections import deque +from dataclasses import dataclass +from fractions import Fraction +from math import gcd, isqrt +from pathlib import Path +from typing import Iterable + +Point = tuple[int, int] +NN: tuple[Point, ...] = ((1, 0), (-1, 0), (0, 1), (0, -1)) + + +def det(a: Point, b: Point) -> int: + return a[0] * b[1] - a[1] * b[0] + + +def add(a: Point, b: Point) -> Point: + return a[0] + b[0], a[1] + b[1] + + +def sub(a: Point, b: Point) -> Point: + return a[0] - b[0], a[1] - b[1] + + +@dataclass(frozen=True) +class Torus: + u: Point + v: Point + + def __post_init__(self) -> None: + if det(self.u, self.v) <= 0: + raise ValueError("The ordered integer period basis must have positive determinant") + + @property + def n(self) -> int: + return det(self.u, self.v) + + def reduce(self, z: Point) -> Point: + # Half-open fundamental parallelogram; floor is correct also at negative z. + i = det(z, self.v) // self.n + j = det(self.u, z) // self.n + return z[0] - i*self.u[0] - j*self.v[0], z[1] - i*self.u[1] - j*self.v[1] + + def vertices(self) -> tuple[Point, ...]: + seen = {(0, 0)} + queue = deque([(0, 0)]) + while queue: + z = queue.popleft() + for d in ((1, 0), (0, 1)): + nxt = self.reduce(add(z, d)) + if nxt not in seen: + seen.add(nxt) + queue.append(nxt) + if len(seen) != self.n: + raise AssertionError("Quotient cardinality mismatch") + return tuple(sorted(seen)) + + def period_coordinates(self, z: Point) -> Point: + a, b = det(z, self.v), det(self.u, z) + if a % self.n or b % self.n: + raise AssertionError("Graph cycle displacement is not a period") + return a // self.n, b // self.n + + +@dataclass(frozen=True) +class Rectangle: + x0: int + x1: int + y0: int + y1: int + direction: str + + def __post_init__(self) -> None: + if self.x0 >= self.x1 or self.y0 >= self.y1 or self.direction not in ("h", "v"): + raise ValueError("Nondegenerate h/v crossing rectangle required") + + def points(self) -> tuple[Point, ...]: + return tuple((x, y) for x in range(self.x0, self.x1+1) + for y in range(self.y0, self.y1+1)) + + def translated(self, z: Point) -> Rectangle: + return Rectangle(self.x0+z[0], self.x1+z[0], self.y0+z[1], self.y1+z[1], self.direction) + + +def staircase(u: Point, scale: int) -> tuple[Point, ...]: + """Exact-endpoint integer staircase; zero steps omitted, signs allowed.""" + if scale < 1 or u == (0, 0): + raise ValueError("Positive scale and nonzero period required") + k = (max(abs(u[0]), abs(u[1])) + scale - 1) // scale + centers: list[Point] = [(0, 0)] + for j in range(k): + x0, y0 = (j*u[0])//k, (j*u[1])//k + x1, y1 = ((j+1)*u[0])//k, ((j+1)*u[1])//k + for z in ((x1, y0), (x1, y1)): + if z != centers[-1]: + centers.append(z) + assert centers[-1] == u + assert all((a[0] == b[0]) ^ (a[1] == b[1]) for a, b in zip(centers, centers[1:])) + assert all(max(abs(a[0]-b[0]), abs(a[1]-b[1])) <= scale + for a, b in zip(centers, centers[1:])) + return tuple(centers) + + +def corridor(u: Point, scale: int) -> tuple[Rectangle, ...]: + """Two crossing directions in each hub; one connector per cyclic step.""" + z = staircase(u, scale) + events: list[Rectangle] = [] + for x, y in z[:-1]: + for d in ("h", "v"): + events.append(Rectangle(x-scale, x+scale, y-scale, y+scale, d)) + for a, b in zip(z, z[1:]): + if a[1] == b[1]: + events.append(Rectangle(min(a[0], b[0])-scale, max(a[0], b[0])+scale, + a[1]-scale, a[1]+scale, "h")) + else: + events.append(Rectangle(a[0]-scale, a[0]+scale, + min(a[1], b[1])-scale, max(a[1], b[1])+scale, "v")) + return tuple(events) + + +def support(events: Iterable[Rectangle]) -> set[Point]: + return {z for r in events for z in r.points()} + + +def rectangle_crossing(torus: Torus, rect: Rectangle, occupied: set[Point]) -> bool: + """Planar-lift BFS. Periodic occupancy, but no extra edges across box sides.""" + allowed = {z for z in rect.points() if torus.reduce(z) in occupied} + starts = [z for z in allowed if (z[0] == rect.x0 if rect.direction == "h" else z[1] == rect.y0)] + seen = set(starts) + queue = deque(starts) + while queue: + z = queue.popleft() + if z[0] == rect.x1 if rect.direction == "h" else z[1] == rect.y1: + return True + for d in NN: + nxt = add(z, d) + if nxt in allowed and nxt not in seen: + seen.add(nxt) + queue.append(nxt) + return False + + +def compiled_crossing(torus: Torus, rect: Rectangle, ids: dict[Point, int]): + """Compile box incidence, retaining a distinct planar vertex per box point.""" + pts = rect.points() + pos = {z: i for i, z in enumerate(pts)} + weights = [1 << ids[torus.reduce(z)] for z in pts] + starts = [i for i, z in enumerate(pts) + if (z[0] == rect.x0 if rect.direction == "h" else z[1] == rect.y0)] + ends = {i for i, z in enumerate(pts) + if (z[0] == rect.x1 if rect.direction == "h" else z[1] == rect.y1)} + adjacency = [[pos[add(z, d)] for d in NN if add(z, d) in pos] for z in pts] + + def check(mask: int) -> bool: + queue = [i for i in starts if mask & weights[i]] + seen = set(queue) + for i in queue: + if i in ends: + return True + for j in adjacency[i]: + if j not in seen and mask & weights[j]: + seen.add(j) + queue.append(j) + return False + return check + + +def winding_vectors(torus: Torus, occupied: set[Point]) -> list[Point]: + """Independent spanning-forest detector on the physical quotient NN graph.""" + potential: dict[Point, Point] = {} + gains: list[Point] = [] + for start in sorted(occupied): + if start in potential: + continue + potential[start] = (0, 0) + queue = deque([start]) + while queue: + z = queue.popleft() + for d in NN: + nxt = torus.reduce(add(z, d)) + if nxt not in occupied: + continue + proposed = add(potential[z], d) + if nxt not in potential: + potential[nxt] = proposed + queue.append(nxt) + else: + gain = sub(proposed, potential[nxt]) + if gain != (0, 0): + gains.append(torus.period_coordinates(gain)) + return gains + + +def rank(torus: Torus, occupied: set[Point]) -> int: + gains = winding_vectors(torus, occupied) + if not gains: + return 0 + return 2 if any(det(gains[0], z) for z in gains[1:]) else 1 + + +def squared_distance_to_segment(z: Point, u: Point) -> Fraction: + length2 = u[0]**2 + u[1]**2 + dot = z[0]*u[0] + z[1]*u[1] + if dot < 0: + return Fraction(z[0]**2+z[1]**2) + if dot > length2: + d = sub(z, u) + return Fraction(d[0]**2+d[1]**2) + return Fraction(det(u, z)**2, length2) + + +def circle_packing(modulus: int, pattern: set[int]) -> tuple[list[int], set[int]]: + """Greedy disjoint translates, used only to check the finite-group lemma.""" + if modulus < 1 or not pattern: + raise ValueError("Positive group order and nonempty pattern required") + pattern = {x % modulus for x in pattern} + differences = {(x-y) % modulus for x in pattern for y in pattern} + available = set(range(modulus)) + centers: list[int] = [] + while available: + c = min(available) + centers.append(c) + available.difference_update((c+d) % modulus for d in differences) + return centers, differences + + +def exhaustive_case(u: Point, v: Point, scale: int = 1) -> dict: + torus = Torus(u, v) + if torus.n > 16: + raise ValueError("Exhaustive control limited to 16 sites; no implicit large census") + vertices = torus.vertices() + ids = {z: i for i, z in enumerate(vertices)} + events = corridor(u, scale) + injective = all(len({torus.reduce(z) for z in r.points()}) == len(r.points()) for r in events) + if not injective: + raise ValueError("These test rectangles do not inject; choose a different control geometry") + checks = [compiled_crossing(torus, r, ids) for r in events] + ring_counts = [0]*(torus.n+1) + failures: list[int] = [] + for mask in range(1 << torus.n): + if all(f(mask) for f in checks): + occ = {vertices[i] for i in range(torus.n) if mask >> i & 1} + ring_counts[mask.bit_count()] += 1 + if rank(torus, occ) == 0: + failures.append(mask) + return {"u": u, "v": v, "scale": scale, "N": torus.n, + "rectangles": len(events), "all_rectangles_inject": injective, + "configurations": 1 << torus.n, "ring_configurations": sum(ring_counts), + "ring_counts_by_occupation": ring_counts, "implication_failures": failures} + + +def execute_controls() -> dict: + # Tiny controls verify the geometric gluing, not the asymptotic ell>=64s constants. + small = [exhaustive_case((4, 0), (1, 3)), + exhaustive_case((4, 1), (0, 4)), + exhaustive_case((4, 2), (0, 4))] + geometry = [] + for u, v, s in [((64,0),(17,83),1), ((63,16),(-33,127),1), + ((32,57),(-130,85),1), ((48,48),(-80,112),1), + ((128,33),(-50,263),2), ((-65,17),(-100,-171),1)]: + # Require a reduced basis with shortest u; this criterion is exact in dimension two. + uu = u[0]**2+u[1]**2 + vv = v[0]**2+v[1]**2 + dot = u[0]*v[0]+u[1]*v[1] + if uu > vv or 2*abs(dot) > uu: + q = (2*dot+uu)//(2*uu) + v = v[0]-q*u[0], v[1]-q*u[1] + vv = v[0]**2+v[1]**2 + dot = u[0]*v[0]+u[1]*v[1] + assert uu <= vv and 2*abs(dot) <= uu + torus = Torus(u,v) + ev = corridor(u,s) + pts = support(ev) + maxdist = max(squared_distance_to_segment(z,u) for z in pts) + assert maxdist <= 16*s*s + assert uu >= (64*s)**2 + assert all(len({torus.reduce(z) for z in r.points()}) == len(r.points()) for r in ev) + occ = {torus.reduce(z) for z in pts} + assert all(rectangle_crossing(torus,r,occ) for r in ev) + assert rank(torus,occ) > 0 + # No floating square roots enter these diameter/area inequalities. + assert len(ev)**2 * s*s <= 64*uu # event count <= 8 ell/s + geometry.append({"u":u,"v":v,"scale":s,"ell_squared":uu,"N":torus.n, + "ambient_gcd_u":gcd(abs(u[0]),abs(u[1])), + "events":len(ev),"lifted_support_size":len(pts), + "max_distance_squared_to_segment":str(maxdist), + "rank_of_full_corridor_support":rank(torus,occ)}) + packing = [] + for n, pattern in [(23,{0,1,4}), (64,{0,2,5,11}), (101,{0,1,2,8,9})]: + centers, diff = circle_packing(n,pattern) + packed = [{(x+c)%n for x in pattern} for c in centers] + assert all(not(a & b) for i,a in enumerate(packed) for b in packed[i+1:]) + assert len(centers)*len(diff) >= n + packing.append({"group_order":n,"pattern":sorted(pattern),"difference_size":len(diff), + "centers":centers,"cover_inequality":len(centers)*len(diff)>=n}) + return {"schema":"matching-one.oblique-corridor.v1", "date":"2026-09-13", + "scope":"deterministic NN staircase gluing and finite-group packing; not an RSW computation", + "tiny_exhaustive":small,"geometry":geometry,"packing":packing, + "total_exhaustive_configurations":sum(c["configurations"] for c in small), + "all_implications_hold":not any(c["implication_failures"] for c in small)} + + +def main() -> None: + p = argparse.ArgumentParser(description=__doc__) + p.add_argument("--output", type=Path, required=True) + args = p.parse_args() + if args.output.exists(): + raise SystemExit("Refusing to replace an existing result; use a new --output path") + result = execute_controls() + args.output.parent.mkdir(parents=True,exist_ok=True) + args.output.write_text(json.dumps(result,indent=2,ensure_ascii=False)+"\n",encoding="utf-8") + print(json.dumps({"configurations":result["total_exhaustive_configurations"], + "all_implications_hold":result["all_implications_hold"]})) + + +if __name__ == "__main__": + main() diff --git a/tests/test_oblique_winding_corridor.py b/tests/test_oblique_winding_corridor.py new file mode 100644 index 00000000..ac6d28ee --- /dev/null +++ b/tests/test_oblique_winding_corridor.py @@ -0,0 +1,54 @@ +import sys +import unittest +from pathlib import Path + +sys.path.insert(0,str(Path(__file__).resolve().parents[1]/"scripts")) +from oblique_winding_corridor import (Torus, corridor, staircase, support, + squared_distance_to_segment, rectangle_crossing, rank, circle_packing, exhaustive_case) + + +class ObliqueCorridorTests(unittest.TestCase): + def test_staircase_endpoint_and_step_bound(self): + for u in [(17,5),(16,8),(3,0),(-17,6),(0,-9),(-11,-13)]: + for s in [1,2,3,5]: + path=staircase(u,s) + self.assertEqual(path[0],(0,0)) + self.assertEqual(path[-1],u) + for a,b in zip(path,path[1:]): + self.assertLessEqual(max(abs(a[0]-b[0]),abs(a[1]-b[1])),s) + for z in support(corridor(u,s)): + self.assertLessEqual(squared_distance_to_segment(z,u),16*s*s) + + def test_full_oblique_rings_close(self): + for u,v in [((4,1),(0,4)),((4,2),(0,4)),((-5,2),(-3,-7))]: + t=Torus(u,v); ev=corridor(u,1) + occ={t.reduce(z) for z in support(ev)} + self.assertTrue(all(rectangle_crossing(t,r,occ) for r in ev)) + self.assertGreater(rank(t,occ),0) + + def test_small_twisted_exhaustive(self): + result=exhaustive_case((4,0),(1,3)) + self.assertEqual(result["implication_failures"],[]) + self.assertGreater(result["ring_configurations"],0) + self.assertLess(result["ring_configurations"],result["configurations"]) + + def test_greedy_translation_packing(self): + for n in range(5,26): + for pattern in [{0},{0,1},{0,1,3},{0,2,4}]: + centers,diff=circle_packing(n,pattern) + packed=[{(x+c)%n for x in pattern} for c in centers] + self.assertGreaterEqual(len(centers)*len(diff),n) + for i,a in enumerate(packed): + for b in packed[i+1:]: self.assertFalse(a&b) + + def test_quotient_and_seam_robustness(self): + t=Torus((4,2),(-1,5)) + self.assertEqual(len(t.vertices()),t.n) + for z in [(-9,7),(0,0),(13,-8)]: + for u in [t.u,t.v]: + self.assertEqual(t.reduce(z),t.reduce((z[0]+u[0],z[1]+u[1]))) + self.assertEqual(rank(t,set()),0) + self.assertEqual(rank(t,set(t.vertices())),2) + + +if __name__ == "__main__": unittest.main() From 2f63cbb1735fa799c620ece51568287128c9dac0 Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 13:01:33 +0800 Subject: [PATCH 02/17] Add the two exponential-rectangle birth centres to the geometric-balance manuscript Additive continuation of this same manuscript; four new files, no existing result, freeze, navigation document, or old PR touched. On axial w-by-m tori with w -> infinity, m >= w and log(m)/w -> d in (0,inf), the two rank births converge in probability to the unique inverse- correlation-length values a(d) = kappa_NN^{-1}(d), b(d) = 1 - kappa_matching^{-1}(d), and the birth mixture converges to (delta_a + delta_b)/2. Non-median quantiles go to the corresponding centre; the matching median root is still placed at p_c by the existing root theorem and is not replaced by (a+b)/2. Files added: - docs/manuscripts/geometric-balance/exponential-birth-centres.md - scripts/winding_rate_centres.py (stdlib only) - tests/test_winding_rate_centres.py - results/geometric-consistency/winding-rate-centres.json Executed here (2026-09-13, author-supplied; NOT an independent referee check): - nine local tests pass in this repository tree - both added files byte-identical to the packaged copies - the deterministic result JSON regenerates byte-identically - 140,288 graph/configuration checks over 3x3, 3x4, 4x4 on both adjacencies, 91,668 lifted first-span witnesses cross-checked by an independent planar BFS Not established here: no numerical centre estimate, no Gumbel law, no 1/w shift coefficient, no all-oblique centre formula, no Ornstein-Zernike prefactor transfer, no Monte Carlo, no new p_c. The finite controls are not a proof of the external asymptotic inputs. The source precondition is the previous PR739 comment 5650436953 handoff, which is still absent from this branch; this patch does not depend on it at file level. Full Matching-One repository CI has not been run for this commit. --- .../exponential-birth-centres.md | 514 +++++++++++++++++ .../winding-rate-centres.json | 524 ++++++++++++++++++ scripts/winding_rate_centres.py | 342 ++++++++++++ tests/test_winding_rate_centres.py | 78 +++ 4 files changed, 1458 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/exponential-birth-centres.md create mode 100644 results/geometric-consistency/winding-rate-centres.json create mode 100644 scripts/winding_rate_centres.py create mode 100644 tests/test_winding_rate_centres.py diff --git a/docs/manuscripts/geometric-balance/exponential-birth-centres.md b/docs/manuscripts/geometric-balance/exponential-birth-centres.md new file mode 100644 index 00000000..7633e147 --- /dev/null +++ b/docs/manuscripts/geometric-balance/exponential-birth-centres.md @@ -0,0 +1,514 @@ +# Locating the two births in exponential rectangles + +2026-09-13. Continuation of the same geometric-balance manuscript, #739. +This note replaces the previously unspecified axial birth centres by two +well-defined inverse-correlation-length equations. It is a proof supplied in +this analysis, using the published inputs below; it is not an independent +acceptance of the older manuscript or a claim of literature priority. + +## 1. Main statement + +Let G4 be the nearest-neighbour square lattice and G8 its matching lattice, +with steps (+-1,0), (0,+-1), (+-1,+-1). All random variables are independent +**site** occupations. Write pc(G) for the infinite-volume site threshold. +For either graph define the horizontal inverse correlation length (mass) + +\[ + \kappa_G(p)=\lim_{n\to\infty}-\frac1n + \log\Pr_p^G(0\leftrightarrow(n,0)),\quad 00 satisfies + \[ + \mu_p(A)>\epsilon,\quad + q-p\ge\rho\frac{\log(1/(2\epsilon))}{\log N} + \quad\Longrightarrow\quad\mu_q(A)>1-\epsilon. \tag{7} + \] + Parameters must lie in [0,1] and 0x), including occupation of both endpoints, and put +s_p(x)=tau_p(x)/p. In particular s_p(0)=1. Conditioning a common endpoint to +be occupied leaves a product measure. Harris correlation on that measure gives + +\[ + \tau_p(x+y)\ge \frac{\tau_p(x)\tau_p(y)}p,\qquad + s_p(x+y)\ge s_p(x)s_p(y). \tag{8} +\] + +This argument is valid even when the connection events overlap away from the +common endpoint. It does not incorrectly treat them as independent. +Fekete's lemma along e1 gives + +\[ + \kappa(p)=\inf_{n\ge1}-\frac1n\log s_p(ne_1),\qquad + s_p(ne_1)\le e^{-n\kappa(p)}. \tag{9} +\] + +The limit in (1) is the same because log p/n tends to zero. A straight occupied +path gives kappa(p)<=-log p. AV supplies kappa(p)>0 for p0) on the axial torus. For m>=w>=2, + +\[ + \boxed{f_G(w,m;p)\le 2p\,m w^3\,e^{-(w-1)\kappa_G(p)}.} \tag{11} +\] + +**First-span proof.** A nonzero-homology closed walk has a planar lift whose +endpoint differs by (aw,bm), with (a,b) not both zero. Follow the lift until +its x-range or y-range first reaches w-1. Each physical step changes either +coordinate by at most one, so the entire retained prefix has both ranges at +most w-1. It lies in a translate of the square of vertices {0,...,w-1}^2. +This square injects into the w-by-m torus. The walk within it connects two +opposite sides of the square in one coordinate. + +The relevant edges are the planar edges of this cut square. Extra periodic +edges joining its opposite sides are NOT admitted to this event. The product +law on its vertices is exactly the planar finite-box law. There are at most +N=wm translations, two crossing directions, and w^2 pairs of endpoints. +Equation (10), followed by the union bound, proves (11). + +This argument retains the full distance w-1. An embedded radius-w/2 arm bound +would lose a factor two in the exponential rate and could not identify the +same centre. Backtracking, vertical winding and winding with both projections +nonzero are included. Planarity of the matching graph is not required: the +walk and its lifted displacement remain well defined despite crossing edges. + +## 4. Closing finite connecting seeds into a winding ring + +For fixed p in (0,pc), every epsilon>0 has a finite integer height D and w0 +such that, for every w>=w0 and m>=D, + +\[ + \boxed{1-f_G(w,m;p)\le + \exp\{-\lfloor m/D\rfloor e^{-(\kappa_G(p)+\epsilon)w}\}.} \tag{12} +\] + +Here D and w0 may depend on p and epsilon, but not on w or m. + +**Selecting a finite seed.** By (9), choose an integer L such that +-log s_p(Le1)/L < kappa(p)+epsilon/4. Connections in finite boxes +[-R,L+R] x [-R,R] increase to the full-plane connection as R grows. Choose a +fixed R for which the conditional finite-box probability + +\[ + q=\Pr_p(0\leftrightarrow Le_1\text{ in the box}\mid0\text{ occupied}) + >e^{-(\kappa(p)+\epsilon/2)L}. \tag{13} +\] + +No asymptotic theorem about the shape of a connecting cluster is needed for +this exhaustion step. Set D=2R+1, enlarging R if necessary to at least one. + +**Closing the seam.** Write w=kL+r, 0<=rL+2R, and all seeds use sites in the same D-row band. Require their +connection events and, when r>0, the remaining straight occupied path from +kL to w. A connected walk then runs from 0 to w e1 in the lift, including the +last endpoint, which is the translate of the first. Its projection has +nonzero homology. This is a closed ring, not an open cut-side crossing. + +The first seed has probability p q. If the first j seeds occur, the common +endpoint is occupied. Applying conditional Harris as in (8) to the next seed +shows inductively that the first k seeds have probability at least p q^k. +The remainder path contributes at worst p^r by the same common-endpoint +argument. Therefore the ring probability is at least + +\[ + p^{r+1}q^k\ge e^{-(\kappa(p)+\epsilon)w} \tag{14} +\] + +for all sufficiently large w. For r=0 there is no remainder requirement; +retaining the factor p is a harmless conservative lower bound. Possible +additional overlaps at the final seam only increase the Harris lower bound. + +Pack floor(m/D) disjoint D-row site bands. Their events are independent +because their SITE supports are disjoint. Unused edges between bands are +irrelevant. One successful ring already forces r_G>0, proving (12). + +### The rate statement, before making any inverse in p + +If log m/w -> d with d>=0, (11)--(12) imply for every fixed pd the event vanishes exponentially. For kappa(p)=min{x/2,1-exp(-1)}. Then let epsilon in +(12) decrease to zero AFTER taking liminf. The upper rate follows from (11) +and f<=1. This proves (15) without continuity, strict monotonicity, or an +assumed limit of the birth medians. + +## 5. Why the mass equation has exactly one solution + +The following arguments avoid silently importing a bond-only +inverse-correlation-length theorem. + +### 5.1 Continuity in the subcritical interval + +Monotone coupling makes kappa nonincreasing. Moreover it is the infimum over +L,R of the continuous functions -log q_(L,R)(p)/L, with q the conditional +finite seed probability above. Hence kappa is upper semicontinuous; together +with monotonicity this proves left continuity. + +For right continuity fix pn)<=C0 exp(-c0 n); the prefactor may be taken one in the cited +formulation but is immaterial. For any p<=q<=p0 and connected finite set A +containing 0, + +\[ + \frac{\Pr_q(C_0=A)}{\Pr_p(C_0=A)} + =(q/p)^{|A|}[(1-q)/(1-p)]^{|\partial A|}\le(q/p)^{|A|}. \tag{16} +\] + +Restrict the connection 0<->ne1 to clusters of size at most A0*n. The +remaining probability is bounded by the p0 tail. Using (9), + +\[ + \tau_q(ne_1)\le (q/p)^{\lceil A_0n\rceil}\tau_p(ne_1) + +C_0e^{-c_0 A_0n+O(1)}. +\] + +Taking exponential rates yields + +\[ + \kappa(q)\ge\min\{\kappa(p)-A_0\log(q/p),\ c_0 A_0\}. \tag{17} +\] + +Choose A0 large enough that c0 A0>kappa(p), and then q close to p. This proves +right continuity. The use of a CLUSTER-VOLUME exponential tail here is +essential; a one-arm radius tail alone does not justify this cutoff. + +### 5.2 Strict monotonicity from transitive sharp thresholds + +Suppose 00. Choose d in (0,A) so close to A +that rho*(A-d)/d0 small enough that +rho*(A-d+zeta)/d\epsilon_w:=e^{-(A-d+\zeta)w} +\] + +for large w. The event r_G>0 is increasing and invariant under all torus +translations, which act transitively on the N=wm sites. Also + +\[ + \rho\frac{\log(1/(2\epsilon_w))}{\log(wm)} + \longrightarrow\rho\frac{A-d+\zeta}{d}1-epsilon_w ->1. But (11) and kappa(q)=A>d force +f_G(w,m;q)->0. This contradiction proves strict decrease. Notice the logical +order: (15) was proved without strict monotonicity, so there is no circular +use of the desired inverse-centre formula. + +### 5.3 Range of the mass + +For p sufficiently small, counting nonbacktracking paths of length at least n +on the degree-z graph gives kappa(p)>=-log((z-1)p). Together with the straight +path bound, + +\[ + \max\{0,-\log((z-1)p)\}\le\kappa(p)\le-\log p. \tag{18} +\] + +Thus kappa(p)->infinity as p decreases to zero. If kappa were bounded below +by c>0 as p increases to pc, (10) would imply the uniform susceptibility bound + +\[ + \chi(p)=\sum_x\tau_p(x) + \le p+8p\sum_{n\ge1}n e^{-cn}<\infty. +\] + +This contradicts AV Proposition 5 and Theorem 2, which give +chi(p)->infinity as p increases to pc. Therefore kappa(p)->0. +Together with continuity and strict decrease, this makes kappa a bijection +from (0,pc) to (0,infinity), with a continuous strictly decreasing inverse. + +## 6. Deducing the two centres and the rank-one phase + +Let c_G(d)=kappa_G^{-1}(d). For p0. For +c_G(d)1. Monotonicity extends the latter conclusion +to all larger p. These pointwise limits imply concentration of the first +positive-rank birth at c_G(d) and convergence of its median there. + +Apply this first to G4. For G8, use its OWN site parameter t, its OWN +critical threshold, and its OWN mass kappa_G8. Finite digital duality gives + +\[ + P_2^{G4}(p)=1-f_{G8}(w,m;1-p). \tag{19} +\] + +Consequently the second G4 birth concentrates at 1-c_G8(d), not at +1-c_G4(d). The matching critical relation places pc(G4) strictly between the +two limits. The three off-boundary phases are + +\[ +\begin{array}{c|c} + pb(d)&P_2\to1. +\end{array} \tag{20} +\] + +Joint concentration follows from the union bound on the two actual coupled +births. Mixture weak convergence, fixed nonmedian quantiles and (6) follow +without a finite-birth independence assumption. Since all times lie in [0,1], +convergence of the moments follows by boundedness. + +The plateau in (20) contains pc. The balance root can select pc inside it +while the two births remain separated. Its location is governed by rare-sector +odds that disappear from the limiting unscaled CDF. No average of the two +centres is substituted for that root. + +## 7. A small exact computation improves the old quantitative brackets + +For d=log 4, the previous nonbacktracking/full-row bounds gave + + a in [1/12,1/4], b in [3/4,27/28]. + +A 3-by-3 planar seed already improves them, without estimating kappa itself. +Condition its left-middle vertex occupied and ask for a connection to the +right-middle vertex using only planar edges of the box. Denote this +conditional probability by q_G(p). By (9), + +\[ + \kappa_G(p)\le-\tfrac12\log q_G(p). \tag{21} +\] + +The seed probabilities have elementary independent derivations: + +\[ +\begin{aligned} + q_{G4}(p) + &=p\{p+(1-p)(2p^3-p^6)\} + =p^2+2p^4-2p^5-p^7+p^8,\\ + q_{G8}(p)&=p[1-(1-p)^3]=3p^2-3p^3+p^4. \tag{22} +\end{aligned} +\] + +For G4, if the centre vertex is closed, either the complete top or complete +bottom three-site detour must be occupied. For G8, any occupied vertex in the +middle column connects the two occupied endpoints. These derivations and an +independent enumeration of the eight remaining bits give identical polynomials. + +Let alpha and beta be the unique roots of q_G4(alpha)=1/16 and +q_G8(beta)=1/16. Exact rational bisection gives + +\[ + \alpha=0.239805566880062101\ldots,\qquad + \beta=0.156394361442176291\ldots. +\] + +Then (21) and the strict monotonicity of kappa give the rigorous bounds + +\[ + \boxed{\frac1{12}\le a(\log4)\le\alpha,\qquad + 1-\beta\le b(\log4)\le\frac{27}{28}.} \tag{23} +\] + +In particular 1-beta=0.843605638557823708... . The JSON stores rational +outward endpoints rather than relying on rounded decimals. Alpha and beta are +roots of FINITE SEED polynomials; they are NOT computed values or point +estimates of the two infinite centre locations. The gain here is a certified +bound: actual centres still require the actual planar mass functions. + +A finite seed is a one-sided certificate. Larger boxes can improve that upper +bound, but this note does not open a box/width sweep or claim an efficient +algorithm for arbitrary precision at a centre. + +## 8. Relation to prior work and remaining question + +The competition between a crossing's exponential cost and the number of +attempts is not a new mechanism. Grimmett's wedge/sponge programme, as recalled +and sharpened by Damron--Lam [DL], already connects a logarithmic geometry to +inverse correlation length. Their Section 1.1.2 records an inverse- +correlation-length formula for a BOND percolation wedge threshold; Section 2 +studies open-boundary rectangle crossings and uses stronger two-point +asymptotics. Those are close antecedents, not a site-torus theorem that can be +copied without examining the event and boundary conditions. + +Here the distinction is two periodic, ambient-rank births in the actual +square SITE / matching SITE pair. Equations (11)--(14) explicitly close the +seam and provide matching exponential rates using elementary product +arguments. We proved the mass properties needed for inversion with the +published AV and FK inputs, rather than asserting unverified site extensions +of Ornstein--Zernike results. No claim is made that inverse correlation length, +its qualitative properties, or this general entropy-cost mechanism are new. +A systematic literature priority conclusion has not been established. + +What remains at fixed d is NOT whether these two axial limiting centres +exist or how to characterize them: (3) settles that in this argument. What is +not determined is the finite-width displacement and fluctuation law at +kappa(p)=d. At equality, polynomial prefactors in a true winding probability, +periodic closure weights, and subexponential factors in m can matter. Neither +(15) nor an uncalibrated FK constant supplies a Gumbel law, a 1/w shift or its +coefficient. Changing to genuinely oblique growing periods may also require +a directional mass, not merely substituting the Euclidean systole in (3). +These are possible extensions of the same paper, not new automatic queues. + +## 9. Finite controls actually executed + +`scripts/winding_rate_centres.py` uses only the Python standard library. +It does not import an older automaton or transfer certificate. + +* Both 3-by-3 conditional seed polynomials are enumerated exactly; their roots + against 1/16 are isolated to dyadic intervals of width 2^-64. +* An independent graph-potential traversal on 3x3, 3x4 and 4x4, for BOTH + adjacencies, checks 140,288 graph/configuration pairs. For each of the 91,668 + nonzero-winding configurations it produces a lifted closed walk, extracts + the first-span prefix, and independently checks a crossing on the cut planar + square. Extra seam edges are excluded from that square. +* At 3x3 the finite seed-ring event, winding probability and conditional-Harris + lower bound are compared at p=1/100,1/4,1/2 with Fraction arithmetic. + The ring event is only sufficient, not all windings. +* The cluster-size truncation comparison used in right continuity is checked + directly on the finite 3x3 product law, at p=1/5 and q=1/4. +* Tests separately compare (22) to enumeration, check strict dyadic endpoint + signs, seam witnesses, probability inequalities and invalid inputs. + +These finite checks do not prove the infinite AV/FK inputs, the asymptotic +inversion or the all-size geometry. Those claims rest on the arguments above. +Full Matching-One repository CI was not run. No Monte Carlo, new numerical pc, +critical exponent fit, or external compute was used. + +## References and exact use + +[AV] T. Antunovic and I. Veselic, *Sharpness of the phase transition and +exponential decay of the subcritical cluster size for percolation on +quasi-transitive graphs*, Journal of Statistical Physics 130 (2008), 983--1009. +https://arxiv.org/html/0707.1089v3 +Theorems 2--3, Proposition 5, model definitions and Section 6 read in primary +HTML. Supplies subcritical site cluster-volume tails and susceptibility +divergence; no matrix or torus-centre formula is attributed to this source. + +[FK] E. Friedgut and G. Kalai, *Every monotone graph property has a sharp +threshold*, Proceedings of the AMS 124 (1996), 2993--3002. +https://www.ams.org/journals/proc/1996-124-10/S0002-9939-96-03732-X/ +The precise statement used here is the published theorem as reproduced in +[DKS] Theorem 6. No new direct reading of the original PDF is claimed. + +[DKS] P. Duncan, M. Kahle and B. Schweinhart, *Homological percolation on a +torus: plaquettes and permutohedra*. +https://arxiv.org/html/2011.11903v4 +Section 1.3, Theorems 5--6, read in primary HTML. Used for (7) and the Harris +formulation, NOT to assert that their model-specific torus theorem is the +square-site statement (3). + +[DL] M. Damron and W.-K. Lam, *Asymptotics for first passage percolation on +logarithmic subgraphs of Z^2*, arXiv:2502.18235v1 (2025). +https://arxiv.org/html/2502.18235v1 +Section 1.1.2 and Section 2 read in primary HTML. Context and closest mechanism: +bond wedges, correlation length and rectangle crossings, not a claimed proof +of our two site-rank birth locations. Original older wedge sources were not +independently read in this delivery. + +[D] Matching critical relation and finite digital-Alexander identity are +specified, sourced and used in the parent geometric-balance manuscript at +PR739 head 758800f92fd84ee036e9b10e91a7facd54d3712c. diff --git a/results/geometric-consistency/winding-rate-centres.json b/results/geometric-consistency/winding-rate-centres.json new file mode 100644 index 00000000..c7f92344 --- /dev/null +++ b/results/geometric-consistency/winding-rate-centres.json @@ -0,0 +1,524 @@ +{ + "schema": "matching-one.winding-rate-centres.v1", + "date": "2026-09-13", + "claim_boundary": "Finite deterministic controls only; no infinite correlation length numerically computed.", + "seed_bounds": { + "NN": { + "conditional_origin": true, + "seed_vertex_count": 9, + "independent_remaining_sites": 8, + "source": [ + 0, + 1 + ], + "target": [ + 2, + 1 + ], + "conditional_connection_counts_by_occupied_other_sites": [ + 0, + 0, + 1, + 6, + 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"p": "1/4", + "winding_probability": { + "numerator": "147889201", + "denominator": "4294967296" + }, + "cut_upper_using_path_mass_lower": { + "numerator": "54", + "denominator": "1" + } + }, + { + "p": "1/2", + "winding_probability": { + "numerator": "28977", + "denominator": "65536" + } + } + ] + }, + { + "width": 4, + "height": 4, + "graph": "matching", + "configurations": 65536, + "nonzero_winding_configurations": 56491, + "first_span_cut_witnesses_checked": 56491, + "winding_counts": [ + 0, + 0, + 0, + 0, + 160, + 1504, + 5496, + 10352, + 12662, + 11424, + 8008, + 4368, + 1820, + 560, + 120, + 16, + 1 + ], + "ring_counts": null, + "rational_probability_checks": [ + { + "p": "1/100", + "winding_probability": { + "numerator": "155794089847908319405147243", + "denominator": "100000000000000000000000000000000" + }, + "cut_upper_using_path_mass_lower": { + "numerator": "686", + "denominator": "390625" + } + }, + { + "p": "1/4", + "winding_probability": { + "numerator": "994873963", + "denominator": "4294967296" + } + }, + { + "p": "1/2", + "winding_probability": { + "numerator": "56491", + "denominator": "65536" + } + } + ] + } + ], + "total_graph_configuration_checks": 140288, + "total_cut_witnesses": 91668 +} diff --git a/scripts/winding_rate_centres.py b/scripts/winding_rate_centres.py new file mode 100644 index 00000000..bba7be18 --- /dev/null +++ b/scripts/winding_rate_centres.py @@ -0,0 +1,342 @@ +#!/usr/bin/env python3 +"""Finite controls for the microscopic winding-cost / two-centre theorem. + +Only deterministic finite product measures are computed. No number returned +here is an estimate of the infinite-lattice inverse correlation length. +Run from the repository root; output files are never overwritten. +""" +from __future__ import annotations +import argparse +from collections import deque +from fractions import Fraction +import json +from math import comb +from pathlib import Path +from typing import Sequence + +STEPS4 = ((1, 0), (-1, 0), (0, 1), (0, -1)) +STEPS8 = STEPS4 + ((1, 1), (1, -1), (-1, 1), (-1, -1)) + + +def steps(kind: str) -> tuple[tuple[int, int], ...]: + if kind not in ('NN', 'matching'): + raise ValueError('kind must be NN or matching') + return STEPS4 if kind == 'NN' else STEPS8 + + +def fraction_record(value: Fraction) -> dict[str, str]: + return {'numerator': str(value.numerator), 'denominator': str(value.denominator)} + + +def decimal(value: Fraction, places: int = 24) -> str: + from decimal import Decimal, localcontext + with localcontext() as ctx: + ctx.prec = places + 12 + return format(Decimal(value.numerator) / Decimal(value.denominator), f'.{places}f') + + +def bernstein_count_value(counts: Sequence[int], p: Fraction) -> Fraction: + """counts[k] is a count, not an already normalized Bernstein coefficient.""" + if not 0 <= p <= 1: + raise ValueError('probability outside [0,1]') + n = len(counts) - 1 + return sum((Fraction(c) * p**k * (1-p)**(n-k) for k, c in enumerate(counts)), Fraction()) + + +def finite_adjacency(width: int, height: int, kind: str) -> tuple[tuple[int, ...], ...]: + result = [] + for y in range(height): + for x in range(width): + result.append(tuple(yy * width + xx for dx, dy in steps(kind) + if 0 <= (xx := x+dx) < width + and 0 <= (yy := y+dy) < height)) + return tuple(result) + + +def reached(mask: int, adjacency: Sequence[Sequence[int]], source: int) -> set[int]: + if not (mask >> source) & 1: + return set() + seen = {source} + stack = [source] + while stack: + u = stack.pop() + for v in adjacency[u]: + if ((mask >> v) & 1) and v not in seen: + seen.add(v) + stack.append(v) + return seen + + +def seed_counts(kind: str) -> list[int]: + """P((0,1)<->(2,1) inside 3x3 | (0,1) occupied). + + Eight independent bits remain. This is an explicit finite witness and + hence only provides an UPPER bound -log(q)/2 on the planar mass. + """ + adjacency = finite_adjacency(3, 3, kind) + source, target = 3, 5 + others = [i for i in range(9) if i != source] + counts = [0] * 9 + for word in range(1 << 8): + mask = 1 << source + for j, i in enumerate(others): + mask |= ((word >> j) & 1) << i + if target in reached(mask, adjacency, source): + counts[word.bit_count()] += 1 + return counts + + +def seed_root_interval(counts: Sequence[int], target: Fraction = Fraction(1, 16), + bits: int = 64) -> tuple[Fraction, Fraction]: + """Enclose the finite seed root q(p)=target; NOT the torus birth centre.""" + lo, hi = Fraction(0), Fraction(1) + for _ in range(bits): + mid = (lo+hi)/2 + if bernstein_count_value(counts, mid) < target: + lo = mid + else: + hi = mid + assert bernstein_count_value(counts, lo) <= target <= bernstein_count_value(counts, hi) + return lo, hi + + +def torus_adjacency(width: int, height: int, kind: str): + """Keep parallel periodic edges and their actual lifted steps.""" + return tuple(tuple((((x+dx) % width) + width*((y+dy) % height), dx, dy) + for dx, dy in steps(kind)) + for y in range(height) for x in range(width)) + + +def winding_walk(mask: int, width: int, height: int, kind: str, + adjacency=None) -> list[tuple[int, int]] | None: + """Independent graph-potential traversal returning a lifted closed walk.""" + adjacency = adjacency or torus_adjacency(width, height, kind) + potentials: dict[int, tuple[int, int]] = {} + parents: dict[int, int | None] = {} + for root in range(width*height): + if not ((mask >> root) & 1) or root in potentials: + continue + potentials[root] = (root % width, root // width) + parents[root] = None + queue = deque([root]) + while queue: + u = queue.popleft() + ux, uy = potentials[u] + for v, dx, dy in adjacency[u]: + if not ((mask >> v) & 1): + continue + candidate = (ux+dx, uy+dy) + if v not in potentials: + potentials[v] = candidate + parents[v] = u + queue.append(v) + continue + vx, vy = potentials[v] + gain = (candidate[0]-vx, candidate[1]-vy) + if gain == (0, 0): + continue + assert gain[0] % width == gain[1] % height == 0 + up = [] + node = u + while node is not None: + up.append(node) + node = parents[node] + walk = [potentials[node] for node in reversed(up)] + walk.append(candidate) + node = parents[v] + while node is not None: + nx, ny = potentials[node] + walk.append((nx+gain[0], ny+gain[1])) + node = parents[node] + assert walk[-1] != walk[0] + assert ((walk[-1][0]-walk[0][0]) % width == 0 and + (walk[-1][1]-walk[0][1]) % height == 0) + return walk + return None + + +def cut_witness(walk: Sequence[tuple[int, int]], width: int): + """First range reaching w-1, not a radius-w/2 estimate. + + The entire retained prefix lies in ONE injecting w-by-w vertex square. + All used edges are its planar edges, never the extra torus seam edges. + """ + if width < 2 or not walk: + raise ValueError('nonempty walk and width >=2 required') + x0, y0 = walk[0] + xmin = xmax = x0 + ymin = ymax = y0 + for j, (x, y) in enumerate(walk): + xmin, xmax = min(xmin, x), max(xmax, x) + ymin, ymax = min(ymin, y), max(ymax, y) + if max(xmax-xmin, ymax-ymin) == width-1: + return list(walk[:j+1]), (xmin, ymin), ('x' if xmax-xmin == width-1 else 'y') + assert max(xmax-xmin, ymax-ymin) < width-1 + raise AssertionError('nonzero winding did not span the injecting square') + + +def verify_cut(mask: int, width: int, height: int, kind: str, walk) -> None: + prefix, (xmin, ymin), axis = cut_witness(walk, width) + for x, y in prefix: + assert xmin <= x < xmin+width and ymin <= y < ymin+width + assert (mask >> ((x % width)+width*(y % height))) & 1 + for (x, y), (xx, yy) in zip(prefix, prefix[1:]): + assert (xx-x, yy-y) in steps(kind) + # Independent planar finite-box connectivity check on the projected mask. + boxmask = 0 + for by in range(width): + for bx in range(width): + v = ((xmin+bx) % width)+width*((ymin+by) % height) + boxmask |= ((mask >> v) & 1) << (by*width+bx) + adjacency = finite_adjacency(width, width, kind) + side0 = [width*j for j in range(width)] if axis == 'x' else list(range(width)) + side1 = {width*j+width-1 for j in range(width)} if axis == 'x' else set(range(width*(width-1), width*width)) + assert any(reached(boxmask, adjacency, source) & side1 for source in side0) + + +def rectangle_connection(mask: int, width: int, height: int, kind: str, + xleft: int, length: int, row: int = 1) -> bool: + """A length+1 by 3 planar seed, with coordinates mapped periodically.""" + localmask = 0 + for y in range(3): + for x in range(length+1): + v = (xleft+x) % width + width*((row-1+y) % height) + localmask |= ((mask >> v) & 1) << (y*(length+1)+x) + adj = finite_adjacency(length+1, 3, kind) + return 2*(length+1)-1 in reached(localmask, adj, length+1) + + +def seed_ring(mask: int, width: int, height: int, kind: str) -> bool: + """Concatenate length-2 seeds, then force at most one leftover edge.""" + if width < 3 or height < 3: + raise ValueError('individual 3x3 seeds must inject') + k, remainder = divmod(width, 2) + if not all(rectangle_connection(mask, width, height, kind, 2*j, 2) for j in range(k)): + return False + if remainder: + return bool((mask >> (width+width-1)) & 1 and (mask >> width) & 1) + return True + + +def seed_and_cluster_controls() -> dict: + result = {} + for kind in ('NN', 'matching'): + counts = seed_counts(kind) + lo, hi = seed_root_interval(counts) + # A direct finite-cluster comparison, independent of the asymptotic proof. + adj = finite_adjacency(3, 3, kind) + p, q, cap = Fraction(1, 5), Fraction(1, 4), 5 + tau_p = Fraction() + tau_q_small = Fraction() + connected_count = 0 + for mask in range(1 << 9): + cluster = reached(mask, adj, 3) + if 5 not in cluster: + continue + connected_count += 1 + n = mask.bit_count() + tau_p += p**n*(1-p)**(9-n) + if len(cluster) <= cap: + tau_q_small += q**n*(1-q)**(9-n) + assert tau_q_small <= (q/p)**cap*tau_p + qquarter = bernstein_count_value(counts, Fraction(1,4)) + assert qquarter > Fraction(1,16) + result[kind] = { + 'conditional_origin': True, + 'seed_vertex_count': 9, + 'independent_remaining_sites': 8, + 'source': [0,1], 'target': [2,1], + 'conditional_connection_counts_by_occupied_other_sites': counts, + 'q_at_one_quarter': fraction_record(qquarter), + 'seed_root_target': '1/16', + 'seed_root_interval': [fraction_record(lo), fraction_record(hi)], + 'seed_root_midpoint_decimal': decimal((lo+hi)/2), + 'strict_upper_bound_on_first_centre_at_d_log4': decimal(hi), + 'cluster_comparison': {'cap': cap, 'p': str(p), 'q': str(q), + 'lhs': fraction_record(tau_q_small), + 'rhs': fraction_record((q/p)**cap*tau_p), + 'passed': True}, + } + # NN centre a is <= its seed root; b is >= one minus matching seed root. + result['certified_d_log4_centre_brackets'] = { + 'a_lower': '1/12', + 'a_upper': result['NN']['seed_root_interval'][1], + 'b_lower': fraction_record(1-Fraction(int(result['matching']['seed_root_interval'][1]['numerator']), + int(result['matching']['seed_root_interval'][1]['denominator']))), + 'b_upper': '27/28', + 'warning': 'Seed roots bound the infinite centres; they are NOT centre estimates.' + } + return result + + +def census_control(width: int, height: int, kind: str, ring_check: bool = False) -> dict: + n = width*height + if height < width: + raise ValueError('cut control assumes w<=m') + adj = torus_adjacency(width, height, kind) + winding_counts = [0]*(n+1) + ring_counts = [0]*(n+1) + windings = 0 + for mask in range(1 << n): + walk = winding_walk(mask, width, height, kind, adj) + if walk is not None: + windings += 1 + winding_counts[mask.bit_count()] += 1 + verify_cut(mask, width, height, kind, walk) + if ring_check and seed_ring(mask, width, height, kind): + assert walk is not None + ring_counts[mask.bit_count()] += 1 + finite_checks = [] + for p in (Fraction(1,100), Fraction(1,4), Fraction(1,2)): + f = bernstein_count_value(winding_counts, p) + item = {'p': str(p), 'winding_probability': fraction_record(f)} + branch = len(steps(kind))-1 + if branch*p < 1: + # kappa >= -log(branch*p) from nonbacktracking path counting. + upper = 2*n*width**2*p*(branch*p)**(width-1) + assert f <= upper + item['cut_upper_using_path_mass_lower'] = fraction_record(upper) + if ring_check: + qseed = bernstein_count_value(seed_counts(kind), p) + k, remainder = divmod(width,2) + lower = p**(remainder+1)*qseed**k + ring = bernstein_count_value(ring_counts, p) + assert lower <= ring <= f + item['conditional_harris_lower'] = fraction_record(lower) + item['ring_probability'] = fraction_record(ring) + finite_checks.append(item) + return {'width': width, 'height': height, 'graph': kind, + 'configurations': 1 << n, 'nonzero_winding_configurations': windings, + 'first_span_cut_witnesses_checked': windings, + 'winding_counts': winding_counts, + 'ring_counts': ring_counts if ring_check else None, + 'rational_probability_checks': finite_checks} + + +def run_controls() -> dict: + seeds = seed_and_cluster_controls() + censuses = [census_control(w,h,kind,ring_check=(w==3 and h==3)) + for w,h in ((3,3),(3,4),(4,4)) for kind in ('NN','matching')] + return { + 'schema': 'matching-one.winding-rate-centres.v1', + 'date': '2026-09-13', + 'claim_boundary': 'Finite deterministic controls only; no infinite correlation length numerically computed.', + 'seed_bounds': seeds, + 'censuses': censuses, + 'total_graph_configuration_checks': sum(c['configurations'] for c in censuses), + 'total_cut_witnesses': sum(c['first_span_cut_witnesses_checked'] for c in censuses), + } + + +def main() -> None: + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output', type=Path, required=True) + args = parser.parse_args() + args.output.parent.mkdir(parents=True, exist_ok=True) + # Reserve before computing to ensure no result file is silently overwritten. + with args.output.open('x', encoding='utf-8') as handle: + json.dump(run_controls(), handle, ensure_ascii=False, indent=2) + handle.write('\n') + +if __name__ == '__main__': + main() diff --git a/tests/test_winding_rate_centres.py b/tests/test_winding_rate_centres.py new file mode 100644 index 00000000..70452a31 --- /dev/null +++ b/tests/test_winding_rate_centres.py @@ -0,0 +1,78 @@ +"""Mathematical controls, not tests of document wording.""" +from fractions import Fraction as F +from pathlib import Path +import sys +import unittest +sys.path.insert(0, str(Path(__file__).resolve().parents[1] / 'scripts')) +import winding_rate_centres as w + +class WindingRateControls(unittest.TestCase): + def test_seed_enumeration_against_independent_formulas(self): + for p in (F(1,100),F(1,5),F(1,4),F(1,2),F(4,5),F(1)): + self.assertEqual(w.bernstein_count_value(w.seed_counts('NN'),p), + p*(p+(1-p)*(2*p**3-p**6))) + self.assertEqual(w.bernstein_count_value(w.seed_counts('matching'),p), + p*(1-(1-p)**3)) + + def test_seed_dyadic_signs(self): + for kind in ('NN','matching'): + counts = w.seed_counts(kind) + lo,hi = w.seed_root_interval(counts) + self.assertEqual(hi-lo,F(1,2**64)) + self.assertLess(w.bernstein_count_value(counts,lo),F(1,16)) + self.assertGreater(w.bernstein_count_value(counts,hi),F(1,16)) + self.assertLess(hi,F(1,4)) + + def test_all_tiny_cut_witnesses(self): + for kind in ('NN','matching'): + data=w.census_control(3,3,kind,ring_check=True) + self.assertEqual(data['configurations'],512) + self.assertEqual(data['first_span_cut_witnesses_checked'], + data['nonzero_winding_configurations']) + + def test_seam_ring_is_sufficient_not_identical(self): + for kind in ('NN','matching'): + # The top full row winds, but the seed requires the middle source. + top=7 + self.assertIsNotNone(w.winding_walk(top,3,3,kind)) + self.assertFalse(w.seed_ring(top,3,3,kind)) + middle=7<<3 + self.assertTrue(w.seed_ring(middle,3,3,kind)) + self.assertIsNotNone(w.winding_walk(middle,3,3,kind)) + + def test_parallel_edges_keep_short_winding(self): + # Circumference two has distinct lifted edges between the same vertices. + for kind in ('NN','matching'): + walk=w.winding_walk(3,2,3,kind) + self.assertIsNotNone(walk) + w.verify_cut(3,2,3,kind,walk) + + def test_normalized_seed_counts_not_double_binomial(self): + for kind in ('NN','matching'): + counts=w.seed_counts(kind) + self.assertEqual(w.bernstein_count_value(counts,F(1,2)),F(sum(counts),256)) + self.assertEqual(w.bernstein_count_value(counts,F(0)),0) + self.assertEqual(w.bernstein_count_value(counts,F(1)),1) + + def test_finite_cluster_likelihood_bound(self): + data=w.seed_and_cluster_controls() + for kind in ('NN','matching'): + row=data[kind]['cluster_comparison'] + lhs=F(int(row['lhs']['numerator']),int(row['lhs']['denominator'])) + rhs=F(int(row['rhs']['numerator']),int(row['rhs']['denominator'])) + self.assertLessEqual(lhs,rhs) + + def test_backtracking_first_span(self): + path=[(0,0),(1,0),(0,0),(0,1),(1,1),(2,1),(3,1),(4,1),(4,0)] + prefix,origin,axis=w.cut_witness(path,4) + self.assertEqual(axis,'x') + self.assertEqual(prefix[-1],(3,1)) + self.assertEqual(origin,(0,0)) + + def test_invalid_arguments(self): + with self.assertRaises(ValueError): w.steps('bond') + with self.assertRaises(ValueError): w.bernstein_count_value([1,1],F(3,2)) + with self.assertRaises(ValueError): w.census_control(4,3,'NN') + with self.assertRaises(ValueError): w.seed_ring(0,2,3,'NN') + +if __name__=='__main__': unittest.main() From 162db6eb55fe3f6d5ba35a78b31d8ae80ad71f4e Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 13:01:43 +0800 Subject: [PATCH 03/17] Add winding-cluster intensities, Poisson windows, and the birth fluctuations Second additive continuation of this same manuscript; four further new files. Independent of the previous commit at file level. No existing result, freeze, navigation document, or old PR touched. 1. A once-per-COMPONENT winding intensity nu_w on the infinite cylinder, anchored at each full winding component's lowest row with a tie-broken site; window height w^2 with two guard rows gives nu_w - nu_{w,H} <= C w exp(-c H), and -log(nu_w)/w -> kappa_G(p). 2. Poisson approximation of the two winding-component counts near each centre, with the black lower-window and white upper-window processes jointly independent two-type Poisson on the same uniform labels. BK is applied to disjoint increasing winding witnesses, NOT to the nonmonotone anchors, which retain a positive short-range covariance. 3. At all d outside an at-most-countable exceptional set, the finite-median centred 1/w fluctuations are jointly independent, oppositely oriented Gumbels, with an explicit cylinder cluster-volume tail and semiconvexity of log(nu_{w,w^2})/w. No Ornstein-Zernike prefactor is assumed for this. 4. Explicit Gumbel means, variances and vanishing covariance, and the scale-free archive-facing ratio [E(T2-T1) - IQR(mixture)] / [IQR(T1) + IQR(T2)] -> [EulerGamma + log(log 2)] / log[log(4)/log(4/3)] = 0.133989344651100063543034... 5. The boundary CDF is not determined by d alone: subexponential changes in m keep log(m)/w -> d while realizing every boundary probability in [0,1]. Files added: - docs/manuscripts/geometric-balance/poisson-birth-windows.md - scripts/winding_poisson_controls.py (stdlib only) - tests/test_winding_poisson_controls.py - results/geometric-consistency/winding-poisson-controls.json Executed here (2026-09-13, author-supplied; NOT an independent referee check): - sixteen local tests pass in this repository tree - all four added files byte-identical to the packaged copies - the deterministic result JSON regenerates byte-identically - 75,776 graph/configuration controls, 6,561 shared-label assignments, 8,192 fixed row masks, 18 exact component-activity derivative comparisons Not established here: the displacement from the infinite centres a(d), b(d) still needs a subexponential prefactor for nu_w which is not proved here; a planar two-point w^{-1/2} is not transplanted onto periodic component counts; no all-oblique fluctuation formula, no d -> 0 uniform crossover, no Monte Carlo, no new p_c. The H=1/2 controls explicitly retain void/rank discrepancies and do not prove the asymptotic statement. Full Matching-One repository CI has not been run for this commit. --- .../poisson-birth-windows.md | 787 ++++++++++++++++++ .../winding-poisson-controls.json | 774 +++++++++++++++++ scripts/winding_poisson_controls.py | 464 +++++++++++ tests/test_winding_poisson_controls.py | 125 +++ 4 files changed, 2150 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/poisson-birth-windows.md create mode 100644 results/geometric-consistency/winding-poisson-controls.json create mode 100644 scripts/winding_poisson_controls.py create mode 100644 tests/test_winding_poisson_controls.py diff --git a/docs/manuscripts/geometric-balance/poisson-birth-windows.md b/docs/manuscripts/geometric-balance/poisson-birth-windows.md new file mode 100644 index 00000000..c0c8e741 --- /dev/null +++ b/docs/manuscripts/geometric-balance/poisson-birth-windows.md @@ -0,0 +1,787 @@ +# Winding-cluster intensities, Poisson windows, and the two birth fluctuations + +2026-09-13. Continuation of the SAME geometric-balance paper, PR #739. +Author-supplied proof, not an independent referee acceptance or priority claim. +The earlier axial mass/centre argument is used explicitly. The new step is a +local, once-per-component intensity and a two-colour Poisson approximation. +No fixed-width continuum identification or unproved Ornstein--Zernike +prefactor is used. + +## 1. What is settled, and what is conditional + +Let G4 be the NN square graph and G8 its eight-neighbour matching graph. All +occupations are independent SITES. On the axial w-by-m torus suppose + + w -> infinity, log(m)/w -> d in (0,infinity). + +Use the earlier manuscript's mass + + kappa_G(p) = lim_n -log Pr_p^G(0 <-> n e1)/n, + +and its continuous strictly decreasing inverse on the subcritical interval. +Write a=kappa_4^{-1}(d), c=kappa_8^{-1}(d), b=1-c. The earlier result gives +T1 -> a and T2 -> b, with 0 kappa_G(p). A window of w^2 interior rows approximates + it to an error exp[-c_I w^2], up to polynomial factors, uniformly on a fixed + compact subcritical parameter interval I. +* The two colour counts, using the SAME uniform labels at the two separated + birth windows, converge jointly to INDEPENDENT Poisson laws. +* Consequently, birth CDFs at specified intensity levels have opposite Gumbel + forms for every d>0. A separate uniform cluster-volume and semiconvexity + argument upgrades this to median-centred affine 1/w Gumbel limits for all + d outside an at-most-countable exceptional set. This does not require an OZ + prefactor. At exceptional d we give a convex-log-intensity subsequential + classification rather than pretending that a unique affine law is proved. +* Even in the actual site model, d alone does not fix the CDF at p=a(d). + By changing only the subexponential length factor, any boundary CDF value in + [0,1] is possible. + +The exceptional set refers to possible nondifferentiability of the mass; it +is NOT asserted to be nonempty. All limits here fix d>0 and send w to infinity, +with m growing exponentially. The fixed-width 2/3/4 laboratory is not this limit. + +A later section derives the displacement FROM THE INFINITE centre, including +a possible log(w)/w term, UNDER a separate prefactor hypothesis. That +hypothesis is not proved for G4/G8 here. The affine fluctuation theorem around +the TRUE FINITE MEDIAN is proved at regular d without this hypothesis. The +distinction between locating a centre and resolving fluctuations around it is +part of the mathematical conclusion. + +## 2. External tools and reused inputs + +[AV] supplies uniform subcritical one-arm decay on compact p intervals and +states Harris/FKG and BK for both site and bond product spaces. We use its +site statement, not a square-bond numerical critical probability. + +[AGG], Theorem 2, supplies Poisson PROCESS approximation for locally dependent +indicators. In this note d_TV is sup_A |P(A)-Q(A)| (half the L1 convention). +For indicators I_i with means pi_i, dependency neighbourhoods B_i including i, +put + + b1 = sum_i sum_{j in B_i} pi_i pi_j, + b2 = sum_i sum_{j in B_i,j != i} E(I_i I_j). + +When I_i is independent of the joint family outside B_i (b3=0), the published +process bound implies + + d_TV(Law((I_i)), product_i Poi(pi_i)) <= 2(b1+b2). (2.1) + +Its contractions give the same bound for sums or finitely many typed sums. +This factor of two has been checked against the paper's doubled-TV convention, +not inferred from OCR of a formula. Zero-mean coordinates may be discarded. + +The preceding `exponential-birth-centres.md` proves two model-specific inputs: +for a cylinder strip of t rows the positive-winding event has upper bound + + 2p w^3 (t+2w+4) exp[-(w-1) kappa_G(p)], (2.2) + +with an inessential enlargement of the boundary count; and a fixed-height, +seam-closed ring can be built with probability >=exp[-(kappa_G(p)+eps)w] +for each fixed subcritical p and eps>0. The first-span proof uses only planar +edges of an injecting w-by-w vertex square. The matching diagonals do not +invalidate it. We recall where these enter below. + +Digital Alexander duality is the earlier finite identity + + r_4(omega)+r_8(omega^c)=2. (2.3) + +No correlation-length differentiability, OZ amplitude, Gumbel hypothesis, or +independence of the finite birth times is included among these inputs. + +## 3. One anchor per full winding component + +Work first on the infinite cylinder C_w x Z, periodic horizontally and free +vertically. Because a whole empty row has probability (1-p)^w>0 and separated +rows are independent, every occupied component is vertically bounded almost +surely for fixed w and p<1, on BOTH graphs (all steps have vertical increment +at most one). + +For any component with nonzero horizontal winding, let j be its lowest row. +Choose as its anchor (j,x), where x is the smallest label in {0,...,w-1} +among its vertices in row j. This arbitrary horizontal tie-break is only a +counting convention. In particular, individual anchor probabilities need NOT +be equal in x. The sum over x is independent of the choice of tie-break. +Define + + nu_w^G(p) = E[number of winding-component anchors in row 0]. (3.1) + +This is NOT an event probability when more than one anchor is possible. +It is a mean count per unit vertical length and is at most w. + +For an integer H>=1, retain only components occupying at most H consecutive +rows. Their intensity is nu_{w,H}. The event that (j,x) is such an anchor can +be decided on the window + + C_w x {j-1,j,...,j+H} (3.2) + +alone. Compute the component of (j,x) in this window; require that it meets +neither guard row j-1 nor j+H, that its lowest row is j, that it has nonzero +horizontal winding, and that x is its bottom-row tie-break. + +The two guard rows are essential: without them a locally winding cluster might +join a larger component outside the window and get counted more than once. +Since vertical jumps are at most one, not meeting a guard row proves that the +computed component is the FULL infinite-cylinder component. + +Thus nu_{w,H}(p) is a finite polynomial in the product measure on w(H+2) sites. +Its events include closed sites and are generally NOT increasing. No use of +Harris or BK below is made directly on these anchor events. + +On a vertically periodic torus with m>4H+4, the same local test defines +indicators I_{j,x}. A component confined to at most H consecutive cyclic rows +has a unique bottom after the complementary gap, so it contributes exactly +one anchor. Translation in j, not an incorrect equality of all x marginals, +gives + + E Z_{w,m,H} = m nu_{w,H}, Z=sum_{j,x} I_{j,x}. (3.3) + +## 4. Uniform vertical localization on the cylinder + +Fix a compact subcritical interval I. By domination at sup I, the local +one-arm probability a_R is bounded by A exp(-cR), uniformly on I. Set + + R=w/64, h=floor(w/8), + +and take w sufficiently large. A path crossing a cylinder band of h rows +must run from its first to its last row and hence produce an R-arm from one +of the w possible entry sites. A first-exit arm uses only the radius +R+sqrt(2) neighbourhood, which injects into C_w x Z. Hence + + Pr(cross a specified h-row band) <= w a_R <= exp(-c0 w) (4.1) + +for some c0>0 and all large w, uniformly on I. + +To cross a vertical distance H-1, a path must cross floor(H/h) disjoint h-row +bands. Take the last entrance before the first exit in each band to handle +backtracking. Band crossing events use disjoint SITE sets, so are independent. +For H>=w, floor(H/h)>=4H/w, after an immaterial adjustment for endpoints. +Absorb that adjustment in constants. There are C_I,c_I>0 such that + + Pr(a specified H-row strip has a bottom/top crossing) + <= C_I exp(-c_I H). (4.2) + +This is a cylinder estimate derived from planar local arms, not an unjustified +application of an infinite-plane cluster law to a periodic graph. + +If a torus component cannot fit inside H consecutive cyclic rows, a lifted +path contains such a strip crossing. A component with any vertical homology +also does so. Union over the m strip positions gives + + Pr(Bad_H) <= C_I m exp(-c_I H). (4.3) + +On Bad_H^c all nonzero homology is horizontal and every winding component is +counted by Z. In particular + + {r_G=0} = {Z=0} on Bad_H^c. (4.4) + +For the infinite-cylinder anchor at row zero, height greater than H forces +the specified strip crossing. At most w anchors can lie in a row, so + + 0 <= nu_w - nu_{w,H} <= C_I w exp(-c_I H). (4.5) + +We henceforth set H=w^2. There is no unknown fitted cutoff constant. The +window has O(w^3) site variables but length polynomial rather than exponential +in w. Exact enumeration of it is NOT claimed to be computationally cheap. + +For every fixed w, local-polynomial exhaustion and uniform empty-row tails on +compact subsets of p<1 also prove continuity of nu_w(p). We do not assume +that this component density is globally monotone in p. + +## 5. Density and planar mass have the same exponential rate + +The upper bound follows from (2.2) applied to an H-row window: + + nu_{w,H} <= poly(w) exp[-(w-1) kappa_G(p)]. (5.1) + +Equation (4.5), with H=w^2, is superexponentially smaller. For the lower bound, +use the earlier fixed-height D ring with probability +exp[-(kappa_G(p)+eps)w]. Unless its full component has vertical height greater +than H, a component anchor lies in one of at most H+D nearby rows. The chance +that the component containing the ring travels a distance H/2 is at most +poly(H+D) exp(-c_I H/2), by (4.2). Therefore + + (H+D) nu_w(p) + >= exp[-(kappa_G(p)+eps)w] - poly(H+D) exp(-c_I H/2). (5.2) + +Together these prove + + -log nu_w^G(p)/w -> kappa_G(p). (5.3) + +The same conclusion holds for p_w -> p in a compact subcritical interval. +For the lower bound use a fixed seed at p-delta and monotonicity of the seed +event, subtracting a uniform long-component tail at p+delta. For the upper +bound use monotonicity of the enclosing winding event. Let delta decrease +after the size limit, using the already proved continuity of kappa. This +avoids falsely treating nu itself as an increasing event. + +## 6. Poisson approximation with a vanishing explicit error + +For a fixed (j,x), the dependency neighbourhood consists of all anchors whose +windows (3.2) overlap. Its size is at most + + D_w = w(2H+3). (6.1) + +Disjoint windows are functions of disjoint product variables, so b3=0 exactly. +If two windows overlap, their union lies in a cylinder band of at most 2H+3 +rows (including an appropriate lift across the vertical seam). Let + + B_w(p) = 2p w^3 (4H+4w+4) exp[-(w-1) kappa_G(p)]. (6.2) + +By first-span counting this bounds the probability of any winding in that +union; it also bounds each anchor probability. If I_i=I_j=1 for distinct +anchors, they are DIFFERENT full components. Each contains a winding witness, +and these occupied witness sets are disjoint. Let E be the increasing event +that the enlarged band contains a horizontal winding. Then + + {I_i=I_j=1} subset E square E, + E(I_i I_j) <= Pr(E square E) <= Pr(E)^2 <= B_w(p)^2. (6.3) + +The middle inequality is the site BK inequality. Anchors themselves can be +positively correlated because they share CLOSED guard sites; treating them +as increasing would be wrong. The executable controls exhibit that effect. + +There are mw anchor indices. Thus b1,b2 are each at most mw D_w B_w^2. +Equation (2.1), contraction to the count, localization, and a coupling of +Poisson laws with nearby means give + + d_TV(Law(actual winding-component count), Poi(m nu_w)) + <= C_I m(w+1) exp(-c_I w^2) + + 4 m w^2(2w^2+3) B_w(p)^2. (6.4) + +For the count outside Bad_H, define it as the number of components with any +nonzero ambient homology; its value on Bad_H changes the comparison by at +most Pr(Bad_H). This makes (6.4) an assertion about the true finite graph. + +Choose a compact interval I around c_G(d)=kappa_G^{-1}(d) so small that + + 2 inf_{p in I} kappa_G(p) > d. (6.5) + +If log m/w -> d, the RHS of (6.4) tends to zero exponentially in w, uniformly +on I. Polynomial factors in (6.4) are harmless; H=w^2 and the first term is +superexponentially small. In particular + + Pr_p(r_G=0) = exp[-m nu_w^G(p)] + o(1), uniformly on I. (6.6) + +No limit of m nu_w is needed for this absolute-error assertion. When +m nu_w(p_w)->lambda in (0,infinity), the count is Poi(lambda) asymptotically. +When the mean tends to zero/infinity, the corresponding void probabilities +follow as well. A finite real-rate numerical constant is not supplied here. + +The PROCESS version of (2.1) yields more: after projecting anchor row j to +j/m on the unit circle, the winding-component positions converge to a +homogeneous Poisson process of intensity lambda. For a fixed arc the mean is +its number of discrete rows times nu, tending to lambda times its length; +disjoint arcs have independent Poisson limits. No uniform distribution of the +horizontal tie-break x is asserted. + +## 7. Why the two actual births decouple in these windows + +Use the SAME independent uniform site labels U_v. At a lower parameter p_1 +call U_v<=p_1 black. At an upper parameter p_2 call U_v>p_2 white and use G8. +Choose p_1 near a and p_2 near b; then p_1 lambda_1, + m nu_w^8(1-p_2,w) -> lambda_2, (7.3) + +the two winding-component counts converge to independent Poi(lambda_1) and +Poi(lambda_2), despite being coupled through the same U_v. + +Duality identifies T1>p1 with no black winding, and T2<=p2 with no white +winding. Hence + + Pr(T1<=p1, T2<=p2) + -> (1-exp(-lambda_1)) exp(-lambda_2). (7.4) + +This is an asymptotic independence result for the two separated transition +windows, not a statement that finite birth times are independent, not a +process-in-p convergence theorem, and not a result uniform as d decreases to +zero and the two windows merge. + +## 8. The intensity-clock Gumbel law, without guessing a prefactor + +Put Lambda_w^G(p)=m nu_w^G(p). For each fixed lambda>0 define p_w^G(lambda) +as the FIRST parameter in a fixed small interval around c_G(d) where +Lambda_w^G reaches lambda. It exists for large w: the mass rate makes Lambda +tend to zero at the lower endpoint and infinity at the upper endpoint. +Continuity makes its value exactly lambda. Its first-hit definition is +monotone in lambda even if nu is not globally monotone. Bounds using the +monotone enclosing winding event show p_w^G(lambda)->c_G(d). + +Define deterministic level coordinates + + l_w(x)=p_w^4(exp(x)), + u_w(y)=1-p_w^8(exp(-y)). (8.1) + +Both increase with their displayed argument. Equations (6.6)--(7.4) give + + Pr(T1<=l_w(x)) -> 1-exp[-exp(x)], + Pr(T2<=u_w(y)) -> exp[-exp(-y)], + Pr(T1<=l_w(x), T2<=u_w(y)) + -> (1-exp[-exp(x)]) exp[-exp(-y)]. (8.2) + +Thus the intensity-level CDFs have opposite Gumbel forms and factorize. The +clock here is a cylinder component density determined in polynomial-height +windows, not -log of the full-torus survival CDF defined tautologically. + +Equivalent fixed quantile calibrations are + + first-birth u-quantile: Lambda_4 = -log(1-u), + second-birth u-quantile: Lambda_8 = -log(u), + either marginal median: Lambda = log(2). (8.3) + +For the equal mixture, at each fixed u != 1/2 the local approximation is + + u<1/2: Lambda_4 = -log(1-2u), + u>1/2: Lambda_8 = -log(2u-1). (8.4) + +In the lower window the whole rank law is + + P0=exp(-Lambda_4)+o(1), P1=1-exp(-Lambda_4)+o(1), P2=o(1), + +because localization excludes vertical homology. In the upper window duality +instead gives + + P0=o(1), P1=1-exp(-Lambda_8)+o(1), P2=exp(-Lambda_8)+o(1). + +These identify both edges of the rank-one plateau; they are not a replacement +for the rare-odds analysis at the matching median in its interior. + +Here 'quantile calibration' means the CDF at the specified intensity inverse +converges to u. A quantitative p-error requires control of the local inverse +clock; (8.3) is NOT silently called an affine quantile expansion. + +The matching median lies inside the rank-one plateau, not in either window. +It remains a rare-odds balance problem and is not determined by averaging +(8.3). This paper's root theorem is still a separate input to its pc limit. + +## 9. A genuine non-universality at the boundary + +Fix a subcritical p0 and d=kappa_G(p0). For any t>0 choose + + m_w(t)=floor(t/nu_w^G(p0)). (9.1) + +By (5.3), log m_w(t)/w -> d and m_w(t)>=w eventually. By (6.6), + + Pr_{p0}(r_G>0) -> 1-exp(-t). (9.2) + +Thus every boundary probability in (0,1) occurs in the ACTUAL site model +while keeping the same exponential geometry rate d. Taking t_w=exp(-sqrt w) +or exp(sqrt w) similarly gives 0 or 1, respectively. The absolute Poisson +error still vanishes since log m/w -> d. + +This is not a toy with a different percolation rule. It proves that d by +itself cannot select the boundary CDF or a universal finite-centre correction. +It does not show that any pair of black and white boundary probabilities can +be prescribed simultaneously by the one common length m; their density ratio +would additionally matter. + +At a first-birth median a_w,m in I, (6.6) does prove + + m nu_w^4(a_w,m) -> log(2), (9.3) + +and similarly for the white intensity at the second-birth median. This is a +sharper and correctly normalized finite-centre equation than kappa=d alone. + +## 10. Affine fluctuations at all but countably many d, without OZ + +We can go further than a conditional statement about p-scaling. The missing +regularity can be obtained at every differentiability point of kappa using a +uniform cylinder cluster-volume bound. This section supplies that argument. + +### 10.1 Uniform subcritical cluster-volume tails on finite-width cylinders + +**Lemma.** On every fixed compact I subset (0,pc(G)), there are C,c>0 and w0 +such that for all w>=w0, p in I and cylinder vertices v, + + Pr_p^{C_w x Z}(|C_v|>=n) <= C exp(-cn). (10.1) + +This does not follow by relabelling the plane graph. Here is an explicit +coarse-block proof using only the local one-arm bound already assumed. + +Choose a fixed integer scale s>=4. Partition the horizontal circle into cells +of integer widths between s and 2s, and the vertical line into cells of s rows. +For w>=32s this can be done with at least sixteen horizontal cells. Each block +contains at most 2s^2 vertices. Call a block bad if some occupied vertex in it +has an occupied arm of Euclidean length at least s. The event is decided in +the block enlarged by s+sqrt(2), and its probability is at most + + delta_s <= 2s^2 A exp(-c0 s), (10.2) + +uniformly on I. These supports inject into the cylinder. Their dependency +graph has a bounded degree independent of w and s; a conservative bound of +D=289 including each block itself suffices (only blocks within eight coarse +steps in each coordinate can overlap). + +The set of blocks visited by a connected site cluster is connected in a graph +of maximum degree eight. If it contains more than 81 blocks, it cannot be +contained in the Euclidean (s+sqrt(2))-enlargement of ANY visited block: such an enlargement meets +fewer than 81 blocks. Thus every visited block is bad. There are at most +64^k connected k-block sets through a specified block: encode a canonical +spanning tree by a depth-first walk of length 2(k-1), with at most eight +choices per step. Any k-set contains at least k/D mutually independent bad +block events (greedy selection in the dependency graph). If the full visited +set is larger than k, extract a connected k-set through the root block, all +of whose blocks are still bad. Therefore + + Pr(cluster visits at least k blocks) <= 64^k delta_s^(k/D). + +Take s once and for all so delta_s<=128^(-D); then this is at most 2^(-k). +Since a block holds at most 2s^2 vertices, absorb the finitely many k<=81 +cases in C to prove (10.1). The constants may be extremely conservative and +are not a numerical algorithm. Horizontal periodicity, diagonal matching +steps, uneven cells, and arbitrary vertical length are all covered. + +### 10.2 Cluster activities cancel the irrelevant bulk sites + +A horizontally winding finite cylinder component C with lowest row zero +contributes + + p^{n(C)} (1-p)^{b(C)}, + n(C)=|C|, b(C)=|external vertex boundary of C|. + +The boundary has distinct sites; no edge multiplicity is substituted for +b(C). Sum over such connected sets C of height at most H to get exactly +nu_{w,H}. This is a sum of component probabilities, NOT a probability that +only one component exists. Each full component is represented once. + +Under the probability measure on component shapes proportional to these +activities, denote expectation by E_*. Then with q=1-p, + + S_C=n(C)/p-b(C)/q, + (log nu_{w,H})'=E_* S_C, + (log nu_{w,H})''=Var_*(S_C)-E_*[n(C)/p^2+b(C)/q^2]. (10.3) + +These identities are exact. For logit parameter z they become + + partial_z log nu=E_*[(1-p)n-pb], + partial_z^2 log nu=Var_*[(1-p)n-pb]-p(1-p)E_*[n+b]. (10.4) + +They do not involve an artificial O(wH) independent bulk occupation count. + +The isolated full horizontal row has activity p^w(1-p)^{2w}; it belongs to +nu_{w,H} for every H>=1. Hence nu_{w,H}>=exp(-K_I w) uniformly on I. +Meanwhile (10.1) gives + + Pr_*(n(C)>=n) <= C w exp(K_I w-cn), + +because at most w bottom-row sites can represent such a component. Splitting +the tail sum at a sufficiently large multiple of w yields + + E_* n(C) = O_I(w), E_* b(C) = O_I(w), (10.5) + +uniformly in H and large w; use b(C)<=8n(C). In particular for H=w^2, + + F_w(p)=(1/w)log nu_{w,w^2}(p) + satisfies F_w''(p)>=-C_I on I. (10.6) + +Thus the scaled log-intensities are uniformly SEMICONVEX. Discarding the +nonnegative variance in (10.3) proves the lower curvature bound; we do not +need a variance asymptotic or cluster renewal theorem. + +### 10.3 Consequences for the mass and its slopes + +Section 5 gives F_w(p)->-kappa_G(p). Equation (10.6) says that +F_w(p)+C_I p^2/2 is convex. The finite pointwise limit is convex too, so +-kappa is locally semiconvex (equivalently kappa is locally semiconcave). +Convex secant bounds imply local uniform convergence. The same bounds imply: +at every differentiability point a of kappa, for EVERY sequence p_w->a, + + F_w'(p_w) -> -kappa_G'(a). (10.7) + +For completeness, bound the derivative of the convex function at p_w between +its secant slopes with endpoints p_w+-delta. Use local uniform convergence, +then let delta decrease to zero. This works for moving p_w, not merely fixed p. +In one dimension a convex function has at most countably many derivative +jumps. Hence kappa is differentiable outside an at-most-countable set of p. + +The slope does not vanish at a differentiability point. The earlier FK +argument actually gives the quantitative comparison + + kappa(p)-kappa(q) >= (q-p) kappa(q)/rho, p= kappa_G(a)/rho >0 (10.9) + +where the derivative exists. No numerical rho is calibrated. These arguments +do not assert analyticity of kappa or rule out every possible corner. + +### 10.4 Actual median-centred Gumbel theorem + +Let a_w,m and b_w,m be the TRUE medians of T1 and T2. For a regular d, meaning +that both kappa_4 at a(d) and kappa_8 at c(d)=1-b(d) are differentiable, put +v4=-kappa_4'(a)>0 and v8=-kappa_8'(c)>0. Then + + X_w=v4 w (T1-a_w,m), Y_w=v8 w (T2-b_w,m) + +converge jointly to INDEPENDENT variables with CDFs + + Pr(X<=x)=1-2^(-exp(x)), + Pr(Y<=y)=2^(-exp(-y)). (10.10) + +Regular d excludes at most a countable subset of (0,infinity), since each +mass is a bijection and its exceptional p set is countable. + +**Proof.** The centres converge to a and b by the earlier mass argument. +The Poisson formula at an exact marginal median implies +m nu_{w,w^2}(a_w,m)->log 2, and likewise on the matching side at 1-b_w,m. +For fixed x, apply (10.7) throughout the shrinking interval from a_w,m to +p_w=a_w,m+x/(v4 w), and integrate: + + log[nu_{w,w^2}(p_w)/nu_{w,w^2}(a_w,m)] -> x. + +Apply (6.6). For the white side the displacement is negative in its own +occupation probability, giving -y. Apply (7.4) for joint factorization. This +proves (10.10) without a prefactor expansion or a formula for the medians' +distance from a,b. A finite-centre fluctuation theorem and a deterministic +centering correction are genuinely different results. + +### 10.5 Moments and an archive-facing consequence + +The earlier FK concentration about true medians has exponential tails on +scale 1/log N; here log N~dw. Consequently the w-scaled variables have +uniformly bounded moments of every fixed order. Joint convergence therefore +also gives convergence of means, variances, and products. Set + + h0=EulerGamma+log(log 2) = 0.210702744319868533594073... . + +Then at regular d, + + E T1=a_w,m-h0/(v4 w)+o(1/w), + E T2=b_w,m+h0/(v8 w)+o(1/w), + Var(T1)=pi^2/(6v4^2 w^2)+o(w^-2), + Var(T2)=pi^2/(6v8^2 w^2)+o(w^-2), + Cov(T1,T2)=o(w^-2). (10.11) + +The quarter and three-quarter quantiles of the mixture lie within o(1/w) of +a_w,m and b_w,m: the other birth is a fixed positive distance away and the +limiting CDF (10.10) crosses its own median strictly. Hence, for the already +available rank-gap observable G=E(T2-T1), + + G-IQR(F) = [h0/w](1/v4+1/v8)+o(1/w). (10.12) + +This refines the earlier O(1/log N) comparison. It requires regular d, the +correct ambient-rank birth statistics, and the actual mass slopes. It is not +a free numerical prediction at width two or four. + + +### 10.6 A scale-free consequence that does not require numerical mass slopes + +Write J1=IQR(Law(T1)), J2=IQR(Law(T2)), and J=IQR(F) for the mixture. +Let + + D0=log[log(4)/log(4/3)] = 1.57253358368551918078557... . + +The marginal quantile functions of (10.10) are + + x(u)=log[-log(1-u)/log 2], + y(u)=log[log 2/(-log u)]. + +Both marginal IQRs in the scaled coordinate equal D0. Consequently, at every +regular fixed d, + + J1=D0/(v4 w)+o(1/w), J2=D0/(v8 w)+o(1/w), + [G-J]/(J1+J2) -> h0/D0 = 0.133989344651100063543034... , (10.13) + Var(Tj)/Jj^2 -> pi^2/(6 D0^2) + = 0.665194479964364626613157... . (10.14) + +Thus a properly typed exponential-aspect birth archive can test the shape +without knowing A, beta, or the numerical mass slopes. These are statements +about population quantiles/moments, not finite-sample unbiasedness claims. +They are not predictions for fixed-aspect square sequences (d=0) or the +fixed-width 2/3/4 controls. No new data collection is commissioned here. + +## 11. Exceptional d: a constrained crossover, not an arbitrary profile + +The intensity-clock statement (8.2) holds for EVERY d>0. We can also constrain +all subsequential p-scaled profiles at a corner of the mass. + +Let a=c_G(d), let a_w,m be the first positive-rank median for G, and define + + psi_w(x)=log[m nu_{w,w^2}(a_w,m+x/w)]. + +Here psi_w(0)->log(log 2), and (10.6) gives psi_w''>=-C_I/w. Secant bounds +from local uniform convergence to -kappa bound psi_w' on compact x intervals +between the limiting one-sided slopes + + v_-=-kappa'_-(a), v_+=-kappa'_+(a), + 00, A is continuous, beta is fixed, kappa is twice differentiable +near a, and v=-kappa'(a)>0. Suppose + + log m=dw+gamma log w+c0+o(1). (12.2) + +Then a first-birth u-quantile satisfies + + Q_G(u)=a+[(beta-gamma)log w-c0-log A(a)+log(-log(1-u))]/(vw) + +o(1/w). (12.3) + +For the marginal median use log(log 2). Apply the analogous formula to the +white site graph and reflect p=1-q for the second birth. At a location with +log[m nu_w(p)]=x the law is 1-exp[-exp(x)], while around the true median it +is (10.10). The distinct constants are not interchangeable. + +No A or beta value for square-site winding COMPONENT density has been derived +here. In particular, a plane two-point OZ factor w^-1/2 cannot simply be +copied into (12.1): periodic seam closure and once-per-component counting +change the normalization. The existence of a pure polynomial prefactor is itself left unproved, not +only the numerical values of A and beta. This unresolved centering problem does NOT +invalidate the proved regular-d finite-median fluctuation law. + +## 13. What the calculation actually checked + +The accompanying script has NO dependency on earlier state certificates. +It uses physical lifted graph traversal, retaining parallel periodic edges, +and a second, window-restricted implementation of component anchoring. + +* 75,776 graph/configuration pairs across 2x5, 2x6, 3x5 and both graphs. + Every eligible full component gets exactly one local anchor. Void mismatch + is contained in the explicit localization-failure event. +* Local window polynomials independently reproduce the expected number of + anchors divided by m at two rational probabilities. Marginals, neighbour + pairs, count laws, and b1/b2 are exact fractions. +* All 3^8=6,561 low/middle/high assignments on 2x4, with probabilities + 1/4,1/2,1/4, check the common-label coupling and typed count law. The small + system's count covariance is NONZERO (4831/1048576); no finite independence + is inferred. Opposite association is checked for the enclosing winding + events, not falsely asserted for the anchor counts. +* 8,192 deterministic full/empty-row masks on 3x12 check seams, cutoffs and + disjoint-window neighbourhoods without random sampling. +* Component-shape activity sums are checked against full-window occupancy + sums at widths/cutoffs (2,1), (2,2), (3,2), on both graphs. That is 168 + interior candidate shapes and 8,832 surrounding-window configurations. + At three rational p values, 18 exact checks compare first and second + log-derivatives via distinct external boundary counts against derivatives + of the full-window polynomial. Logit chain rules and the curvature lower + bound agree as well. These checks test (10.3)--(10.4), not the asymptotic + uniform cluster-volume proof. +* For one-row isolated loops the exact local intensity is + nu_{w,1}(p)=p^w(1-p)^(2w). Two loops two rows apart have joint probability + p^(2w)(1-p)^(3w), strictly GREATER than the product of their anchor marginals. + This is the closed-guard countercontrol for an invalid direct BK argument. + +These tiny cutoffs are NOT H=w^2 and their void/true-rank discrepancy need not +be small. The output records that discrepancy rather than claiming a tiny +system verifies the asymptotic localization constants. Total variation against +Poisson uses floating exponentials as a diagnostic; all site probabilities +and the published-theorem RHS are stored exactly. No finite check proves the +asymptotic result, its source inputs, or publication novelty. + +## 14. Position within the same paper + +The present closure is: + + geometric consistency -> two mass-defined centres -> local component + intensity -> joint Poisson windows -> natural-clock laws at every d -> + uniform component tails/semiconvexity -> median-centred affine laws at + regular d -> explicit leading fluctuation moments. + +No new width ladder is required for this proof. At regular d, the affine +finite-median law is now proved. The remaining numerical centre displacement +is a local prefactor expansion of the winding-component density; the possible +countable exceptional d need separate regularity information. Merely fitting +a Gumbel curve or copying a plane two-point prefactor would not supply either. +The unconditional and conditional statements above are kept separate. + +This is not presented as a novel Chen--Stein method. [AGG] already explicitly +explains declumping, extremes, and process approximation. [DL] is nearby +bond-wedge/rectangle work with inverse-correlation and Poisson arguments. +The specific model work here is physical SITE winding-component anchoring, +vertical localization on a periodic cylinder, and the opposite-colour +common-label joint limit. No systematic priority certification is claimed. + +## References and actual reading + +[AGG] R. Arratia, L. Goldstein, L. Gordon, *Two Moments Suffice for Poisson +Approximations: The Chen--Stein Method*, Annals of Probability 17 (1989), 9--25. +Author-hosted PDF, Section 2, Theorems 1--2; printed pages 10--11 rendered and +checked. Uses the doubled total-variation convention; (2.1) uses half of it. +https://dornsife.usc.edu/larry-goldstein/wp-content/uploads/sites/221/2023/06/AGG-1.pdf +DOI: 10.1214/aop/1176991491. + +[AV] T. Antunovic, I. Veselic, *Sharpness of the phase transition and exponential +decay of the subcritical cluster size for percolation on quasi-transitive +graphs*, J. Stat. Phys. 130 (2008), 983--1009. Primary HTML, Theorems 2--3 and +Section 3 Fundamental Tools, especially product-site Harris and BK. +https://arxiv.org/html/0707.1089v3 + +[FK/DKS] P. Duncan, M. Kahle, B. Schweinhart, *Homological percolation on a +torus: plaquettes and permutohedra*, Theorem 6 reproduces Friedgut--Kalai. +Used for the earlier two-birth concentration, the quantitative mass-slope +inequality, and uniform integrability of median-centred fluctuations. It is +not the model-specific Poisson proof. +https://arxiv.org/html/2011.11903v4 + +[DL] M. Damron, W.-K. Lam, *Asymptotics for first passage percolation on +logarithmic subgraphs of Z^2*, arXiv:2502.18235v1. Sections 1--2 and the Poisson +comparison in Section 5 read for scope. Bond/open-boundary context, not a +substitute for site-periodic closure. +https://arxiv.org/html/2502.18235v1 + +[Previous working proof] `exponential-birth-centres.md`, owner handoff on +2026-09-13, recorded in #739 comment 5650571466. It supplies the axial mass +rate and its continuity/inversion with named inputs. 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"56357211969/500000000000", + "2": "623340927/250000000000", + "3": "531441/500000000000" + }, + "b1": "34635779137089/2500000000000000", + "b2": "1248276177/250000000000", + "lambda": "5885217/50000000", + "localization_failure_probability": "33381785881/1000000000000", + "nu_per_row": "1961739/100000000", + "p": "1/10", + "poisson_tv_float_diagnostic": 0.008080109995976081, + "rank_void_poisson_error_float_diagnostic": 0.031023344898971406, + "rank_zero_probability": "857935498437/1000000000000", + "void_mismatch_probability": "5371130207/200000000000" + }, + { + "agg_process_tv_bound": "206519625/536870912", + "anchored_count_law": { + "0": "1440323/2097152", + "1": "2331585/8388608", + "2": "147501/4194304", + "3": "729/8388608" + }, + "b1": "130439241/1073741824", + "b2": "297189/4194304", + "lambda": "11421/32768", + "localization_failure_probability": "4859281/16777216", + "nu_per_row": "3807/65536", + "p": "1/4", + "poisson_tv_float_diagnostic": 0.03197517001294048, + "rank_void_poisson_error_float_diagnostic": 0.26889599164081057, + "rank_zero_probability": "7328637/16777216", + "void_mismatch_probability": "4193947/16777216" + } + ], + "configurations": 4096, + "cutoff_rows": 2, + "length": 6, + "local_configurations": 256, + "local_global_anchor_failures": 0, + "local_intensity_bernstein_counts": [ + 0, + 0, + 3, + 6, + 2, + 0, + 0, + 0, + 0 + ], + "matching": true, + "max_anchored_count": 3, + "width": 2 + }, + { + "cases": [ + { + "agg_process_tv_bound": "437397732081/20000000000000000", + "anchored_count_law": { + "0": "199468946420489/200000000000000", + "1": "265333079511/100000000000000", + "2": "387420489/200000000000000" + }, + "b1": "282429536481/40000000000000000", + "b2": "387420489/100000000000000", + "lambda": "531441/200000000", + "localization_failure_probability": "129043485463501/1000000000000000", + "nu_per_row": "531441/1000000000", + "p": "1/10", + "poisson_tv_float_diagnostic": 3.1771609108055543e-06, + "rank_void_poisson_error_float_diagnostic": 0.0023671823893616217, + "rank_zero_probability": "994979139854949/1000000000000000", + "void_mismatch_probability": "295699030937/125000000000000" + }, + { + "agg_process_tv_bound": "25883145/34359738368", + "anchored_count_law": { + "0": "1058910319/1073741824", + "1": "7366545/536870912", + "2": "98415/1073741824" + }, + "b1": "13286025/68719476736", + "b2": "98415/536870912", + "lambda": "3645/262144", + "localization_failure_probability": "593693137/1073741824", + "nu_per_row": "729/262144", + "p": "1/4", + "poisson_tv_float_diagnostic": 8.68696060285501e-06, + "rank_void_poisson_error_float_diagnostic": 0.06482164808865176, + "rank_zero_probability": "989313507/1073741824", + "void_mismatch_probability": "17399203/268435456" + } + ], + "configurations": 32768, + "cutoff_rows": 1, + "length": 5, + "local_configurations": 512, + "local_global_anchor_failures": 0, + "local_intensity_bernstein_counts": [ + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0 + ], + "matching": false, + "max_anchored_count": 2, + "width": 3 + }, + { + "cases": [ + { + "agg_process_tv_bound": "437397732081/20000000000000000", + "anchored_count_law": { + "0": "199468946420489/200000000000000", + "1": "265333079511/100000000000000", + "2": "387420489/200000000000000" + }, + "b1": "282429536481/40000000000000000", + "b2": "387420489/100000000000000", + "lambda": "531441/200000000", + "localization_failure_probability": "269154105310351/1000000000000000", + "nu_per_row": "531441/1000000000", + "p": "1/10", + "poisson_tv_float_diagnostic": 3.1771609108055543e-06, + "rank_void_poisson_error_float_diagnostic": 0.02615049574186168, + "rank_zero_probability": "971195826502449/1000000000000000", + "void_mismatch_probability": "6537226399999/250000000000000" + }, + { + "agg_process_tv_bound": "25883145/34359738368", + "anchored_count_law": { + "0": "1058910319/1073741824", + "1": "7366545/536870912", + "2": "98415/1073741824" + }, + "b1": "13286025/68719476736", + "b2": "98415/536870912", + "lambda": "3645/262144", + "localization_failure_probability": "826346197/1073741824", + "nu_per_row": "729/262144", + "p": "1/4", + "poisson_tv_float_diagnostic": 8.68696060285501e-06, + "rank_void_poisson_error_float_diagnostic": 0.29115113863106357, + "rank_zero_probability": "746294067/1073741824", + "void_mismatch_probability": "78154063/268435456" + } + ], + "configurations": 32768, + "cutoff_rows": 1, + "length": 5, + "local_configurations": 512, + "local_global_anchor_failures": 0, + "local_intensity_bernstein_counts": [ + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0 + ], + "matching": true, + "max_anchored_count": 2, + "width": 3 + } + ], + "nonclaims": [ + "no numerical mass kappa", + "no prefactor A or beta determination", + "no proof by finite enumeration", + "no full repository CI" + ], + "probability_accuracy": "all probabilities, intensities and b1/b2 are exact fractions; TV against exp is a float diagnostic", + "schema": "matching-one.winding-poisson-controls.v1", + "scope": "finite exact anchoring and dependency controls; not asymptotic simulations", + "total_graph_configurations": 75776, + "two_colour": { + "agg_process_tv_bound": "14849/131072", + "b1": "6561/262144", + "b2": "259/8192", + "categorical_configurations": 6561, + "count_covariance": "4831/1048576", + "count_law": { + "0,0": "28067/32768", + "0,1": "1063/16384", + "0,2": "1/512", + "1,0": "1063/16384", + "1,1": "9/1024", + "1,2": "1/2048", + "2,0": "1/512", + "2,1": "1/2048", + "2,2": "1/32768" + }, + "cutoff_rows": 1, + "high_probability": "1/4", + "joint_poisson_tv_float_diagnostic": 0.00688426799249664, + "lambda_high": "81/1024", + "lambda_low": "81/1024", + "length": 4, + "low_probability": "1/4", + "opposite_monotone_joint": "143/2048", + "opposite_monotone_product": "420099583/4294967296", + "width": 2 + } +} diff --git a/scripts/winding_poisson_controls.py b/scripts/winding_poisson_controls.py new file mode 100644 index 00000000..c04d1c27 --- /dev/null +++ b/scripts/winding_poisson_controls.py @@ -0,0 +1,464 @@ +#!/usr/bin/env python3 +"""Finite controls for one-anchor-per-winding-component Poisson reduction. + +Standard library only. Physical lifted edges are retained, including parallel +edges at width two. No simulated samples or fitted percolation constants. +""" +from __future__ import annotations + +import argparse +from collections import Counter +from fractions import Fraction +from itertools import product +import json +import math +from pathlib import Path +from typing import Iterable + + +def steps(matching: bool) -> tuple[tuple[int, int], ...]: + return ((1, 0), (-1, 0), (0, 1), (0, -1)) + ( + ((1, 1), (1, -1), (-1, 1), (-1, -1)) if matching else () + ) + + +def adjacency(w: int, m: int, matching: bool = False, periodic_y: bool = True): + if w < 2 or m < 2: + raise ValueError('width and height must both be at least two') + out = [[] for _ in range(w * m)] + for y in range(m): + for x in range(w): + for dx, dy in steps(matching): + yy = y + dy + if not periodic_y and not 0 <= yy < m: + continue + out[y * w + x].append((((yy % m) * w + (x + dx) % w), dx, dy)) + return out + + +def components(mask: int, w: int, m: int, adj): + """Independent graph-potential traversal; return vertices and lift cycles.""" + unseen = mask + answer = [] + while unseen: + bit = unseen & -unseen + root = bit.bit_length() - 1 + unseen ^= bit + lift = {root: (0, 0)} + todo = [root] + verts = [] + cycles = set() + while todo: + v = todo.pop() + verts.append(v) + vx, vy = lift[v] + for z, dx, dy in adj[v]: + if not mask & (1 << z): + continue + q = (vx + dx, vy + dy) + if z not in lift: + lift[z] = q + unseen &= ~(1 << z) + todo.append(z) + else: + gain = (q[0] - lift[z][0], q[1] - lift[z][1]) + if gain != (0, 0): + cycles.add(gain) + answer.append((tuple(verts), tuple(sorted(cycles)))) + return answer + + +def ambient_rank(comps) -> int: + vecs = [z for _, cyc in comps for z in cyc] + if not vecs: + return 0 + ax, ay = vecs[0] + return 2 if any(ax * by != ay * bx for bx, by in vecs[1:]) else 1 + + +def global_anchors(comps, w: int, m: int, height: int) -> tuple[int, ...]: + """One lexicographic anchor for each full component using <=height rows.""" + found = [] + for verts, cyc in comps: + if not any(dx for dx, _ in cyc): + continue + rows = {v // w for v in verts} + if len(rows) > height or len(rows) == m: + continue + starts = [y for y in rows if (y - 1) % m not in rows] + if len(starts) != 1: + raise AssertionError('connected component has non-contiguous row projection') + bottom = starts[0] + if any((y - bottom) % m >= height for y in rows): + continue + if any(dy for _, dy in cyc): + raise AssertionError('short component has vertical winding') + x = min(v % w for v in verts if v // w == bottom) + found.append(bottom * w + x) + return tuple(sorted(found)) + + +def local_anchor_bits(mask: int, w: int, height: int, local_adj) -> tuple[int, ...]: + """Window rows 0..height+1, bottom row 1, two guard rows. + + Does not call global_anchors and does not use vertical periodic edges. + """ + ans = [] + for verts, cyc in components(mask, w, height + 2, local_adj): + ys = {v // w for v in verts} + if 0 in ys or height + 1 in ys or 1 not in ys: + continue + if not any(dx for dx, _ in cyc): + continue + if any(dy for _, dy in cyc): + raise AssertionError('open vertical window has vertical winding') + ans.append(min(v % w for v in verts if v // w == 1)) + return tuple(sorted(ans)) + + +def local_all_anchors(mask: int, w: int, m: int, height: int, local_adj): + if not 1 <= height < m / 2 or height + 2 >= m: + raise ValueError('need 1 <= cutoff < m/2 and cutoff+2 < m') + rowmask = (1 << w) - 1 + ans = [] + for bottom in range(m): + win = 0 + for k in range(height + 2): + row = (bottom - 1 + k) % m + win |= ((mask >> (w * row)) & rowmask) << (w * k) + ans.extend(bottom * w + x for x in local_anchor_bits(win, w, height, local_adj)) + return tuple(sorted(ans)) + + +def support_rows(anchor: int, w: int, m: int, height: int) -> frozenset[int]: + y = anchor // w + return frozenset((y - 1 + k) % m for k in range(height + 2)) + + +def dependency(w: int, m: int, height: int): + rr = [support_rows(i, w, m, height) for i in range(w * m)] + return [tuple(j for j in range(w * m) if rr[i] & rr[j]) for i in range(w * m)] + + +def bernstein_value(coef: Iterable[int], p: Fraction) -> Fraction: + v = list(coef) + n = len(v) - 1 + return sum((Fraction(c) * p ** k * (1 - p) ** (n - k) + for k, c in enumerate(v)), Fraction()) + + +def local_intensity_coefficients(w: int, height: int, matching: bool): + n = w * (height + 2) + adj = adjacency(w, height + 2, matching, False) + cs = [0] * (n + 1) + for mask in range(1 << n): + cs[mask.bit_count()] += len(local_anchor_bits(mask, w, height, adj)) + return cs + + +def fstr(x: Fraction) -> str: + return str(x.numerator) if x.denominator == 1 else f'{x.numerator}/{x.denominator}' + + +def univariate_tv(law: dict[int, Fraction], lam: Fraction) -> float: + z = float(lam) + overlap = sum(min(float(prob), math.exp(-z) * z ** k / math.factorial(k)) + for k, prob in law.items()) + return max(0.0, 1.0 - overlap) + + +def enumerate_one(w: int, m: int, height: int, matching: bool, ps=(Fraction(1, 10), Fraction(1, 4))): + n = w * m + adj = adjacency(w, m, matching) + ladj = adjacency(w, height + 2, matching, False) + dep = dependency(w, m, height) + singles = [[0] * (n + 1) for _ in range(n)] + pairs = {(i, j): [0] * (n + 1) for i in range(n) for j in dep[i] if j != i} + laws = Counter() + absent = [0] * (n + 1) + mismatch = [0] * (n + 1) + large = [0] * (n + 1) + maxz = 0 + # Cache the independent local-window classification, not global labels. + lcache = {b: local_anchor_bits(b, w, height, ladj) + for b in range(1 << (w * (height + 2)))} + rowmask = (1 << w) - 1 + for mask in range(1 << n): + cc = components(mask, w, m, adj) + aa = global_anchors(cc, w, m, height) + loc = [] + for y in range(m): + win = 0 + for k in range(height + 2): + win |= (((mask >> (w * ((y - 1 + k) % m))) & rowmask) + << (w * k)) + loc.extend(y * w + x for x in lcache[win]) + if aa != tuple(sorted(loc)): + raise AssertionError(('local/global mismatch', w, m, height, matching, mask)) + k = mask.bit_count() + r = ambient_rank(cc) + z = len(aa) + maxz = max(maxz, z) + laws[z, k] += 1 + absent[k] += (r == 0) + is_bad = any(len({v // w for v in verts}) > height for verts, _ in cc) + large[k] += is_bad + mismatch[k] += ((r == 0) != (z == 0)) + if ((r == 0) != (z == 0)) and not is_bad: + raise AssertionError('void mismatch not covered by localization failure') + for i in aa: + singles[i][k] += 1 + for j in aa: + if (i, j) in pairs: + pairs[i, j][k] += 1 + local_cs = local_intensity_coefficients(w, height, matching) + cases = [] + for p in ps: + pp = [bernstein_value(c, p) for c in singles] + lam = sum(pp) + nu = bernstein_value(local_cs, p) + assert lam == m * nu + b1 = sum((pp[i] * pp[j] for i in range(n) for j in dep[i]), Fraction()) + b2 = sum((bernstein_value(c, p) for c in pairs.values()), Fraction()) + law = {z: sum((Fraction(laws[z, k]) * p ** k * (1-p) ** (n-k) + for k in range(n+1)), Fraction()) for z in range(maxz+1)} + assert sum(law.values()) == 1 + assert sum(z * prob for z, prob in law.items()) == lam + actual_void = bernstein_value(absent, p) + miss = bernstein_value(mismatch, p) + bad = bernstein_value(large, p) + assert abs(actual_void - law[0]) <= miss <= bad + tv = univariate_tv(law, lam) + # AGG process bound in sup-event total-variation convention. + bound = min(Fraction(1), 2 * (b1 + b2)) + if tv > float(bound) + 1e-13: + raise AssertionError(('Poisson bound failed', tv, bound)) + cases.append({'p': fstr(p), 'lambda': fstr(lam), 'nu_per_row': fstr(nu), + 'b1': fstr(b1), 'b2': fstr(b2), + 'anchored_count_law': {str(z): fstr(prob) for z, prob in law.items()}, + 'rank_zero_probability': fstr(actual_void), + 'void_mismatch_probability': fstr(miss), + 'localization_failure_probability': fstr(bad), + 'poisson_tv_float_diagnostic': tv, + 'agg_process_tv_bound': fstr(bound), + 'rank_void_poisson_error_float_diagnostic': abs(float(actual_void)-math.exp(-float(lam)))}) + return {'width': w, 'length': m, 'cutoff_rows': height, 'matching': matching, + 'configurations': 1 << n, 'local_configurations': 1 << (w*(height+2)), + 'local_global_anchor_failures': 0, 'max_anchored_count': maxz, + 'local_intensity_bernstein_counts': local_cs, + 'cases': cases} + + +def two_colour_control(w=2, m=4, height=1): + """Exact categorical coupling: low=1/4, middle=1/2, high=1/4.""" + n = w * m + adjs = [adjacency(w, m, False), adjacency(w, m, True)] + deps = dependency(w, m, height) + dset = [set(d) for d in deps] + one = [0] * (2*n) + pair = Counter() + law = Counter() + void = Counter() + total = 4 ** n + lowq = highq = Fraction(1, 4) + tables = [] + for adj in adjs: + table = [] + for mask in range(1 << n): + cc = components(mask, w, m, adj) + table.append((global_anchors(cc, w, m, height), ambient_rank(cc))) + tables.append(table) + for labels in product(range(3), repeat=n): + low = high = 0 + mid = 0 + for i, label in enumerate(labels): + if label == 0: + low |= 1 << i + elif label == 2: + high |= 1 << i + else: + mid += 1 + weight = 1 << mid + al, rl = tables[0][low] + ah, rh = tables[1][high] + aa = list(al) + [n+i for i in ah] + law[len(al), len(ah)] += weight + void[rl == 0, rh == 0] += weight + for i in aa: + one[i] += weight + for j in aa: + if j != i and j % n in dset[i % n]: + pair[i,j] += weight + assert sum(law.values()) == total + pp = [Fraction(v, total) for v in one] + lams = [sum(pp[:n]), sum(pp[n:])] + b1 = sum((pp[i] * pp[j] for i in range(2*n) for j in range(2*n) + if j % n in dset[i % n]), Fraction()) + b2 = Fraction(sum(pair.values()), total) + pmf = {key: Fraction(v, total) for key, v in law.items()} + lp, lq = map(float, lams) + tv = 1 - sum(min(float(prob), math.exp(-lp-lq) * lp**z * lq**v / + (math.factorial(z)*math.factorial(v))) for (z,v),prob in pmf.items()) + assert tv <= float(2*(b1+b2)) + 1e-13 + cov = sum((z*v*pr for (z,v),pr in pmf.items()), Fraction()) - lams[0]*lams[1] + # Coarse positive-winding events, unlike component anchors, are oppositely + # monotone functions of the shared uniform labels, hence negatively associated. + p_lo = sum(Fraction(v,total) for (zl,zh),v in void.items() if not zl) + p_hi = sum(Fraction(v,total) for (zl,zh),v in void.items() if not zh) + p_joint = Fraction(void[False,False],total) + assert p_joint <= p_lo*p_hi + return {'width':w, 'length':m,'cutoff_rows':height,'categorical_configurations':3**n, + 'low_probability':fstr(lowq),'high_probability':fstr(highq), + 'lambda_low':fstr(lams[0]),'lambda_high':fstr(lams[1]), + 'b1':fstr(b1),'b2':fstr(b2),'count_covariance':fstr(cov), + 'count_law': {f'{z},{v}':fstr(pr) for (z,v),pr in sorted(pmf.items())}, + 'opposite_monotone_joint':fstr(p_joint), + 'opposite_monotone_product':fstr(p_lo*p_hi), + 'joint_poisson_tv_float_diagnostic':tv, + 'agg_process_tv_bound':fstr(min(Fraction(1),2*(b1+b2)))} + + +def exact_row_contact_control(w=3, m=5, p=Fraction(1,4)): + """Two isolated full-row clusters may be positively correlated. + + They share a CLOSED guard row; applying BK to the anchors would be wrong. + """ + single = p**w * (1-p)**(2*w) + joint = p**(2*w) * (1-p)**(3*w) + assert joint > single*single + return {'width': w, 'length_at_least':5,'p':fstr(p), + 'single_anchor_probability':fstr(single), + 'two_anchors_two_rows_apart':fstr(joint), + 'product_of_marginals':fstr(single*single), + 'ratio':fstr(joint/(single*single)), + 'bk_applies_to_disjoint_increasing_winding_witnesses_not_anchors':True} + + +def geometry_checks(): + nchecks = 0 + for matching in (False, True): + w,m,h=3,12,2 + adj=adjacency(w,m,matching) + ladj=adjacency(w,h+2,matching,False) + # All rows are either entirely open or entirely closed: tests seam, + # multiple components, and cutoff coverage without random sampling. + for rowpattern in range(1<= -contact + cases.append({'p':fstr(p),'nu':fstr(nu),'mean_component_volume':fstr(mean_n), + 'mean_distinct_boundary_volume':fstr(mean_b), + 'log_nu_derivative':fstr(es),'log_nu_second_derivative':fstr(curvature), + 'curvature_lower_bound':fstr(-contact), + 'logit_log_nu_second_derivative':fstr(zcurv), + 'two_differentiation_routes_agree':True}) + return {'width':w,'cutoff_rows':height,'matching':matching, + 'interior_subsets':1<<(w*height),'surrounding_window_configurations':1< Date: Sun, 13 Sep 2026 13:11:35 +0800 Subject: [PATCH 04/17] Reduce the birth law to two sharp centres, with the existing-data interface Third additive continuation of the same geometric-balance manuscript; four new files. This is the handoff announced in the comment of 2026-09-13 02:53 that was still absent from this branch. The Friedgut--Kalai sharp-threshold theorem for symmetric monotone events is applied to rank>=1 and to rank=2: each birth concentrates about its own median. Any subsequential unscaled birth mixture therefore has at most two equally weighted atoms, the two centres need not coincide, and the matching median is NOT recovered from their average. Interface to data already in the tree: G = E(K2-K1)/(N+1) = integral_0^1 P1 exactly, and |G - (b_N - a_N)| <= 2C/log N. Combined with the geometry theorem this reads the full-law concentration condition as this average rank-one lifetime tending to zero. Explicit brackets from nonbacktracking paths and full rows are given for log(m)/w -> d in (0,inf). They are NOT claimed as exact centre predictions or as measured inverse correlation lengths, and uniqueness of a limiting centre pair at fixed d is NOT proved here. Files added: - docs/manuscripts/geometric-balance/two-birth-reduction.md - scripts/two_birth_reduction.py (requires mpmath) - tests/test_two_birth_reduction.py - results/geometric-consistency/two-birth-reduction.json Executed here (2026-09-13, author-supplied; NOT an independent referee check): - seven local tests pass in this repository tree (mpmath 1.4.1, Python 3.13.12) - all four added files byte-identical to the packaged copies - the deterministic JSON regenerates byte-identically - 336 small configurations, all 720 orders on 2x3, numerical work at 80 digits, with the two endpoint cases recomputed at 110 digits Not established: no uniqueness of a limiting centre pair at fixed d, no exact centre value, no inverse-correlation-length measurement, no Monte Carlo, no new p_c. The sharp-threshold input is prior art; this is its corollary for these observables, not a novelty claim. Full Matching-One repository CI has not been run for this commit. --- .../geometric-balance/two-birth-reduction.md | 361 ++++++++++++++++++ .../two-birth-reduction.json | 238 ++++++++++++ scripts/two_birth_reduction.py | 226 +++++++++++ tests/test_two_birth_reduction.py | 65 ++++ 4 files changed, 890 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/two-birth-reduction.md create mode 100644 results/geometric-consistency/two-birth-reduction.json create mode 100644 scripts/two_birth_reduction.py create mode 100644 tests/test_two_birth_reduction.py diff --git a/docs/manuscripts/geometric-balance/two-birth-reduction.md b/docs/manuscripts/geometric-balance/two-birth-reduction.md new file mode 100644 index 00000000..7ebdd1f4 --- /dev/null +++ b/docs/manuscripts/geometric-balance/two-birth-reduction.md @@ -0,0 +1,361 @@ +# Two sharp births, not an arbitrary broad threshold law + +2026-09-13. A continuation of the **same** geometric-balance manuscript (#739). +The new organizing statement here is a direct application of the published +Friedgut--Kalai sharp-threshold theorem, followed by finite identities and +consequences. It is not presented as a new sharp-threshold theorem or as a +novelty certificate. No new continuum model, source dictionary, or width scan +is introduced. + +## 1. The next question, and what is settled here + +The main manuscript distinguishes the balance root from concentration of the +whole birth-time mixture. It gives the full-law geometric criterion +`log N / ell -> 0`, and endpoint splitting for `log N / ell -> infinity`. +The intermediate regime is not a reason to fit an unrestricted profile. + +For an honest integer-period square-site torus, write + +\[ + f_1(p)=\Pr_p(r\ge1)=1-P_0(p),\qquad f_2(p)=\Pr_p(r=2)=P_2(p). +\] + +Both are increasing nonconstant Boolean-event probabilities on **N independent +site variables**. The translation group `Z^2/Lambda` acts transitively on +those variables and preserves either event. Translation acts trivially on +ambient homology, but preserves its rank; no rotation of the period lattice +or irreducibility assumption on its homology representation is needed. + +Let `T1 <= T2` be the two birth times in the usual uniform-label coupling. +Define their unique medians by + +\[ + f_1(a_N)=\tfrac12,\qquad f_2(b_N)=\tfrac12. +\] + +Finite monotonicity gives `a_N <= b_N`. The mixture CDF and its median are + +\[ + F_N=\tfrac12(f_1+f_2),\qquad q_N=F_N^{-1}(1/2). +\] + +The main conclusion below is uniform over the period shape: + +\[ + W_1\!\left(\mathcal L(T),\tfrac12\delta_{a_N}+\tfrac12\delta_{b_N}\right) + \le \frac{C_{\rm FK}}{\log N}. \tag{1} +\] + +Here `T` is a fair independent selection of T1 or T2. The constant is the +universal constant in the imported theorem; it is NOT estimated by our tiny +controls. At finite precision and small N the bound may be uninformative. +Equation (1) is in the **unscaled probability coordinate p**. It does not imply +an affine closure of standardized quantile shapes or of finite-size response +jets. In particular it does not undo #706's finite nonaffine shape result. + +## 2. Precisely the imported theorem + +Friedgut and Kalai (1996), in the theorem stated on the first page and in the +publisher abstract [FK], prove that for a monotone event A invariant under a +transitive group on N Boolean coordinates, there is an absolute C_FK such that + +\[ + \mu_p(A)>\epsilon,\qquad + q\ge p+C_{\rm FK}\frac{\log(1/(2\epsilon))}{\log N} + \quad\Longrightarrow\quad \mu_q(A)>1-\epsilon, \tag{2} +\] + +when the displayed parameters lie in [0,1]. This exact formulation is also +stated as Theorem 6 of Duncan--Kahle--Schweinhart [DKS]. We read the publisher's +statement and DKS section 1.3; the original AMS PDF was not fetched successfully. +The general theorem, not DKS's model-specific conclusion about synchronized +homological thresholds, is used here. + +Only product measure, event monotonicity and coordinate transitivity are +needed for this application. Site sharpness, RSW and matching duality are +**not** inputs to equations (1)--(8). They remain inputs to the separate +geometric conclusions in the main manuscript. + +## 3. Concentration of each birth, with a rate + +Put `L=log N`, let A be either rank event, and let theta be its median. For x>0 +choose `epsilon = exp(-L*x/C_FK)/2` in (2). Starting from the median bounds +the upper tail. If `mu_(theta-x)(A)>epsilon`, (2) would give +`mu_theta(A)>1-epsilon>1/2`, a contradiction, bounding the lower tail. A tail +whose parameter is outside [0,1] is simply zero. Thus + +\[ + \Pr(T_j\le\theta_j-x),\;\Pr(T_j>\theta_j+x) + \le\tfrac12 e^{-Lx/C_{\rm FK}},\qquad + (\theta_1,\theta_2)=(a_N,b_N). \tag{3} +\] + +Consequently, for every positive integer k, + +\[ + \mathbb E|T_j-\theta_j|^k + \le k!\left(\frac{C_{\rm FK}}L\right)^k. \tag{4} +\] + +This follows by integrating the tail: `E X^k = integral k x^(k-1) Pr(X>x) dx`. +In particular the mean-median discrepancy is at most C_FK/L and the variance +is at most `2 C_FK^2/L^2`. + +These estimates do **not** assert independence of T1 and T2. Their joint law +is within `2 C_FK/L` of the point `(a_N,b_N)` in W1 with the l1 metric, +by their actual common-label coupling. Correlations of rescaled fluctuations +can survive. The finite 2x3 control has covariance `1123/58800`, not zero. + +Couple T to the corresponding median using the same fair selector. Equation +(4) proves (1), and similarly W_k is at most `(k!)^(1/k) C_FK/L`. +Every convergent subsequence `(a_N,b_N)->(a,b)` therefore satisfies + +\[ + \mathcal L(T_1,T_2)\Longrightarrow\delta_{(a,b)},\qquad + \mathcal L(T)\Longrightarrow\tfrac12\delta_a+\tfrac12\delta_b. \tag{5} +\] + +Conversely any subsequential weak limit of the mixture has this form, by +compactness of the two centers. Centers need not converge on an arbitrary +oscillating geometric sequence. The two atoms may coincide. This is not an +assertion about the number of physical fields or Markov states. + +There is also a path statement. If +`r_det(p)=1_{p>=a_N}+1_{p>=b_N}`, then under the same uniform labels + +\[ + \mathbb E\int_0^1 |r(p)-r_{\rm det}(p)|\,dp + \le\frac{2C_{\rm FK}}L. \tag{6} +\] + +This is an integrated-p bound, not a uniform-in-p approximation at jumps. +For `a_N+x <= p <= b_N-x`, (3) gives `P1(p)>=1-exp(-L*x/C_FK)`. +The rank-one plateau is caused by separation of two individually sharp +transitions, not by a broad transition of either Boolean event. + +## 4. A finite data reduction already available in rank-birth archives + +Set `Delta_N=b_N-a_N`, `G_N=integral_0^1 P1(p) dp`, and let K1,K2 be the +first and second ambient-rank birth **occupation indices** in a uniform +random permutation of the N sites. Then + +\[ + G_N=\mathbb E(T_2-T_1) + =\frac{\mathbb E(K_2-K_1)}{N+1},\qquad + |G_N-\Delta_N|\le\frac{2C_{\rm FK}}L. \tag{7} +\] + +The first equality is the survival-function identity. Conditional on the +permutation, the kth uniform order statistic has mean k/(N+1), giving the +second. It handles a simultaneous rank jump. K means occupation index here, +not elapsed physical time, and neither birth is replaced by a directional +wrapping proxy. + +No new source or simulation is needed to read G_N from an archive that +already stores these correctly typed birth indices. For occupation-count +censuses `c_(1,k)`, the same exact identity is + +`G_N = (1/(N+1)) sum_(k=0)^N c_(1,k)/binom(N,k)`. + +The mixture quartiles give another version of the same asymptotic gap: + +\[ + a_N-\frac{C_{\rm FK}\log2}{L}\le Q_N(1/4)\le a_N, + \qquad b_N\le Q_N(3/4)\le b_N+\frac{C_{\rm FK}\log2}{L}, +\] +\[ + \Delta_N\le\operatorname{IQR}(F_N) + \le\Delta_N+\frac{2C_{\rm FK}\log2}{L}. \tag{8} +\] + +For example, `f1/2 <= F <= f1` brackets Q(1/4) between the first-birth +quarter-quantile and median. The symmetric argument brackets Q(3/4). +The factor log2 then comes directly from (2). + +These are asymptotic absolute-error relations, not a license to equate finite +thermal jets, IQR and the rank gap without their finite errors. C_FK is not +a calibrated finite-sample error bar. + +For completeness the mixture variance separates exactly: + +\[ + \operatorname{Var}(T)=\tfrac12\operatorname{Var}(T_1) + +\tfrac12\operatorname{Var}(T_2) + +\tfrac14[\mathbb ET_2-\mathbb ET_1]^2. +\] + +Equations (3)--(4) imply an error of at most +`C_FK*Delta_N/L + 3 C_FK^2/L^2` from `Delta_N^2/4`. +A macroscopically broad mixture need not mean either birth is noisy. + +## 5. What this adds to the geometric main theorem + +For every finite torus, + +\[ + a_N\le q_N\le b_N. \tag{9} +\] + +At a_N, `F<=1/2`; at b_N, `F>=1/2`. Thus when ell_N->infinity, Theorem A +of the manuscript puts p_c between every limiting pair of birth centers. +It does **not** make q_N their midpoint. A plateau discards the exponentially +small tail odds that can determine the finite median. + +Combining the reduction with the geometric theorem in #739 gives, for +N_n->infinity, + +\[ + \frac{\log N_n}{\ell_n}\to0 + \quad\Longleftrightarrow\quad \Delta_n\to0 + \quad\Longleftrightarrow\quad G_n\to0 + \quad\Longleftrightarrow\quad \operatorname{IQR}(F_n)\to0. \tag{10} +\] + +Here is the potentially missing step in the reverse direction: if ell has a +bounded subsequence, #739's forced-path packing gives T1->0 and T2->1 on that +subsequence, so Delta->1. Hence Delta->0 forces ell->infinity. Equation (9) +and root consistency then force both centers to p_c, and Theorem B yields +the geometric criterion. These geometric inputs are author-supplied results +of #739; equation (10) is not an independent validation of its corridor proof. + +At extreme elongation the pair is (0,1). At finite positive log N/ell we now +know the possible shape of every subsequential unscaled law: two equal atoms, +with an undetermined pair of locations. The main remaining problem is the +location of those centers, not an unrestricted limiting profile. + +## 6. Why transitivity does not synchronize the births + +Translation is transitive on sites but acts trivially on homology. It sharpens +each rank event separately. It does not require its two threshold locations +to coincide. DKS explicitly separates transitivity from the extra point-group +symmetry of homology in its introduction and surjectivity argument [DKS]. + +A short finite comparison illustrates the extra symmetry. Suppose the actual +period lattice is invariant under a quarter-turn J. Split rational homology +lines into pairs `{l,Jl}`, choosing one line from each pair for a class A, +and its rotate for class B. Define E_A/E_B to mean the ambient image contains +a line in the respective class. These events are increasing. Their union is +`r>=1`, their intersection is `r=2`, and quarter-turn symmetry gives equal +probabilities. Harris association therefore gives, at every p, + +\[ + P_2(p)\ge\Pr(E_A)\Pr(E_B)\ge\tfrac14[1-P_0(p)]^2. \tag{11} +\] + +This is the two-dimensional elementary instance of DKS's symmetry strategy, +not a new general surjectivity theorem. At a_N it implies `P2>=1/16`. +Applying (2) with epsilon=1/32 shows + +`b_N-a_N <= C_FK log(16)/log N`. + +Such quarter-turn symmetry is real on Gaussian ideal quotients. It is absent +on a generic elongated or tilted period lattice. Changing a period basis does +not create it. This explains why importing a synchronized-threshold theorem +from symmetric tori would answer a different question. + +## 7. A quantitative first step in the unsolved intermediate regime + +For axial periods (w,0),(0,m), let w->infinity and `log m / w -> d` with +`00)\le + N\frac{D}{D-1}\frac{[(D-1)p]^w}{1-(D-1)p}. \tag{12} +\] + +It is an overcount, which is appropriate for this upper bound. It also works +for the matching graph: each diagonal changes either coordinate by at most +one, so a nonzero axial period still requires at least w steps. D is 4 on NN +and 8 on matching. Fully occupied horizontal rows, on disjoint site sets, +give the other bound `P0 <= (1-p^w)^m`. + +Using the exact matching-complement relation between birth medians, every +subsequential pair (a,b) obeys + +\[ + \frac{e^{-d}}3\le a\le\min\{p_c,e^{-d}\},\qquad + \max\{p_c,1-e^{-d}\}\le b\le1-\frac{e^{-d}}7. \tag{13} +\] + +For example d=log4 gives `1/12 <= a <= 1/4` and +`3/4 <= b <= 27/28`. These are proven broad brackets, not predictions of +exact center values. The constants 3 and 7 are walk-count bounds, not measured +surface tensions. The relations do NOT establish convergence of a_n,b_n for +all sequences with the same d, and do not identify the equality case in a +putative cost-versus-volume transition. + +## 8. Finite controls, not additional asymptotic evidence + +The existing width-two local matrix gives + +`x=p(1-p), lambda_±=p(1±sqrt(1+4p(1-p)))/2`, + +`P0=(1-p^2)^m-2*x^m`, + +`P2=lambda_+^m+lambda_-^m-x^m`. + +The companion script uses those already established #705 formulas at +m=2,4,8,16,32,128. This is not a new state engine or a new circumference. +Selected values (full values are in the result JSON): + +| m | a_m | q_m | b_m | integral P1 | +|---:|---:|---:|---:|---:| +| 4 | 0.39543672 | 0.56386499 | 0.71750418 | 0.29523810 | +| 16 | 0.20590464 | 0.56519772 | 0.86713870 | 0.63896753 | +| 128 | 0.07348861 | 0.56519772 | 0.95578173 | 0.87423968 | + +Width 2 is fixed, so q_m is not approaching the infinite square-site p_c. +The example illustrates how separated sharp births and an internal balance +root coexist; it is not an extrapolation test of (10). + +Independent graph-potential censuses of 2x2,2x3,2x4 (336 configurations total) +check the formulas, exact beta integrals and normalization. All 720 orders on +2x3 independently verify `E(K2-K1)/(N+1)=3/14` and retain nonzero birth-time +covariance. Seven targeted tests pass. Numerical roots/quadratures are computed +at 80 digits and selected values are recomputed at 110; they are not interval +certificates. Full repository CI has not been run for this addition. + +## 9. One next research target + +Keep the work in #739. The next target is whether the two centers have limits +on exponentially elongated sequences and, when they do, which **microscopic +winding cost** determines them. Start with the axial finite-d family and only +then ask about orientation dependence. Do not infer these costs from a +fixed-width eigenvalue or from a continuum cusp formula with another order of +limits. Equations (12)--(13) are the completed elementary start, not a solution +to the center-location problem. + +No new production is requested here. Existing ambient-rank birth archives can +already supply a_N,b_N and G_N where both births are recorded; do not substitute +directional wrapping times. The balance-root theorem and its independent +proof review remain intact. Higher source orders, more Jordan examples, and +new generic control machinery are not needed to answer this next question. + +## References and claim scope + +[FK] E. Friedgut and G. Kalai, *Every monotone graph property has a sharp +threshold*, Proc. AMS **124** (1996), 2993--3002, +https://doi.org/10.1090/S0002-9939-96-03732-X . The exact transitive-event theorem +is printed in the publisher abstract: +https://www.ams.org/journals/proc/1996-124-10/S0002-9939-96-03732-X/ . +Publisher theorem statement read; full original PDF not retrieved this round. + +[DKS] P. Duncan, M. Kahle and B. Schweinhart, *Homological percolation on a torus: +plaquettes and permutohedra*, arXiv:2011.11903v4, +https://arxiv.org/html/2011.11903v4 . Theorem 6 is precisely (2); section 1.1 +and the section-3 surjectivity discussion distinguish coordinate transitivity +from the homology point-group argument. Those sections were read; their +model-specific simultaneous-threshold theorem is not imported to arbitrary +integer-period square-site tori. + +The generic sharpness input, phase separation through multiple Boolean +events, order-statistic integral and symmetry mechanism are prior tools. +This note supplies their explicit consequences for the manuscript's exact +observable and narrows the remaining research question. No priority claim, +new critical exponent, original-U identification, or complete finite-profile +closure is made. diff --git a/results/geometric-consistency/two-birth-reduction.json b/results/geometric-consistency/two-birth-reduction.json new file mode 100644 index 00000000..5b8d7cc7 --- /dev/null +++ b/results/geometric-consistency/two-birth-reduction.json @@ -0,0 +1,238 @@ +{ + "schema": "matching-one.two-birth-reduction.v1", + "scope": "Existing width-two exact oracle; no new width, no Monte Carlo", + "precision": { + "working_decimal_digits": 80, + "reported_digits": 24, + "quadrature_status": "high precision numerical control, not interval certification" + }, + "universal_constant": "Friedgut--Kalai constant left unspecified; not fitted or priced", + "exact_controls": { + "censuses": [ + { + "width": 2, + "length": 2, + "configurations": 16, + "counts_by_rank_and_occupation": [ + [ + 1, + 4, + 2, + 0, + 0 + ], + [ + 0, + 0, + 4, + 0, + 0 + ], + [ + 0, + 0, + 0, + 4, + 1 + ] + ], + "mean_first_birth": "7/15", + "mean_second_birth": "3/5", + "integral_P1_exact": "2/15" + }, + { + "width": 2, + "length": 3, + "configurations": 64, + "counts_by_rank_and_occupation": [ + [ + 1, + 6, + 12, + 6, + 0, + 0, + 0 + ], + [ + 0, + 0, + 3, + 14, + 9, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 6, + 6, + 1 + ] + ], + "mean_first_birth": "31/70", + "mean_second_birth": "23/35", + "integral_P1_exact": "3/14" + }, + { + "width": 2, + "length": 4, + "configurations": 256, + "counts_by_rank_and_occupation": [ + [ + 1, + 8, + 24, + 32, + 14, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 4, + 24, + 56, + 48, + 12, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 8, + 16, + 8, + 1 + ] + ], + "mean_first_birth": "127/315", + "mean_second_birth": "44/63", + "integral_P1_exact": "31/105" + } + ], + "total_configurations": 336, + "permutation_control": { + "width": 2, + "length": 3, + "permutations": 720, + "normalized_mean_birth_rank_gap": "3/14", + "birth_time_covariance": "1123/58800" + } + }, + "finite_length_controls": [ + { + "width": 2, + "length": 2, + "sites": 4, + "first_birth_median": "0.458803899853803015600277", + "matching_root": "0.541196100146196984399723", + "second_birth_median": "0.614272431867610451724497", + "median_separation": "0.15546853201380743612422", + "mixture_q25": "0.366025403784438646763723", + "mixture_q75": "0.707106781186547524400844", + "mixture_IQR": "0.341081377402108877637121", + "integral_P1": "0.133333333333333333333333", + "first_birth_MAD": "0.184916542699632925904895", + "second_birth_MAD": "0.165533413388540059841259", + "mixture_W1_to_two_median_atoms": "0.125640559792018812765218", + "component_coupling_W1_upper": "0.175224978044086492873077" + }, + { + "width": 2, + "length": 4, + "sites": 8, + "first_birth_median": "0.395436722146492579302221", + "matching_root": "0.563864986818845832313288", + "second_birth_median": "0.717504182456928314943545", + "median_separation": "0.322067460310435735641324", + "mixture_q25": "0.37580635877208387452519", + "mixture_q75": "0.735662455380684050167617", + "mixture_IQR": "0.359856096608600175642427", + "integral_P1": "0.295238095238095238095238", + "first_birth_MAD": "0.152731960470947226955025", + "second_birth_MAD": "0.132186922221715014370339", + "mixture_W1_to_two_median_atoms": "0.104585785998960442820658", + "component_coupling_W1_upper": "0.142459441346331120662682" + }, + { + "width": 2, + "length": 8, + "sites": 16, + "first_birth_median": "0.288087697543491088866109", + "matching_root": "0.565186915007972908279838", + "second_birth_median": "0.8062743570534467506602", + "median_separation": "0.518186659509955661794091", + "mixture_q25": "0.288020678659088142104401", + "mixture_q75": "0.806342683005338318960378", + "mixture_IQR": "0.518322004346250176855977", + "integral_P1": "0.490331890331890331890332", + "first_birth_MAD": "0.119169122112798731672424", + "second_birth_MAD": "0.0936880608925678662934832", + "mixture_W1_to_two_median_atoms": "0.100715347015863140545303", + "component_coupling_W1_upper": "0.106428591502683298982954" + }, + { + "width": 2, + "length": 16, + "sites": 32, + "first_birth_median": "0.205904636425127329663174", + "matching_root": "0.565197715825889906165507", + "second_birth_median": "0.867138698438055207434922", + "median_separation": "0.661234062012927877771749", + "mixture_q25": "0.205904636399435486977378", + "mixture_q75": "0.867138698480686033844308", + "mixture_IQR": "0.66123406208125054686693", + "integral_P1": "0.638967532846024539955863", + "first_birth_MAD": "0.088236907821274692877401", + "second_birth_MAD": "0.0641893702225797165224883", + "mixture_W1_to_two_median_atoms": "0.0760701055702370555700931", + "component_coupling_W1_upper": "0.0762131390219272046999447" + }, + { + "width": 2, + "length": 32, + "sites": 64, + "first_birth_median": "0.146382847056271813348706", + "matching_root": "0.56519771738363933134003", + "second_birth_median": "0.908410478436269182496879", + "median_separation": "0.762027631379997369148173", + "mixture_q25": "0.146382847056271813348706", + "mixture_q75": "0.908410478436269182496879", + "mixture_IQR": "0.762027631379997369148173", + "integral_P1": "0.745727145682731652042279", + "first_birth_MAD": "0.0638725762471686157164193", + "second_birth_MAD": "0.0437352250473570618738386", + "mixture_W1_to_two_median_atoms": "0.053803742027170950798523", + "component_coupling_W1_upper": "0.053803900647262838795129" + }, + { + "width": 2, + "length": 128, + "sites": 256, + "first_birth_median": "0.0734886148758215767986785", + "matching_root": "0.565197717383639396437528", + "second_birth_median": "0.955781733111481641066709", + "median_separation": "0.88229311823566006426803", + "mixture_q25": "0.0734886148758215767986785", + "mixture_q75": "0.955781733111481641066709", + "mixture_IQR": "0.88229311823566006426803", + "integral_P1": "0.874239684305420680243789", + "first_birth_MAD": "0.0325092171535030132257381", + "second_birth_MAD": "0.0205388473568717407670128", + "mixture_W1_to_two_median_atoms": "0.0265240322551873769963718", + "component_coupling_W1_upper": "0.0265240322551873769963755" + } + ], + "interpretation": "Fixed width 2 does not approach the infinite square-site critical point" +} diff --git a/scripts/two_birth_reduction.py b/scripts/two_birth_reduction.py new file mode 100644 index 00000000..3b3def37 --- /dev/null +++ b/scripts/two_birth_reduction.py @@ -0,0 +1,226 @@ +#!/usr/bin/env python3 +"""Finite controls for the two-birth reduction (not a new width scan). + +The universal concentration constant comes from Friedgut--Kalai and is NOT +estimated here. Width-two formulas are an existing exact calibration model. +Only mpmath is required beyond the standard library. No random sampling. +""" +from __future__ import annotations +import argparse +from fractions import Fraction +from itertools import permutations +import json +from math import comb, factorial +from pathlib import Path +import mpmath as mp + + +def lifted_rank(width: int, length: int, mask: int) -> int: + """Ambient rank of the occupied NN graph, retaining parallel lifted edges.""" + if width < 2 or length < 2 or not 0 <= mask < (1 << (width * length)): + raise ValueError("Expected width,length >= 2 and a legal site mask") + potentials: dict[int, tuple[int, int]] = {} + generator: tuple[int, int] | None = None + for root in range(width * length): + if not (mask >> root & 1) or root in potentials: + continue + potentials[root] = (0, 0) + stack = [root] + while stack: + vertex = stack.pop() + x, y = vertex % width, vertex // width + px, py = potentials[vertex] + for dx, dy in ((1, 0), (-1, 0), (0, 1), (0, -1)): + other = ((y + dy) % length) * width + (x + dx) % width + if not (mask >> other & 1): + continue + expected = (px + dx, py + dy) + if other not in potentials: + potentials[other] = expected + stack.append(other) + else: + old = potentials[other] + gain = (expected[0] - old[0], expected[1] - old[1]) + if gain != (0, 0): + if generator is None: + generator = gain + elif generator[0] * gain[1] != generator[1] * gain[0]: + return 2 + return int(generator is not None) + + +def census(width: int, length: int) -> list[list[int]]: + n = width * length + if n > 16: + raise ValueError("This control deliberately caps exhaustive enumeration at 16 sites") + counts = [[0] * (n + 1) for _ in range(3)] + for mask in range(1 << n): + counts[lifted_rank(width, length, mask)][mask.bit_count()] += 1 + return counts + + +def beta_integral(counts: list[int]) -> Fraction: + """Integral of sum_k counts[k] p^k(1-p)^(N-k), exactly.""" + n = len(counts) - 1 + return sum((Fraction(c, (n + 1) * comb(n, k)) + for k, c in enumerate(counts)), Fraction()) + + +def census_probability(counts: list[int], p): + n = len(counts) - 1 + return sum(c * p**k * (1 - p)**(n - k) for k, c in enumerate(counts)) + + +def sector_probabilities(length: int, p): + """Exact algebraic width-two expressions, evaluated at current mp precision.""" + if length < 2 or not 0 <= p <= 1: + raise ValueError("Expected length >= 2 and p in [0,1]") + p = mp.mpf(p) + if p == 0: + return mp.mpf(1), mp.mpf(0), mp.mpf(0) + if p == 1: + return mp.mpf(0), mp.mpf(0), mp.mpf(1) + x, y = p * (1 - p), p * p + lp = p * (1 + mp.sqrt(1 + 4 * p * (1 - p))) / 2 + lm = -x * y / lp # stable product relation, instead of subtracting square roots + p0 = (1 - y)**length - 2 * x**length + p2 = lp**length + lm**length - x**length + return p0, 1 - p0 - p2, p2 + + +def cdf(length: int, channel: str, p): + p0, _, p2 = sector_probabilities(length, p) + if channel == "first": + return 1 - p0 + if channel == "second": + return p2 + if channel == "mixture": + return (1 - p0 + p2) / 2 + raise ValueError("channel must be first, second, or mixture") + + +def quantile(length: int, channel: str, u): + u = mp.mpf(u) + if not 0 < u < 1: + raise ValueError("Quantile level must be strictly between 0 and 1") + lo, hi = mp.mpf(0), mp.mpf(1) + # Reporting precision is intentionally far below this working precision. + for _ in range(4 * mp.mp.dps): + mid = (lo + hi) / 2 + if mid == lo or mid == hi: + break + if cdf(length, channel, mid) < u: + lo = mid + else: + hi = mid + return (lo + hi) / 2 + + +def exact_small_controls() -> dict: + records = [] + for length in (2, 3, 4): + counts = census(2, length) + n = 2 * length + for k in range(n + 1): + assert sum(row[k] for row in counts) == comb(n, k) + for rational in (Fraction(1, 3), Fraction(1, 2), Fraction(2, 3)): + p = mp.mpf(rational.numerator) / rational.denominator + got = sector_probabilities(length, p) + for rank in range(3): + expected = census_probability(counts[rank], rational) + assert abs(got[rank] - mp.mpf(expected.numerator) / expected.denominator) < mp.mpf("1e-50") + mu1 = beta_integral(counts[0]) + mu2 = 1 - beta_integral(counts[2]) + gap = beta_integral(counts[1]) + assert mu2 - mu1 == gap + records.append({"width": 2, "length": length, "configurations": 1 << n, + "counts_by_rank_and_occupation": counts, + "mean_first_birth": str(mu1), "mean_second_birth": str(mu2), + "integral_P1_exact": str(gap)}) + # Independent occupation-order check of the normalized rank-gap identity. + n, sum_first, sum_second, sum_product = 6, 0, 0, 0 + for order in permutations(range(n)): + mask, k1, k2 = 0, None, None + for k, v in enumerate(order, 1): + mask |= 1 << v + r = lifted_rank(2, 3, mask) + if r >= 1 and k1 is None: + k1 = k + if r == 2: + k2 = k + break + assert k1 is not None and k2 is not None + sum_first += k1 + sum_second += k2 + sum_product += k1 * (k2 + 1) + mu1 = Fraction(sum_first, factorial(n) * (n + 1)) + mu2 = Fraction(sum_second, factorial(n) * (n + 1)) + covariance = Fraction(sum_product, factorial(n) * (n + 1) * (n + 2)) - mu1 * mu2 + assert str(mu2 - mu1) == records[1]["integral_P1_exact"] + assert covariance > 0 # independent births are not assumed, even in tiny controls + return {"censuses": records, "total_configurations": sum(r["configurations"] for r in records), + "permutation_control": {"width": 2, "length": 3, "permutations": factorial(n), + "normalized_mean_birth_rank_gap": str(mu2 - mu1), + "birth_time_covariance": str(covariance)}} + + +def finite_record(length: int, digits: int = 24) -> dict: + a = quantile(length, "first", mp.mpf("0.5")) + b = quantile(length, "second", mp.mpf("0.5")) + q = quantile(length, "mixture", mp.mpf("0.5")) + q25 = quantile(length, "mixture", mp.mpf("0.25")) + q75 = quantile(length, "mixture", mp.mpf("0.75")) + assert q25 <= a <= q <= b <= q75 + f1 = lambda p: cdf(length, "first", p) + f2 = lambda p: cdf(length, "second", p) + f = lambda p: cdf(length, "mixture", p) + mu1 = mp.quad(lambda p: 1 - f1(p), [0, a, b, 1]) + mu2 = mp.quad(lambda p: 1 - f2(p), [0, a, b, 1]) + gap = mp.quad(lambda p: sector_probabilities(length, p)[1], [0, a, b, 1]) + assert abs(mu2 - mu1 - gap) < mp.mpf("1e-45") + mad1 = mp.quad(f1, [0, a]) + mp.quad(lambda p: 1 - f1(p), [a, b, 1]) + mad2 = mp.quad(f2, [0, a, b]) + mp.quad(lambda p: 1 - f2(p), [b, 1]) + w1 = (mp.quad(f, [0, a]) + mp.quad(lambda p: mp.mpf(".5") - f(p), [a, q]) + + mp.quad(lambda p: f(p) - mp.mpf(".5"), [q, b]) + + mp.quad(lambda p: 1 - f(p), [b, 1])) + assert w1 <= (mad1 + mad2) / 2 + mp.mpf("1e-45") + assert abs(gap - (b - a)) <= mad1 + mad2 + mp.mpf("1e-45") + values = {"first_birth_median": a, "matching_root": q, "second_birth_median": b, + "median_separation": b - a, "mixture_q25": q25, "mixture_q75": q75, + "mixture_IQR": q75 - q25, "integral_P1": gap, + "first_birth_MAD": mad1, "second_birth_MAD": mad2, + "mixture_W1_to_two_median_atoms": w1, + "component_coupling_W1_upper": (mad1 + mad2) / 2} + return {"width": 2, "length": length, "sites": 2 * length, + **{key: mp.nstr(value, digits) for key, value in values.items()}} + + +def make_report(dps: int = 80) -> dict: + if dps < 60: + raise ValueError("Use at least 60 working decimal digits") + with mp.workdps(dps): + return {"schema": "matching-one.two-birth-reduction.v1", + "scope": "Existing width-two exact oracle; no new width, no Monte Carlo", + "precision": {"working_decimal_digits": dps, "reported_digits": 24, + "quadrature_status": "high precision numerical control, not interval certification"}, + "universal_constant": "Friedgut--Kalai constant left unspecified; not fitted or priced", + "exact_controls": exact_small_controls(), + "finite_length_controls": [finite_record(m) for m in (2, 4, 8, 16, 32, 128)], + "interpretation": "Fixed width 2 does not approach the infinite square-site critical point"} + + +def main() -> None: + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--output", type=Path, required=True) + parser.add_argument("--dps", type=int, default=80) + args = parser.parse_args() + if args.output.exists(): + parser.error("Refusing to overwrite an existing result file") + report = make_report(args.dps) + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(report, ensure_ascii=False, indent=2) + "\n", encoding="utf-8") + print(f"Wrote {args.output}") + + +if __name__ == "__main__": + main() diff --git a/tests/test_two_birth_reduction.py b/tests/test_two_birth_reduction.py new file mode 100644 index 00000000..07b7186e --- /dev/null +++ b/tests/test_two_birth_reduction.py @@ -0,0 +1,65 @@ +"""Targeted controls; these do not determine the universal sharp-threshold constant.""" +from fractions import Fraction +from pathlib import Path +import sys +import unittest +import mpmath as mp +sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "scripts")) +import two_birth_reduction as t + + +class TwoBirthReductionTests(unittest.TestCase): + def test_lifted_parallel_edges(self): + self.assertEqual(t.lifted_rank(2, 2, 0), 0) + self.assertEqual(t.lifted_rank(2, 2, 1), 0) + self.assertEqual(t.lifted_rank(2, 2, 3), 1) + self.assertEqual(t.lifted_rank(2, 2, 7), 2) + self.assertEqual(t.lifted_rank(2, 2, 15), 2) + + def test_exact_census_and_permutation_integral(self): + with mp.workdps(70): + data = t.exact_small_controls() + self.assertEqual(data["total_configurations"], 336) + self.assertEqual(data["permutation_control"]["permutations"], 720) + self.assertEqual(data["permutation_control"]["birth_time_covariance"], "1123/58800") + + def test_beta_integral(self): + self.assertEqual(t.beta_integral([1, 4, 6, 4, 1]), Fraction(1)) + self.assertEqual(t.beta_integral([0, 0, 6, 0, 0]), Fraction(1, 5)) + + def test_median_and_quartile_bracketing(self): + with mp.workdps(65): + for m in (2, 4, 16): + a = t.quantile(m, "first", ".5") + b = t.quantile(m, "second", ".5") + q = t.quantile(m, "mixture", ".5") + self.assertLessEqual(t.quantile(m, "mixture", ".25"), a) + self.assertLessEqual(a, q) + self.assertLessEqual(q, b) + self.assertLessEqual(b, t.quantile(m, "mixture", ".75")) + + def test_probability_range(self): + with mp.workdps(65): + for m in (2, 8, 128): + for p in (mp.mpf(".01"), mp.mpf(".5"), mp.mpf(".99")): + probs = t.sector_probabilities(m, p) + self.assertLess(abs(sum(probs) - 1), mp.mpf("1e-60")) + self.assertTrue(all(-mp.mpf("1e-60") <= x <= 1 + mp.mpf("1e-60") for x in probs)) + + def test_fixed_length_coupling_and_gap(self): + with mp.workdps(65): + data = t.finite_record(4) + self.assertGreater(float(data["integral_P1"]), 0) + self.assertLessEqual(float(data["mixture_W1_to_two_median_atoms"]), float(data["component_coupling_W1_upper"])) + + def test_illegal_inputs(self): + with self.assertRaises(ValueError): + t.lifted_rank(1, 4, 0) + with self.assertRaises(ValueError): + t.quantile(2, "first", 1) + with self.assertRaises(ValueError): + t.sector_probabilities(1, mp.mpf(".5")) + + +if __name__ == "__main__": + unittest.main() From 419791e29efcfa6aaf79129dcf629041f54bea5f Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 13:12:14 +0800 Subject: [PATCH 05/17] Add the parallel consolidation and move the roadmap off its missing-lemma text Fourth handoff for this branch: an independent write-up of the same theorem, prepared by a session without repository write access and delivered as a package. It is adopted alongside the existing consolidation rather than replacing it; the two are kept deliberately in parallel. Added: - docs/manuscripts/geometric-consistency/README.md full statement and proof - scripts/oblique_winding_necklace.py - tests/test_oblique_winding_necklace.py - results/geometric-consistency/oblique-necklace-controls.json Changed: - docs/ROADMAP.md the active-package paragraph now points at the consolidation instead of describing the uniform oblique-corridor lemma as an unproved strengthening. That text was written before #739 and is now stale. What this version adds over docs/manuscripts/geometric-balance/manuscript.md: 1. Damron--Lam, Section 2, is compared explicitly as the closest retrieved quantitative mechanism for tall thin rectangles, with the pointer to Grimmett's 1981 sponge-dimension work. The #613 roadmap asks for a strip-percolation prior-art comparison and the balance write-up has none. 2. Necessity in the bounded-ell case is argued by greedy packing of disjoint translates (a k-site support has at most k^2 intersecting translates, hence at least floor(N/k^2) disjoint ones) rather than by an exponential union bound. Same mechanism, different bookkeeping. 3. Ten sections including a separate discussion of what lies outside the result, and a nearest-source table with hypotheses and limit orders. The two write-ups prove the same statement and cite #736 at the same commit 64d809b4404f80ff3f9adf9713337cc76008e92d. Keeping both is an explicit allocation choice, not an oversight: this is an exploratory repository and the parallel text carries prior-art material the other lacks. Neither is presented as an independently refereed acceptance, and no novelty is certified by non-retrieval. Executed here (2026-09-13): - six local tests pass in this repository tree - all five applied files byte-identical to the packaged copies, ROADMAP included - the geometric control regenerates byte-identically: 8 period cases and 12,990 crossing-event checks - the applied ROADMAP blob base f6868c5c98a8a715b1d06154848f6f0ca7b31910 matches the pre-patch file exactly Not established: no new p_c value, no near-critical rate, no exponent, no external peer review. Full Matching-One repository CI has not been run. --- docs/ROADMAP.md | 50 +- .../geometric-consistency/README.md | 581 ++++++++++++++++++ .../oblique-necklace-controls.json | 358 +++++++++++ scripts/oblique_winding_necklace.py | 354 +++++++++++ tests/test_oblique_winding_necklace.py | 6 + 5 files changed, 1323 insertions(+), 26 deletions(-) create mode 100644 docs/manuscripts/geometric-consistency/README.md create mode 100644 results/geometric-consistency/oblique-necklace-controls.json create mode 100644 scripts/oblique_winding_necklace.py create mode 100644 tests/test_oblique_winding_necklace.py diff --git a/docs/ROADMAP.md b/docs/ROADMAP.md index f6868c5c..ccf7dc88 100644 --- a/docs/ROADMAP.md +++ b/docs/ROADMAP.md @@ -13,32 +13,30 @@ Delivered, not to be reassigned: #613/#670 supplied the full-law sufficient condition; #718/#735 removed aspect restrictions for the balance root; #736 now supplies the missing axial full-law necessity. #276 remains completed. -First consolidate and independently audit #735's arbitrary-period root proof, -including oblique entry counts, support injectivity and disjoint bands. Return -one theorem manuscript and a closest-prior-theorem comparison to #735. Use -#736's axial iff as the precise full-law boundary chapter, with its own RSW -input and independent seam/quantifier check. This is one probability paper, -not two new dispatches. - -A possible strengthening INSIDE this programme, not a second active task, is -the **uniform oblique-corridor lemma**: -for every eta>0, find a fixed p_eta=exp(-eta|u|), -uniformly in its orientation and ambient primitivity. Pack disjoint corridors -in an arbitrary integer-period torus, or exhibit the geometric obstruction. -The physical NN interaction must not be rotated by a change of period basis. - -As part of that same acceptance package, independently examine #736's -seam construction and quantifier order once, then compare its exact statement -with strip-percolation, RSW finite-size criteria and homological-percolation -prior art. A negative search is not an originality certificate. - -**Deliverable:** one consolidated theorem manuscript, independent lemma-level -audit and precise novelty comparison. Treat a proved oblique-corridor extension -as a strengthening, or record its exact missing lemma; do not make it a new -prerequisite for the already supplied narrower results. **Not useful:** another enormous -thin-torus simulation, another proof that a fixed-width root is not p_c, -or re-enumeration of the tiny checks already delivered. +The working consolidation is now +[**When the matching root locates criticality but the threshold law does not**](manuscripts/geometric-consistency/README.md). +It contains the earlier root proof, the full-law sufficient argument, and a +new author proof of the arbitrary-period necessity direction. The resulting +criterion is `log(N)/ell -> 0` for the entire law; roots require only `ell -> infinity` +as a sufficient condition. No new numerical critical point or exponent is claimed. + +The previously missing oblique corridor is constructed with axis-aligned +circuits and connectors, including a closed periodic seam. Integer translations +pack disjoint supports even when the shortest period is nonprimitive in Z^2. +This replaces the missing-lemma placeholder as a research result, not as an +assertion of independent referee acceptance or literature novelty. + +Keep further edits within this one probability manuscript and return the +consolidation to #735, with #613/#736 as its source thread. The focused code +controls support geometry and seam closure; they do not substitute for the +proof. The literature comparison now includes Damron--Lam's tall-thin rectangle +crossing estimates and their "sponge dimensions" antecedents. Do not commission +another general survey or thin-torus census merely to restate those inputs. + +**Next useful work:** resolve any concrete objection to the argument and make +the hypothesis/conclusion comparison precise enough for a stand-alone paper. +Do not add the Jordan controls, local-source hierarchy, or sampling variants as +new prerequisites or chapters. Preserve the older narrower proofs in history. ## Representation reserve: #636 — source-visible topology and structural closure diff --git a/docs/manuscripts/geometric-consistency/README.md b/docs/manuscripts/geometric-consistency/README.md new file mode 100644 index 00000000..5ee35afe --- /dev/null +++ b/docs/manuscripts/geometric-consistency/README.md @@ -0,0 +1,581 @@ +# When the matching root locates criticality but the threshold law does not + +**Consolidated research manuscript — 13 September 2026.** + +This document replaces a dispersed reading path with one mathematical argument. +It consolidates the probability results of #613, #735 and #736, and supplies a +new oblique winding construction for the necessity direction of the full-law +criterion. It does not merge those PRs or import their unrelated representation, +Jordan, source-response or sampling results. The construction and proofs below +are this delivery's author analysis; the finite controls are not an independent +referee's proof acceptance. Priority in the literature is not established. + +## Abstract + +We study independent square-lattice site percolation on finite tori obtained +from arbitrary full-rank integer period lattices. Let N be the number of sites, +ell the length of the shortest nonzero period, and r the rank of the occupied +ambient first-homology image. The matching root is the unique solution of +P(r=2)=P(r=0). Its convergence to the infinite-lattice critical probability is +uniform over all period lattices with ell tending to infinity, without aspect +or shear restrictions. In contrast, the mixture of the two ambient-homology +birth times converges to a point mass at the critical probability if and only +if log(N)/ell tends to zero. + +The necessity statement, previously obtained only for axial rectangles in +#736, follows from an explicit necklace of axis-aligned occupied circuits and +connecting crossings around an arbitrary shortest period. Critical box crossing +and finite-product continuity make its exponential probability cost arbitrarily +small at a fixed subcritical parameter. Lattice translations pack order N/ell +independent necklaces. The construction never rotates the square-lattice +interaction and applies when the shortest period is nonprimitive in Z^2. + +The inputs are site sharpness on the nearest-neighbour and matching graphs, +their critical-point relation, critical square-site box crossing, and finite +matching duality. No critical exponent, conformal limit, transfer-matrix +representation, new numerical critical point, or near-critical rate is assumed. + +## 1. Model, observables and statements + +Let Lambda be a rank-two sublattice of Z^2. On the flat torus + + T_Lambda = R^2/Lambda, V_Lambda = Z^2/Lambda, + N = [Z^2:Lambda], ell = min_{u in Lambda, u != 0} |u|_2, + +occupy each vertex independently with probability p. Use the physical nearest- +neighbour square-grid edges; the ambient embedding, including edge lifts, is +part of the model. All tori considered are honest square-cell tori: each unit +square is embedded with four distinct corners. Every sufficiently large ell +has this property. Parallel lifted edges are not identified merely because +they have the same endpoints. + +For the induced occupied graph G_omega, define, over Q, + + r(omega) = rank im[H_1(G_omega) -> H_1(T_Lambda)] in {0,1,2}, + P_j(p) = Pr_p(r=j), + M(p) = P_2(p)-P_0(p), F(p) = E_p[r]/2 = (1+M(p))/2. + +A rank-one diagonal/spiral class remains rank one. Nonzero projection on two +coordinate axes is not rank two. + +Give the sites independent uniform [0,1] labels, and let T_j be the first p +at which r reaches j, for j=1,2. A single insertion can create more than one +rank, so T_1=T_2 is allowed. Pathwise, + + r(p) = 1_{T_1 <= p} + 1_{T_2 <= p}. + +Thus F is the distribution function of T_J, where J is independent and +uniform on {1,2}. It is not the law of r at a fixed p. Let Q=F^{-1} on (0,1). + +Both {r>0} and {r=2} are increasing nonconstant events. A positive pivotal +configuration on an empty-to-full chain, together with the finite Bernoulli +Russo formula, makes their derivatives positive at each interior p. Therefore +F and M are strictly increasing there. The endpoint values M(0)=-1, M(1)=1 +give a unique root q_Lambda, and q_Lambda=Q(1/2). + +### Theorem A — root consistency, consolidated from #735 + +For the actual nearest-neighbour square-site model, + + lim_{L -> infinity} sup_{Lambda: ell(Lambda) >= L} + |q_Lambda - p_c^site(Z^2)| = 0. (A) + +More precisely, for each fixed p0 and L_p=L_p. (A1) + +At each fixed p>p_c, the reciprocal ratio satisfies the corresponding bound. + +### Theorem B — sharp geometry for the whole law + +Let Lambda_n be any sequence of honest integer-period tori with N_n tending +to infinity. The following are equivalent: + +1. Q_n(u) tends to p_c for every fixed u in (0,1). +2. This convergence is uniform on each compact subinterval of (0,1). +3. F_n(p) tends to 0 for pp_c. +4. The mixture law of T_1,T_2 converges weakly to delta_{p_c}. +5. log(N_n)/ell_n tends to zero. (B) + +The result includes arbitrary shear, orientation and Smith class. It does +not assert that ell tending to infinity is *necessary* for a balance-root +sequence to converge: Theorem A is a uniform sufficient statement for roots. +The necessity in Theorem B is for the complete set of fixed interior +quantiles, not for its median alone. + +The new work in this document is the general-period necessity in Theorem B. +Its key estimate is given in Proposition 6 below. + +## 2. Probability and topology inputs + +We use the following established inputs with their actual model types. + +**S: Site sharpness.** At fixed subcritical p, the probability that an occupied +origin connects to Euclidean distance R is at most C(p) exp[-c(p) R], on both +NN Z^2 and the eight-neighbour matching graph. Duminil-Copin–Tassion [DT], +Theorem 1.1(3), is printed in bond language; their Section 1.2 explicitly gives +the adaptation to site percolation on transitive graphs and refers to +Aizenman–Barsky. We use that site result, not square-bond p_c=1/2. + +**D: Matching criticality.** p_c^site(NN)+p_c^site(NN+NNN)=1. The source chain +is Grimmett–Li [GL], Eq. (1.3), with p_u=p_c in the amenable case. Their main +strict-inequality theorem is not being relabelled as this identity. + +**R: Critical box crossing.** For any fixed aspect ratio, critical NN +square-site crossing probabilities of axis-aligned rectangles are bounded +away from zero uniformly in scale. Zeng [Z], Theorem 1.1, states the specific +site version. Only its lower bound, at aspect ratios 4 and 14, is needed for +the new argument. The retrieved source is an arXiv preprint; journal status +and a quantitative value for its constant are not assumed. + +**T: Finite matching duality.** For every configuration on an honest torus, + + r_NN(omega)+r_matching(omega^c)=2. (T) + +For completeness, a topological route to T is as follows. Take a closed regular +neighbourhood U of the occupied NN graph. In a face, a white matching diagonal +can be replaced by a white boundary path unless its endpoints are the only +two white corners. The remaining diagonals do not cross, and facewise replacement +does not change the ambient homology image. The resulting white graph represents +the image of the complementary subsurface V=closure(T_Lambda\U). For +complementary subsurfaces, relative cohomology, excision and Poincare–Lefschetz +duality identify im H_1(V) with the intersection-orthogonal complement of +im H_1(U). The nondegenerate torus intersection form has dimension two, proving +T. This is the repository's digital-Alexander bridge, not a new result here. + +We also use Harris positive association for increasing or decreasing events +under a product Bernoulli law, and continuity of probabilities of finite +cylinder events in p. The latter is simply finite polynomial continuity. + +## 3. Period geometry without rotating the interaction + +Choose a shortest nonzero u in Lambda and put S=|u|^2=ell^2. It is primitive +in Lambda: u=kz with z in Lambda and |k|>1 would contradict shortest length. +It need not be primitive in Z^2. Complete it to a basis (u,v), with + + det(u,v)=N>0, |u dot v| <= ell^2/2. + +Subtracting a nearest integer multiple of u from v achieves the second +condition. Since |v|>=ell, the transverse height satisfies + + h = N/ell >= sqrt(3) ell/2. (3.1) + +Write n=(-u_y,u_x)/ell. The transverse coordinate on the continuous torus is +n dot x modulo h. On vertices it is det(u,x) modulo N, divided by ell. +Every physical matching edge has Euclidean length at most sqrt(2), so its +transverse displacement is at most sqrt(2). These coordinates do not change +the interaction graph. + +Let g=gcd(|u_x|,|u_y|). The possible transverse coordinates of integer +translations have spacing + + delta = g/ell <= 1. (3.2) + +Indeed, Bezout gives z_0 in Z^2 with det(u,z_0)=g, and g divides N. +Thus the translation levels are delta times integers modulo h. This is the +point where ambient nonprimitivity must be retained, rather than assuming g=1. + +A tube of transverse half-width W around R u / Z u embeds in T_Lambda when +2W=64 set rho=ell/64. Let a=a_rho(p) be the probability that an occupied +origin in the infinite graph has an occupied path to Euclidean distance rho. +Stop at first exit. Its support is contained in radius rho+sqrt(2), whose +diameter is less than ell. It therefore injects into every relevant quotient. +Let A_x be its translate to site x. + +A nonzero ambient cycle has a lifted path escaping this ball. Consequently, +if no A_x occurs, r=0. Harris association applies to the overlapping decreasing +events A_x^c and gives + + P_0 >= Pr(intersection_x A_x^c) >= (1-a)^N. (4.1) + +The local balls are not independent. The lower bound depends on positive +association, not on pretending their supports are disjoint. + +### Lemma 2 — independent bands constrain rank two + +In the transverse circle of circumference h, select + + k = floor(8N/ell^2) + +successive half-open bands of physical width ell/8, shifting their boundaries +off vertices. Their vertex sets are disjoint; any leftover strip is unused. +In band j, define B_j to be an occupied path using only that band's sites, +from the lower sqrt(2) layer to the upper sqrt(2) layer. + +Rank two implies a closed occupied walk with nonzero transverse winding. +Repeat its lift if necessary. For each band, take a first passage above its +upper boundary and the last preceding passage below its lower boundary. +Discard the endpoint edges crossing the boundaries. The intervening path is +in the band and starts/ends within sqrt(2) of the corresponding boundary. +This last-entry/first-exit construction permits arbitrary backtracking. +Hence {r=2} is contained in the intersection of all B_j. + +Each B_j uses only its own site's variables, so these events are independent. +Physical edges between different bands are unused and do not alter this fact. + +To count entrance vertices, centre a unit square at each lattice site. These +squares tile the flat torus with total area N. The squares of vertices in a +transverse layer of width sqrt(2) lie in a layer of width 2sqrt(2). The fibres +have length ell, so there are at most 2sqrt(2)ell entrance sites. We may use + + B = 4 ceil(ell). + +A B_j crossing has transverse separation at least ell/8-2sqrt(2)>rho, so its +initial site witnesses a local arm. A union bound and independence give + + P_2 <= (B a)^k. (4.2) + +Equation (3.1) implies k>=4N/ell^2. Both (4.1) and (4.2) apply to NN and to +the matching graph, with their respective local-arm probabilities. + +### Proof of Theorem A + +Fix pp_c(NN), D makes 1-p subcritical for the matching graph; +T exchanges its ranks 0 and 2 with the original ones. Apply the same estimate. +The two fixed parameters p_c-epsilon and p_c+epsilon then trap the unique +root uniformly over all Lambda with sufficiently large ell, proving A. + +The conditional function H=P_2/(P_0+P_2) also has fixed interior quantiles +converging uniformly, since H<=P_2/P_0 below p_c and 1-H<=P_0/P_2 above it. +This conditional function is distinct from F. + +A finite arithmetic consequence is useful but not an infinite-volume bound: +if an actual upper bound abar for a satisfies + + (4 ceil(sqrt(S)) abar)^4 < (1-abar)^S, + +then P_2/P_0<1 on that finite torus. No near-critical arm bound or numerical +p_c enclosure is computed in this manuscript. + +## 5. An oblique necklace of axis-aligned crossings + +This is the new deterministic construction. It addresses the previously +missing necessity direction without assuming rotated RSW, conformal +invariance, a primitive vector in ambient Z^2, or a particular Smith type. + +### Lemma 3 — explicit geometric construction + +Let s>=8 be even and ell>=64s. Set + + n_0 = ceil(4ell/s), + z_i = nearest_integer_coordinatewise(i u/n_0), 0<=i<=n_0. + +Use rounding satisfying round(x+k)=round(x)+k for integer k, and extend +z_{i+n_0}=z_i+u. In particular z_0=0 and z_{n_0}=u. Successive centres satisfy + + |z_{i+1}-z_i|_infinity <= s/4+1 <= s/2. (5.1) + +For z_i=(x_i,y_i), require four occupied NN crossings: + +- horizontal crossings of [x_i-2s,x_i+2s] times [y_i+s,y_i+2s] + and [x_i-2s,x_i+2s] times [y_i-2s,y_i-s]; +- vertical crossings of [x_i-2s,x_i-s] times [y_i-2s,y_i+2s] + and [x_i+s,x_i+2s] times [y_i-2s,y_i+2s]. + +The four crossings meet in the four corner squares. Their union contains an +occupied circuit C_i surrounding the inner square z_i+[-s,s]^2. This is the +usual planar annulus gluing: choose crossing subpaths joining the consecutive +corner intersections. The resulting closed walk travels successively through +top, right, bottom and left strips and has winding one about z_i; extracting +simple cycles leaves one enclosing the inner square's interior. + +Connect C_i to C_{i+1} by requiring one more horizontal crossing of + + R_i = [min(x_i,x_{i+1})-3s, max(x_i,x_{i+1})+3s] + times + [max(y_i,y_{i+1})-s/2, min(y_i,y_{i+1})+s/2]. (5.2) + +Its height is at least s/2 and width at most 13s/2, hence aspect ratio at most +13. A left-right crossing of R_i has endpoints outside both outer squares, +and at x=x_i and x=x_{i+1} passes strictly inside their inner squares. +Planarity of NN edges forces it to meet both C_i and C_{i+1} in occupied +vertices. This works even when the progression of centres is nearly vertical: +the connector is still horizontal, with a common central vertical interval. + +There are exactly 5 n_0 crossing events. Every individual rectangle has +diameter less than 7s0. It is a +sufficient event, not an equality with all winding configurations. Multiple +intersections, side contacts and path backtracking cannot destroy it. + +### Lemma 4 — arbitrarily low exponential cost at a fixed subcritical p + +There is c in (0,1), independent of s, orientation and u, bounding all these +individual crossing probabilities below at p_c. For the annuli use aspect 4 +in R. For connectors compare in the infinite grid with a 7s-by-s/2 crossing +(aspect 14) and restrict to the shorter rectangle. The relevant event on the +torus has exactly the infinite-grid law because its own support injects. + +Fix s. Only finitely many integer rectangle dimensions occur. Finite-product +continuity gives a single p_s in (0,p_c) at which each has probability at least +c/2. Integer translations do not change these probabilities. Consequently +Harris association on the overlapping crossing events gives + + Pr_{p_s}(necklace) >= (c/2)^(5 n_0) + >= exp[-25 log(2/c) ell/s]. (5.3) + +For every eta>0, choose one even s large enough that 25 log(2/c)/s<=eta, +and then choose p_eta=p_s. For every shortest u with ell>=64s, + + Pr_{p_eta}(necklace around u) >= exp(-eta ell). (5.4) + +The quantifier order is eta -> s -> p_eta -> arbitrary large tori. +p_eta is a fixed parameter below p_c, not a size-dependent sequence. +No numerical RSW constant or estimate of p_c-p_eta is supplied. + +## 6. Packing independent necklaces despite shear and nonprimitivity + +### Lemma 5 — integer translations supply enough transverse space + +Let delta=g/ell from (3.2), and take + + D = ceil((12s+2)/delta), Delta = D delta. + +Then 12s+2<=Delta<12s+3. Let z_0 have det(u,z_0)=g, and translate the whole +necklace by jD z_0 for j=0,...,b-1, where + + b=floor(h/Delta). + +Their transverse centres are j Delta modulo h. Successive centres, including +the last-to-first circular gap, are at least Delta apart. Their tubes have +full width 12s, so the complete event supports have disjoint site sets. +No claim of independence is made for the crossings inside one necklace. + +Since h>=sqrt(3)ell/2>=32sqrt(3)s and s>=8, h/Delta>2, and + + b >= h/(2Delta) >= h/(30s) = N/(30s ell). (6.1) + +This explicit use of delta handles u=(w,0), u=(an,bn), and primitive ambient +vectors in the same construction. Taking a period basis with small display +entries is not required. A large longitudinal shift of a translated necklace +is immaterial to disjoint transverse support. + +### Proposition 6 — a uniform upper bound for the rank-zero probability + +For every eta>0 there are s and p_eta=64s satisfies + + P_0^Lambda(p_eta) + <= [1-exp(-eta ell)]^b + <= exp[-N exp(-eta ell)/(30s ell)]. (6.2) + +Only one successful necklace is needed to prevent rank zero. Disjoint support +makes the b necklace events independent. Neither a success nor their union is +being asserted to imply rank two. + +The appearance of both a geometric opportunity count and an exponential +cost is essential. An upper union bound failing to vanish would not yield +this lower-quantile obstruction. + +## 7. Proof of the sharp full-law criterion + +### Sufficiency + +Suppose log(N_n)/ell_n -> 0. In particular ell_n -> infinity. At fixed p0) <= C(p) N_n exp[-c(p) ell_n] -> 0. + +Apply the same statement to the matching complement at fixed p>p_c and use T +to obtain Pr_p(r_n<2)->0. Hence F_n tends to the threshold step. This is the +previous sufficient argument of #613, reproduced to close the theorem. + +### Necessity when ell is unbounded along a witnessing subsequence + +If log(N_n)/ell_n does not tend to zero, choose a subsequence with +log(N_n)>=d ell_n for some fixed d>0. First suppose that along a further +subsequence ell_n tends to infinity. In Proposition 6 choose eta=d/2. +Then + + N_n exp(-eta ell_n)/(30s ell_n) + >= exp(d ell_n/2)/(30s ell_n) -> infinity. + +Therefore P_0(p_eta)->0 at the fixed subcritical p_eta, and + + F_n(p_eta) >= (1-P_0(p_eta))/2 -> at least 1/2. + +For every fixed u<1/2, eventually Q_n(u)<=p_etainfinity, independent repetition again gives P_0(p)->0 and the same +lower-quantile obstruction. + +Every subsequence witnessing failure of the geometric condition has one of +these two further subsequences. This proves necessity without assuming ell_n +already tends to infinity. + +### Equivalence of law and quantile formulations + +Threshold-step convergence traps all Q_n(u), u in [epsilon,1-epsilon], +between p_c-delta and p_c+delta for sufficiently large n. It thus implies +uniform compact-quantile convergence, which implies pointwise convergence. +Conversely, if all fixed quantiles converge, then for fixed p0, eventually p 1/sqrt(2). Theorem A gives q_{Lambda_n}->p_c, while Theorem B +shows that the whole law does not concentrate; each fixed lower quantile is +bounded away from p_c along the witnessing construction. Both conclusions +concern the same microscopic site model, and u_n is nonprimitive in ambient +Z^2. No value of a displaced limiting quantile is asserted. + +### Standard square and Gaussian-square families remain inside the good regime + +For a square Gaussian period lattice generated by g and ig, ell=sqrt(N). +Thus log(N)/ell tends to zero, including nonprimitive Gaussian representatives. +Both root and full-law consistency hold. The obstruction is excessive volume +relative to the shortest period, not arithmetic nonprimitivity by itself. + +### What remains outside the result + +There is no quantitative near-critical rate, no value of a shifted quantile +on exponential-aspect sequences, and no L^-4 correction law. Critical RSW +contributes a qualitative, scale-uniform lower bound, followed by continuity +at a fixed block size. This argument does not identify a continuum operator, +original-U source map, generic-q tangent, or width-uniform spectral expansion. + +Theorems A and B concern independent site occupation. Positive dependent +marks, arbitrary random-cluster measures and large externally imposed sources +need their own hypotheses; they are not inserted as corollaries here. + +## 9. Relation to prior results and contribution boundary + +The relevant comparison is between precise models and limit orders, not +whether a paper uses the name "matching" or "homological". + +| Source / existing asset | Statement used or read | Relation to this manuscript | +|---|---|---| +| Mertens–Ziff [MZ], Section IV and matching identities | Finite matching function, square-sequence root/step convergence; empirical rapid root shift | Observable and square-sequence conclusions are prior art; not an all-period iff quoted from this source | +| Duncan–Kahle–Schweinhart [DKS], Section 1, Theorems 1–2 | Ambient-homology transitions on the specified growing cubical/permutohedral tori; i=1 cubical model is bond, 2D permutohedral site model is triangular | Prior art for ambient-image observables and homological transitions; not automatically the arbitrary-shape NN square-site law | +| Zeng [Z], Theorem 1.1 | Critical NN square-site box crossing | Imported lower bound; no new RSW theorem is claimed | +| Duminil-Copin–Tassion [DT], Theorem 1.1(3), Section 1.2 | Subcritical decay and explicit site adaptation | Imported sharpness, not relabelled bond criticality | +| Grimmett–Li [GL], introduction Eqs. (1.1)–(1.3) | Matching critical relation through uniqueness and amenability | Imported infinite-graph relation | +| Damron–Lam [DL], Section 2, Proposition 2.5 / Corollary 2.7 | Fixed-subcritical bond crossings of tall thin free rectangles, controlled by opportunity count times exp[-short side / correlation length] | Closest retrieved quantitative mechanism; ordinary crossing is not a seam-closed NN-site winding event on an arbitrary period quotient | +| #735 | Uniform root comparison on arbitrary integer-period tori | Consolidated, not counted as a new theorem in this delivery | +| #736 | Axial full-law iff | Extended by the oblique necklace and integer-translation packing | + +[DL] explicitly relates its result to Grimmett's 1981 "Critical sponge +dimensions" and later subcritical-connectivity work. This is a meaningful +prior-art lead, not a claim that the broad opportunity-versus-cost mechanism +originates here. The original 1981 article was not independently read in this +delivery. No exhaustive citation graph or novelty certification was performed. +The direct primary texts read for [Z], [DT], [GL], [MZ] and [DKS] were their +stated theorem/definition sections; [DL] was read at its introduction and +Section 2 crossing statements. Full original proofs of all imported results +were not re-proved or independently refereed. + +## 10. Finite controls and integration + +The new script `scripts/oblique_winding_necklace.py` is standalone standard +Python. It uses exact integer centres and period cosets. Free-rectangle crossing +BFS is separate from its physical lifted-graph winding detector. Eight periods +include an axis shortest vector with ambient gcd 512, an oblique vector with +gcd 128, primitive oblique vectors, two genuinely reduced HNF examples and a +near-diagonal short-period choice. Every example is in ell>=64s with s=8. + +Executed controls verify 12,990 specified crossing events, local rectangle +injectivity bounds, tube support, seam closure with primitive u-winding, and +disjoint *whole event supports* for selected first/second/last packed translates. +These are deterministic occupied witnesses, not Monte Carlo samples. A tiny +separate Fraction calculation checks Harris arithmetic but supplies no RSW +constant. Six local tests pass. No full repository CI was run. + +Read this document as one probability manuscript, not as a new dispatch tree. +The working research boundary is now the validity and positioning of this +single combined statement. Keep the older narrow proofs and original raw +results in their branches; they remain useful if any proposed extension +needs revision. The repository's representation and original-U work is not +made dependent on this document and is not reopened by it. + +## References + +[Z] X. Zeng, *A Russo Seymour Welsh Theorem for critical site percolation on +Z^2*, arXiv:1309.2273v1, Theorem 1.1. +https://arxiv.org/html/1309.2273 + +[DT] H. Duminil-Copin and V. Tassion, *A new proof of the sharpness of the phase +transition for Bernoulli percolation and the Ising model*, arXiv:1502.03050v3, +Theorem 1.1 and Section 1.2. +https://arxiv.org/html/1502.03050v3 + +[GL] G. Grimmett and Z. Li, *Percolation critical probabilities of matching +lattice-pairs*, arXiv:2205.02734v3, introduction Eqs. (1.1)–(1.3), with the +companion proof identified there. +https://arxiv.org/html/2205.02734v3 + +[MZ] S. Mertens and R. M. Ziff, *Percolation in Finite Matching Lattices*, +arXiv:1603.07289v2. Equation numbering can differ between HTML and PDF. +https://arxiv.org/html/1603.07289v2 + +[DKS] P. Duncan, M. Kahle and B. Schweinhart, *Homological percolation on a +torus: plaquettes and permutohedra*, arXiv:2011.11903v4, Section 1. +https://arxiv.org/html/2011.11903v4 + +[DL] M. Damron and W.-K. Lam, *Asymptotics for first passage percolation on +logarithmic subgraphs of Z^2*, arXiv:2502.18235v2, Section 2, especially +Corollary 2.7. +https://arxiv.org/html/2502.18235v2 + +Repository inputs read: #735 at 9d29d014df28af7c635e6859d98a95ffe2b34d06; +#736 at 64d809b4404f80ff3f9adf9713337cc76008e92d; main AGENTS.md, +RESEARCH-FRONTIER.md and ROADMAP.md after #738. 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0, + 512 + ], + [ + -1200, + 51 + ] + ], + "reduced_basis": [ + [ + 0, + 512 + ], + [ + -1200, + 51 + ] + ], + "N": 614400, + "squared_shortest_period": 262144, + "ambient_gcd": 512, + "block_scale": 8, + "blocks": 256, + "crossing_events": 1280, + "support_vertices": 25088, + "occupied_witness_vertices": 19746, + "winding_gcd": [ + 1, + 0 + ], + "packed_bands": 12, + "checked_translates": [ + 0, + 1, + 11 + ], + "conditions_pass": true + }, + { + "input_basis": [ + [ + 384, + 512 + ], + [ + -1536, + 1152 + ] + ], + "reduced_basis": [ + [ + 384, + 512 + ], + [ + -1536, + 1152 + ] + ], + "N": 1228800, + "squared_shortest_period": 409600, + "ambient_gcd": 128, + "block_scale": 8, + "blocks": 320, + "crossing_events": 1600, + "support_vertices": 28928, + "occupied_witness_vertices": 25988, + "winding_gcd": [ + 1, + 0 + ], + "packed_bands": 19, + "checked_translates": [ + 0, + 1, + 18 + ], + "conditions_pass": true + }, + { + "input_basis": [ + [ + 511, + 129 + ], + [ + -387, + 1533 + ] + ], + "reduced_basis": [ + [ + 511, + 129 + ], + [ + -387, + 1533 + ] + ], + "N": 833286, + "squared_shortest_period": 277762, + "ambient_gcd": 1, + "block_scale": 8, + "blocks": 264, + "crossing_events": 1320, + "support_vertices": 20868, + "occupied_witness_vertices": 14930, + "winding_gcd": [ + 1, + 0 + ], + "packed_bands": 16, + "checked_translates": [ + 0, + 1, + 15 + ], + "conditions_pass": true + }, + { + "input_basis": [ + [ + 357, + -407 + ], + [ + 1221, + 1071 + ] + ], + "reduced_basis": [ + [ + 357, + -407 + ], + [ + 1221, + 1071 + ] + ], + "N": 879294, + "squared_shortest_period": 293098, + "ambient_gcd": 1, + "block_scale": 8, + "blocks": 271, + "crossing_events": 1355, + "support_vertices": 24679, + "occupied_witness_vertices": 21782, + "winding_gcd": [ + 1, + 0 + ], + "packed_bands": 16, + "checked_translates": [ + 0, + 1, + 15 + ], + "conditions_pass": true + }, + { + "input_basis": [ + [ + 900, + 0 + ], + [ + 271, + 711 + ] + ], + "reduced_basis": [ + [ + 271, + 711 + ], + [ + -900, + 0 + ] + ], + "N": 639900, + "squared_shortest_period": 578962, + "ambient_gcd": 1, + "block_scale": 8, + "blocks": 381, + "crossing_events": 1905, + "support_vertices": 36787, + "occupied_witness_vertices": 28923, + "winding_gcd": [ + 1, + 0 + ], + "packed_bands": 8, + "checked_translates": [ + 0, + 1, + 7 + ], + "conditions_pass": true + }, + { + "input_basis": [ + [ + 1100, + 0 + ], + [ + 473, + 853 + ] + ], + "reduced_basis": [ + [ + 473, + 853 + ], + [ + -627, + 853 + ] + ], + "N": 938300, + "squared_shortest_period": 951338, + "ambient_gcd": 1, + "block_scale": 8, + "blocks": 488, + "crossing_events": 2440, + "support_vertices": 45226, + "occupied_witness_vertices": 40544, + "winding_gcd": [ + 1, + 0 + ], + "packed_bands": 9, + "checked_translates": [ + 0, + 1, + 8 + ], + "conditions_pass": true + }, + { + "input_basis": [ + [ + 1024, + 0 + ], + [ + 517, + 515 + ] + ], + "reduced_basis": [ + [ + 507, + -515 + ], + [ + 517, + 515 + ] + ], + "N": 527360, + "squared_shortest_period": 522274, + "ambient_gcd": 1, + "block_scale": 8, + "blocks": 362, + "crossing_events": 1810, + "support_vertices": 33002, + "occupied_witness_vertices": 28767, + "winding_gcd": [ + 1, + 0 + ], + "packed_bands": 7, + "checked_translates": [ + 0, + 1, + 6 + ], + "conditions_pass": true + } + ], + "cases_count": 8, + "crossing_event_checks": 12990, + "harris_control": { + "scope": "toy Harris inequality, not a percolation/RSW estimate", + "rows": [ + { + "p": "1/3", + "joint": "7/27", + "product": "125/729" + }, + { + "p": "1/2", + "joint": "1/2", + "product": "27/64" + }, + { + "p": "2/3", + "joint": "20/27", + "product": "512/729" + } + ] + }, + "no_probability_or_novelty_estimate": true, + "full_repository_ci_run": false +} diff --git a/scripts/oblique_winding_necklace.py b/scripts/oblique_winding_necklace.py new file mode 100644 index 00000000..3ba8d5d9 --- /dev/null +++ b/scripts/oblique_winding_necklace.py @@ -0,0 +1,354 @@ +#!/usr/bin/env python3 +"""Finite geometry controls for the oblique RSW necklace construction. + +Only integer arithmetic is used. This tests geometric supports, the periodic +seam, winding of explicit witnesses and disjoint translates. It does not +compute a near-critical probability, prove RSW, or independently prove the +all-size theorem. No old Matching-One implementation is imported. +""" +from __future__ import annotations + +import argparse +from collections import deque +from dataclasses import dataclass +from fractions import Fraction +from math import gcd, isqrt +import json +from pathlib import Path +import unittest + +Point = tuple[int, int] + + +def det(a: Point, b: Point) -> int: + return a[0] * b[1] - a[1] * b[0] + + +def dot(a: Point, b: Point) -> int: + return a[0] * b[0] + a[1] * b[1] + + +def add(a: Point, b: Point) -> Point: + return a[0] + b[0], a[1] + b[1] + + +def nearest(num: int, den: int) -> int: + """Nearest integer, with ties upwards; commutes with integer translation.""" + if den <= 0: + raise ValueError("denominator must be positive") + return (2 * num + den) // (2 * den) + + +def ceil_scaled_sqrt(square: int, multiplier: int, divisor: int) -> int: + """ceil(multiplier * sqrt(square) / divisor), exactly.""" + if square < 0 or multiplier <= 0 or divisor <= 0: + raise ValueError("invalid square-root arguments") + n = isqrt(multiplier * multiplier * square) // divisor + return n + (n * n * divisor * divisor < multiplier * multiplier * square) + + +def reduced_basis(u: Point, v: Point) -> tuple[Point, Point]: + """Two-dimensional Lagrange reduction; u is a shortest lattice vector.""" + if det(u, v) == 0: + raise ValueError("periods must be independent") + for _ in range(256): + if dot(v, v) < dot(u, u): + u, v = v, u + q = nearest(dot(u, v), dot(u, u)) + if q == 0: + break + v = v[0] - q * u[0], v[1] - q * u[1] + else: + raise RuntimeError("reduction did not terminate") + if u[0] < 0 or (u[0] == 0 and u[1] < 0): + u = -u[0], -u[1] + if det(u, v) < 0: + v = -v[0], -v[1] + return u, v + + +def bezout(a: int, b: int) -> tuple[int, int, int]: + """Return g>=0, x, y with ax+by=g=gcd(a,b).""" + r0, r1, x0, x1, y0, y1 = abs(a), abs(b), 1, 0, 0, 1 + while r1: + q = r0 // r1 + r0, r1 = r1, r0 - q * r1 + x0, x1 = x1, x0 - q * x1 + y0, y1 = y1, y0 - q * y1 + return r0, x0 * (1 if a >= 0 else -1), y0 * (1 if b >= 0 else -1) + + +@dataclass(frozen=True) +class Rectangle: + x0: int + x1: int + y0: int + y1: int + direction: str + + def corners(self) -> tuple[Point, ...]: + return ((self.x0, self.y0), (self.x0, self.y1), + (self.x1, self.y0), (self.x1, self.y1)) + + def vertices(self): + for y in range(self.y0, self.y1 + 1): + for x in range(self.x0, self.x1 + 1): + yield x, y + + +def necklace(u: Point, s: int) -> tuple[list[Point], list[Rectangle]]: + """Return n centres plus their translated endpoint, and exactly 5n boxes.""" + if s < 8 or s % 2: + raise ValueError("s must be even and at least 8") + S = dot(u, u) + if S < (64 * s) ** 2: + raise ValueError("this control uses the theorem regime ell >= 64s") + n = ceil_scaled_sqrt(S, 4, s) + centres = [(nearest(i * u[0], n), nearest(i * u[1], n)) + for i in range(n + 1)] + boxes: list[Rectangle] = [] + for i, (x, y) in enumerate(centres[:-1]): + boxes.extend(( + Rectangle(x - 2*s, x + 2*s, y + s, y + 2*s, 'H'), + Rectangle(x - 2*s, x + 2*s, y - 2*s, y - s, 'H'), + Rectangle(x - 2*s, x - s, y - 2*s, y + 2*s, 'V'), + Rectangle(x + s, x + 2*s, y - 2*s, y + 2*s, 'V'), + )) + X, Y = centres[i + 1] + boxes.append(Rectangle(min(x, X) - 3*s, max(x, X) + 3*s, + max(y, Y) - s//2, min(y, Y) + s//2, 'H')) + return centres, boxes + + +def coset_key(x: Point, u: Point, v: Point) -> Point: + N = det(u, v) + if N <= 0: + raise ValueError("basis must be positively oriented") + return det(x, v) % N, det(u, x) % N + + +def crossing_path(box: Rectangle, salt: int) -> set[Point]: + """A deterministic, sometimes backtracking, crossing inside one box.""" + if box.direction == 'H': + y = box.y0 + (salt % (box.y1 - box.y0 + 1)) + pts = {(x, y) for x in range(box.x0, box.x1 + 1)} + if box.y0 < y < box.y1: + # Add a spur; it does not impose any stochastic model. + xm = (box.x0 + box.x1) // 2 + pts.update((xm, yy) for yy in range(y, min(y + 3, box.y1) + 1)) + return pts + x = box.x0 + (salt % (box.x1 - box.x0 + 1)) + return {(x, y) for y in range(box.y0, box.y1 + 1)} + + +def has_crossing(box: Rectangle, occupied, key) -> bool: + """Free-boundary NN crossing BFS, restricted to this one lifted rectangle.""" + if box.direction == 'H': + starts = [(box.x0, y) for y in range(box.y0, box.y1 + 1)] + finish = lambda z: z[0] == box.x1 + else: + starts = [(x, box.y0) for x in range(box.x0, box.x1 + 1)] + finish = lambda z: z[1] == box.y1 + seen = {z for z in starts if key(z) in occupied} + todo = deque(seen) + while todo: + z = todo.popleft() + if finish(z): + return True + for e in ((1, 0), (-1, 0), (0, 1), (0, -1)): + t = add(z, e) + if (box.x0 <= t[0] <= box.x1 and box.y0 <= t[1] <= box.y1 + and t not in seen and key(t) in occupied): + seen.add(t) + todo.append(t) + return False + + +def winding_gcd(occupied: dict[Point, Point], u: Point, v: Point) -> tuple[int, int]: + """Independent graph-potential traversal. Return gcd of u,v windings.""" + N = det(u, v) + visited: set[Point] = set() + all_a = all_b = 0 + for initial in occupied: + if initial in visited: + continue + pot = {initial: (0, 0)} + visited.add(initial) + todo = deque([initial]) + while todo: + k = todo.popleft() + r = occupied[k] + for e in ((1, 0), (-1, 0), (0, 1), (0, -1)): + nb = coset_key(add(r, e), u, v) + if nb not in occupied: + continue + guess = add(pot[k], e) + if nb not in pot: + pot[nb] = guess + visited.add(nb) + todo.append(nb) + else: + c = guess[0] - pot[nb][0], guess[1] - pot[nb][1] + aN, bN = det(c, v), det(u, c) + if aN % N or bN % N: + raise AssertionError("non-period cycle gain") + all_a = gcd(all_a, abs(aN // N)) + all_b = gcd(all_b, abs(bN // N)) + return all_a, all_b + + +def control(u0: Point, v0: Point, s: int = 8, seed: int = 0) -> dict: + u, v = reduced_basis(u0, v0) + S, N = dot(u, u), det(u, v) + centres, boxes = necklace(u, s) + n = len(centres) - 1 + assert centres[0] == (0, 0) and centres[-1] == u + assert 4*N*N >= 3*S*S # h >= sqrt(3)*ell/2 + assert (n*s)**2 >= 16*S and (n-1)**2*s*s < 16*S + assert n*n*s*s <= 25*S + for i in range(n): + assert max(abs(centres[i+1][j]-centres[i][j]) for j in (0, 1)) <= s//2 + for box in boxes: + width, height = box.x1-box.x0, box.y1-box.y0 + assert width > 0 and height > 0 and width*width + height*height < S + long, short = (width, height) if box.direction == 'H' else (height, width) + assert long <= 14*short + assert all(det(u, z)**2 <= 36*s*s*S for z in box.corners()) + + occupied: dict[Point, Point] = {} + key = lambda x: coset_key(x, u, v) + for j, box in enumerate(boxes): + for z in crossing_path(box, seed + 17*j): + occupied.setdefault(key(z), z) + assert all(has_crossing(box, occupied, key) for box in boxes) + ga, gb = winding_gcd(occupied, u, v) + assert (ga, gb) == (1, 0) + + # Check WHOLE event supports for a few translates, not just occupied paths. + support = {key(z): z for box in boxes for z in box.vertices()} + g, a, b = bezout(*u) + transverse_unit = -b, a + assert det(u, transverse_unit) == g and N % g == 0 + D = ceil_scaled_sqrt(S, 12*s+2, g) + gap, bands = D*g, N//(D*g) + assert gap*gap > 144*s*s*S + assert gap < (12*s+3) * (isqrt(S)+1) + assert (30*s*bands)**2*S >= N*N + shifts = sorted(set([0, 1, max(0, bands-1)])) + supports = [] + for j in shifts: + shift = j*D*transverse_unit[0], j*D*transverse_unit[1] + here = {key(add(z, shift)) for z in support.values()} + assert len(here) == len(support) + for previous in supports: + assert not (here & previous) + supports.append(here) + return dict(input_basis=[list(u0), list(v0)], reduced_basis=[list(u), list(v)], + N=N, squared_shortest_period=S, ambient_gcd=g, block_scale=s, + blocks=n, crossing_events=len(boxes), support_vertices=len(support), + occupied_witness_vertices=len(occupied), winding_gcd=[ga, gb], + packed_bands=bands, checked_translates=shifts, + conditions_pass=True) + + +def finite_harris_control() -> dict: + """Independent tiny overlapping-increasing-event Harris arithmetic.""" + # Three overlapping clauses, not a model for the RSW constant. + events = [lambda m: bool(m & 3), lambda m: bool(m & 6), + lambda m: bool(m & 5)] + rows = [] + for p in (Fraction(1, 3), Fraction(1, 2), Fraction(2, 3)): + probs = [Fraction(0)]*3 + joint = Fraction(0) + for m in range(8): + k = bin(m).count('1') + w = p**k * (1-p)**(3-k) + vals = [e(m) for e in events] + for i, ok in enumerate(vals): + probs[i] += w*ok + joint += w*all(vals) + product = probs[0]*probs[1]*probs[2] + assert joint >= product + rows.append(dict(p=str(p), joint=str(joint), product=str(product))) + return dict(scope="toy Harris inequality, not a percolation/RSW estimate", rows=rows) + + +CASES = [ + ((512, 0), (37, 900)), + ((0, 512), (-1200, 51)), + ((384, 512), (-1536, 1152)), + ((511, 129), (-387, 1533)), + ((357, -407), (1221, 1071)), + ((900, 0), (271, 711)), + ((1100, 0), (473, 853)), + ((1024, 0), (517, 515)), +] + + +class GeometryTests(unittest.TestCase): + def test_integer_rounding_and_endpoints(self): + for den in range(1, 16): + for num in range(-40, 41): + k = nearest(num, den) + self.assertLessEqual(abs(k*den-num)*2, den) + self.assertEqual(nearest(num+7*den, den), k+7) + + def test_lagrange_reduction(self): + for u, v in CASES: + a, b = reduced_basis(u, v) + self.assertEqual(det(a, b), abs(det(u, v))) + self.assertLessEqual(dot(a, a), dot(b, b)) + self.assertLessEqual(2*abs(dot(a, b)), dot(a, a)) + for i in range(-5, 6): + for j in range(-5, 6): + if i or j: + x = i*a[0]+j*b[0], i*a[1]+j*b[1] + self.assertGreaterEqual(dot(x, x), dot(a, a)) + + def test_seam_and_nonprimitive_period(self): + r = control((384, 512), (-1536, 1152), seed=3) + self.assertEqual(r['ambient_gcd'], 128) + self.assertEqual(r['winding_gcd'], [1, 0]) + + def test_oblique_primitive_and_packing(self): + r = control((511, 129), (-387, 1533), seed=11) + self.assertEqual(r['ambient_gcd'], 1) + self.assertGreaterEqual(r['packed_bands'], 3) + + def test_reject_too_small_support_regime(self): + with self.assertRaises(ValueError): + necklace((4, 4), 8) + + def test_harris_control(self): + self.assertEqual(len(finite_harris_control()['rows']), 3) + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--output', type=Path) + ap.add_argument('--tests', action='store_true') + args = ap.parse_args() + if args.tests: + suite = unittest.defaultTestLoader.loadTestsFromTestCase(GeometryTests) + result = unittest.TextTestRunner(verbosity=2).run(suite) + raise SystemExit(not result.wasSuccessful()) + if not args.output: + ap.error('--output PATH or --tests is required') + if args.output.exists(): + raise FileExistsError(f"refusing to overwrite {args.output}") + rows = [control(u, v, seed=i*19) for i, (u, v) in enumerate(CASES)] + out = dict(schema='matching-one.oblique-necklace-controls.v1', + nature='deterministic geometric controls, not production evidence', + cases=rows, cases_count=len(rows), + crossing_event_checks=sum(r['crossing_events'] for r in rows), + harris_control=finite_harris_control(), + no_probability_or_novelty_estimate=True, + full_repository_ci_run=False) + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(out, indent=2)+'\n', encoding='utf-8') + print(json.dumps(dict(cases=len(rows), crossing_checks=out['crossing_event_checks'], + output=str(args.output)))) + + +if __name__ == '__main__': + main() diff --git a/tests/test_oblique_winding_necklace.py b/tests/test_oblique_winding_necklace.py new file mode 100644 index 00000000..7f318da9 --- /dev/null +++ b/tests/test_oblique_winding_necklace.py @@ -0,0 +1,6 @@ +"""Expose the focused geometric controls to unittest discovery.""" +from pathlib import Path +import sys + +sys.path.insert(0, str(Path(__file__).resolve().parents[1] / 'scripts')) +from oblique_winding_necklace import GeometryTests # noqa: E402,F401 From fafcc15ec70fa914a0e7343a7b75bad193c95ddf Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 13:26:35 +0800 Subject: [PATCH 06/17] Add the one-frontier winding-intensity transfer and the prefactor handoff Fifth handoff for this branch; five new files, nothing overwritten. This is the engine and the derivation behind the two tickets opened alongside it, #740 and #741. Contents: - docs/manuscripts/geometric-balance/winding-intensity-and-prefactor.md - docs/manuscripts/geometric-balance/prefactor-handoff-20260913.md - scripts/cylinder_winding_intensity.py - tests/test_cylinder_winding_intensity.py - results/geometric-consistency/cylinder-winding-intensity.json What it supplies. An exact one-frontier component-retirement transfer with a winding flag per active component and an integer reward when a winding component is permanently retired; appending an empty row flushes the remainder. The count is once per COMPLETE component, not per row, path, cut or marked vertex. Widths 2/3/4 have 6/14/38 full states and 3/4/7 all-p reward-preserving lumps, giving rational nu_w for both adjacencies; at p = 1/2 the NN intensities are 7/48, 169/1984, 323849/5576960. Also recorded as prior art rather than a new observation definition: W4(omega) - W8(omega^c) = r4(omega) - 1 follows from the wrapping-cluster classification printed in Mertens--Ziff 2016 section II, and yields nu4_w(p) = nu8_w(1-p) -- a complementary-probability identity, NOT equality at the same p. What it does NOT supply, and this is the point of #740. The renewal-loop coefficient calculation is done and gives exp(-kappa w)/sqrt(2 pi D w) with D = sigma^2/mu. What is missing is the sewing or cluster-weight identity relating an ACTUAL SITE COMPONENT to a closed renewal object. The displayed w^{-1/2} is a hypothesis, not an accepted site theorem. Copying the planar two-point OZ amplitude does not supply that identity. Executed here (2026-09-13): - twelve local tests pass in this repository tree - all five added files byte-identical to the packaged copies - the full JSON regenerates with every field identical except elapsed_seconds (here 3.87 s against the packaged 9.13 s report time) Capacity probe recorded, not extrapolated: NN states/transitions are 102/3264, 282/18048, 786/100608, 2214/566784 for w = 5..8. This is a closure and resource probe, not a physical configuration census. Reaching w = 12 needs sparse aggregation or a site-by-site factorisation; a dense operator or a symbolic rational inverse is explicitly not the intended route. No Monte Carlo, no GPU, no new p_c, no merge, no docs/STATUS.md edit. Full Matching-One repository CI has not been run for this commit. --- .../prefactor-handoff-20260913.md | 45 + .../winding-intensity-and-prefactor.md | 555 + .../cylinder-winding-intensity.json | 10123 ++++++++++++++++ scripts/cylinder_winding_intensity.py | 503 + tests/test_cylinder_winding_intensity.py | 109 + 5 files changed, 11335 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/prefactor-handoff-20260913.md create mode 100644 docs/manuscripts/geometric-balance/winding-intensity-and-prefactor.md create mode 100644 results/geometric-consistency/cylinder-winding-intensity.json create mode 100644 scripts/cylinder_winding_intensity.py create mode 100644 tests/test_cylinder_winding_intensity.py diff --git a/docs/manuscripts/geometric-balance/prefactor-handoff-20260913.md b/docs/manuscripts/geometric-balance/prefactor-handoff-20260913.md new file mode 100644 index 00000000..e22af514 --- /dev/null +++ b/docs/manuscripts/geometric-balance/prefactor-handoff-20260913.md @@ -0,0 +1,45 @@ +# 前因子交接：同一篇论文，只剩一个微观映射和一个数值对比 + +2026-09-13。主入口仍为 #739；本批代码尚未推入其分支。阅读 +[winding-intensity-and-prefactor.md](winding-intensity-and-prefactor.md)。 + +## 本轮已完成 + +1. 对实际 NN / NN+NNN 点渗流构造单前沿、带整数提升位移与逐分量绕行标志的转移；只在完整绕行分量永久离开前沿时计数。宽度 2/3/4 的完整状态数为 6/14/38，同权重且保留计数分布的强合并为 3/4/7。两张图的全 p 密度有理式均已求出。 +2. 将 Mertens–Ziff 已有的有限绕行分量分类写成 W4−W8=r4−1，推导互补概率下的圆柱密度与计数压力关系。同一 p 的两个颜色不能独立泊松化：独立模型给不可能的 (0,0) 事件正概率。这是窗口合并的实质约束，不是另开模型。 +3. 对明确指定的有限范围 renewal-loop 模型，证明 w^{-1/2} 前因子及振幅 1/sqrt(2πD)。这还不是 SITE 分量的前因子定理；从两点连接到闭合、完整且每簇只计一次的缝合映射仍然缺失。 +4. 修正上一批条件中心展开：只有可微性时，必须保留精确 κ^{-1}；直接把 log(w)/w 位移线性化并宣称 o(1/w) 余项，需要额外正则性。有限中位数处 O(1/w) 的 Gumbel 结论不因这个修正而改变。 + +## 已直接开出的两项工作 + +**#740 — 理论／定向检索。** +https://github.com/LightChainr/Matching-One/issues/740 + +确定实际 SITE 绕行分量是否具有 A(p)w^{-1/2}exp[−wκ(p)]，或给出正确幂次、周期修正与振幅。必须落实每簇一次、切口数量、w/n 因子、闭合缝与完整外边界权重。已完成的 renewal 系数计算无需重做。先读并追踪 CI/CIV 的实际适用模型及 SITE 扩展；2026 年 D’Alimonte–Manolescu v3 的主要模型为 BOND FK，Theorem 1.1 是两点函数的两侧比较，不能直接当作本题振幅公式。结果返回 #739。 + +**#741 — 有界 CPU 计算。** +https://github.com/LightChainr/Matching-One/issues/741 + +只计算两组固定参数：NN p=1/4、matching p=1/8；每组宽度 4、8、12。输出 + + R4 = nu4*nu12/nu8^2, + beta_eff = log(R4)/log(4/3). + +此对比精确消去所假设渐近式中的质量和振幅；三个宽度不能证明极限。不得换成无绕行生存特征值或长环面直接抽样。记录实际状态数、运行时间、内存与 log(nu) 误差，优先达到 1e-8，达不到就报告实际界。现成的空行复位给出从平稳残差到密度误差的严格界。 + +参考脚本只是有保护上限的原型：width<=10、默认 state_cap=5000。它不是开箱即可计算宽度 12 的优化实现；应根据 #741 做稀疏聚合或逐站点因式分解，而非默认删除上限后建立巨型稠密逆矩阵。已经实际运行的 NN 容量探测到宽度 8：2214 状态、566784 条掩码转移、约 6.7 秒；这不等于宽度 12 的价格估计。 + +## 复现 + +从仓库根执行： + +```sh +python -m unittest discover -s tests -p 'test_cylinder_winding_intensity.py' -v +python scripts/cylinder_winding_intensity.py --output /tmp/cylinder-intensity-new.json +``` + +脚本拒绝覆盖输出。完整报告的符号部分使用 SymPy；转移、精确平稳求解、数值残差证书与测试使用标准库。表内匹配关系为 nu_NN(p)=nu_matching(1-p)，不是同一 p 上的两图相等。 + +本批 12 项局部测试通过。139776 个开放圆柱图／配置逐一对照，66064 对有限环面互补配置检查；18 个有理数参数点与符号函数核对，3 个全 p 互补恒等式。完整 Matching-One CI 没有运行。小系统控制不替代渐近证明，也不是新 Monte Carlo 样本。 + +本轮不要继续派发大角度梯队、generic Jordan 例子、任意高阶源或固定宽度谱重建。下一次新增分析应由 #740 的微观映射或 #741 的这个特定对比改变结论。 diff --git a/docs/manuscripts/geometric-balance/winding-intensity-and-prefactor.md b/docs/manuscripts/geometric-balance/winding-intensity-and-prefactor.md new file mode 100644 index 00000000..318cb6ec --- /dev/null +++ b/docs/manuscripts/geometric-balance/winding-intensity-and-prefactor.md @@ -0,0 +1,555 @@ +# Count the component before assigning its prefactor + +2026-09-13. Continuation of the **same** geometric-balance paper (#739). + +The earlier fixed-d Poisson/Gumbel note leaves the cylinder intensity +`nu_w^G(p)` as the microscopic object controlling the finite centre. This +note makes that object explicitly computable, identifies an exact +complementary-count constraint on the crossover, and separates a proved +renewal-loop calculation from the still missing SITE-cluster sewing theorem. +The current delivery does **not** certify a site prefactor exponent or a +near-critical Gumbel limit. + +## 1. Results and one correction + +**Actual site model, completed here.** A one-frontier transfer retains the +connectivity, horizontal lift gains, and existing winding flag of each active +component. A reward is issued only when a winding component is permanently +retired. For widths 2,3,4 this has respectively 6,14,38 reachable states for +each of G4=NN and G8=NN+NNN. Complete successor tables close exactly. Common +homogeneous row-weight/reward lumpings have 3,4,7 states. Their stationary +rewards give rational functions of p for the actual infinite-cylinder +component intensities. These are sufficient representations, not claimed +minimal positive or linear realizations. + +**Prior topology, with useful consequences.** The classification already +printed in Mertens--Ziff [MZ, section II] gives, for an honest finite torus, + + W4(omega)-W8(omega^c)=r4(omega)-1, (1.1) + +where W counts components with nonzero ambient homology. It implies +`nu4_w(p)=nu8_w(1-p)`, equality of complementary count pressures, and a +rigorous obstruction to independent two-colour Poisson counts at one common +parameter. Neither the classification nor the idea of matching topology is +claimed new. + +**Explicit renewal object, completed here.** For a positive-length, +aperiodic, transversely symmetric finite-range renewal kernel, the +once-per-closed-loop coefficient with the correct horizontal translation +factor has prefactor `1/sqrt(2*pi*D*w)`. This is a direct coefficient/Laplace +calculation. It is NOT yet an identification of percolation components with +those renewal loops. That identification is the precise external task #740. + +**Correction to the preceding conditional displacement formula.** Mere +C1 regularity at a does not justify an `o(1/w)` remainder after linearizing +at a displacement of size `log(w)/w`. The exact mass-inverse formula in +section 7 is valid under the previous prefactor hypothesis and a local +inverse-Lipschitz bound. Its linear version additionally needs a Taylor +remainder of order `o(1/w)` there, for instance C^{1,alpha} regularity. This +does not affect a finite-median-centred window of size `1/w`, which only +uses differentiability. A concrete C1, locally semiconcave countercontrol is +included. + +## 2. A component-retirement transfer for the actual site graph + +### 2.1 Boundary data + +Process spatial rows from bottom to top on `C_w x Z`. Only the current row +is exposed; there is no stored first-row torus seam. Record: + +* which current sites are occupied and their connected-component partition; +* in a component with no horizontal winding, the integer horizontal cut + gain from its first current vertex to each other current vertex; +* a Boolean flag on a component once a nonzero horizontal cycle has appeared. + +Within a winding component, relative gains may be replaced by zero: every +future component meeting it is already winding. This replacement is NOT +permitted in a nonwinding component. It would destroy the distinction, for +example, between row histories `[13,5,13,0]` and `[7,5,13,0]` at width four. +The first has zero horizontal-winding components, the second has one. + +For the new row, add its occupied NN ring edges. Across the old/new interface +add vertical edges (G4), or vertical and both diagonal edges (G8). A cut +gain for an edge from column i to column i+dx is `floor((i+dx)/w)`. Lifted +parallel edges at width two are retained, not collapsed. + +When a component has no representative on the new row, it can never meet +any later row: both graphs have vertical step at most one. Give reward 1 +if that component winds and reward 0 otherwise. Then forget the old row. +Appending a deterministic empty row at the end flushes all active components. + +### 2.2 Why it retains the right count for every continuation + +Replace each processed nonwinding component by a gain-labelled tree between +its exposed vertices. Any future closed path can substitute an old path for +a tree path or vice versa; the difference is a previously existing closed +path. In a nonwinding component that difference has zero horizontal gain. +A winding component stays winding under arbitrary continuation, so its flag +is enough when it merges. Components never split when processed vertices are +forgotten, because the forgotten connecting paths are retained by the +partition. A completely unexposed component cannot reappear. + +Induction on appended rows therefore proves: the cumulative reward plus the +number of active winding components is the number of complete winding +components of the entire finite free-height cylinder. The empty-row flush +returns the exact complete count. This is a statement about component counts, +not just the rank of their union. + +The encoding works whenever its reachable-state exploration terminates. The +returned complete successor table proves termination/closure at each tested +width. It is not an all-width asymptotic state-count formula. For widths +2/3/4, both graphs give the stated 6/14/38 states, all gains in {-1,0,1}, and +1488 total row-transition entries over the six tables. + +### 2.3 Row probabilities and the source-marked kernel + +For row mask b, let `k(b)` be its occupation count and +`a_b(p)=p^{k(b)}(1-p)^{w-k(b)}`. If `tau(s,b)` is the new state and `R(s,b)` +the retirement reward, define + + K_z(s,t)=sum_{b:tau(s,b)=t} a_b(p) exp[z R(s,b)]. (2.1) + +At z=0 the matrix is stochastic. Every state jumps to the empty state with +probability at least `(1-p)^w`; every listed state is reachable from empty. +Hence for 0infinity} Var(W_m)/m=psi''(0). (2.3) + +Changing the initial frontier or flushing the last components has only a +bounded boundary effect on the log generating function. Perron simplicity +near z=0 gives analyticity and the derivatives in (2.3) for these finite +chains. No claim is made that psi''(0)=nu at a fixed small width. + +The implementation computes the variance with exact rational linear algebra. +Writing K_j=partial_z^j K_z|0 and 1 for the constant vector, solve + + (I-K_0)h=K_1*1-nu*1, pi*h=0. + +Then + + psi''(0)=pi*K_2*1+2*pi*K_1*h-nu^2. (2.4) + +### 2.4 What the smaller matrices preserve + +Partition refinement compares, for each old state, the exact integer counts +of row masks for every triple + + (new-row occupied count, reward, next block). + +Equality is a polynomial identity in p, not a numerical test at a few p's. +The resulting 3/4/7-state kernels preserve the joint count/reward law for +any sequence of *row-wise homogeneous* probabilities. + +They need NOT give the same answer configuration by configuration for an +unchanged sequence of fixed row masks. One can relabel/mix same-weight masks +inside a lump. Consequently the verification compares the unreduced transfer +with each physical configuration and the reduced transfer with the complete +occupation/count polynomial. Treating a stochastic lump as a deterministic +pathwise quotient would be a mistake. + +## 3. Exact intensities, not a fitted prefactor + +The NN functions for widths two and three are + + nu_2(p)= p^2(1-p)^2 (p^2+p+1)/(p^2-p+1), (3.1) + + nu_3(p)= p^3(1-p)^3 (p^6+p^3+2p^2+2p+1) + /(p^6-3p^5+3p^4+p^3-p^2-p+1). (3.2) + +The width-four rational function has a degree-27 numerator and degree-19 +denominator. Its exact factored expression and all integer coefficients, +together with both matching-graph functions and the generating state tables, +are stored in `results/geometric-consistency/cylinder-winding-intensity.json`. +They are generated by the actual transfer, not supplied as fitted data. + +At p=1/2: + +| width | NN nu_w | count variance per vertical row | +|---|---:|---:| +| 2 | 7/48 | 343/6912 | +| 3 | 169/1984 | 4769011/122023936 | +| 4 | 323849/5576960 | 186754153229427053/6098108298338304000 | + +For example the width-two asymptotic count Fano ratio is `49/144`, not one. +The finite-width count process is not exactly Poisson. A small-width control +cannot be called evidence for an exact Poisson process at that width. + +Similarly, the no-horizontal-winding probability for a free strip at width +two is `(1-p^2)^m`, because a full row is necessary and sufficient for +horizontal winding there. Its decay rate is `-log(1-p^2)`, not (3.1). +At p=1/2 these are approximately 0.287682 and 0.145833 respectively. The two +objects can have the same leading rare-event rate as w grows without being +identical at fixed w. The component density is the one needed by the +preceding Poisson-window analysis. + +All-p complement identities are checked symbolically. At p=1/4,1/2,3/4, +exact Fraction stationary solves independently agree with the symbolic +functions and with complementary-graph variance rates. + +## 4. Complementary counts constrain the window merger + +### 4.1 The finite topology already in the literature + +[MZ] explicitly states that single-wrapping black and white matching clusters +occur in equal numbers, including spirals, and that a cross-wrapping cluster +is unique and occurs only when the other colour has no wrapping cluster. +On any honest torus the same elementary subsurface argument gives: + + (W4,W8) is (1,0), (0,1), or (k,k) with k>=1. (4.1) + +For completeness, in the rank-one case take a regular neighbourhood of the +occupied graph. Its essential boundary curves are parallel primitive circles. +An essential connected neighbourhood has exactly two essential boundaries; +any further holes bound discs in the torus. Cutting along all essential +boundaries yields an alternating cyclic order of black and white essential +regions. Each has two such boundaries, so their numbers agree. Contractible +regions do not affect this count. The embedded white reduction uses the same +facewise diagonal replacement as the digital-Alexander argument. Rank-two +and rank-zero cases follow from the complement-rank identity and disjointness +of independent essential curves. This proves (4.1) and (1.1). + +This is a reformulation/use of the published classification, not a claim to +have discovered a new homology observable. + +### 4.2 Exact intensity and pressure duality + +Divide the expectation of (1.1) by m and take m->infinity at fixed width. +Closing an open cylinder by its vertical torus seam adds at most 3w edges; +each added edge changes the essential-component count by at most one. Thus +the open-cylinder and torus mean count densities have the same limit. Hence + + nu_w^4(p)=nu_w^8(1-p), 0= exp[-lambda4-lambda8]. (4.5) + +The TV convention is sup over events. This nonvanishing obstruction applies +at every finite size and any limit with both means bounded. It does not +contradict the earlier two-window theorem: that theorem measures BLACK at +one parameter and WHITE at a different, separated parameter. It does prove +that their independence cannot be extended to a common critical window merely +by sending d to zero inside the fixed-d theorem. + +Even when each marginal admits some approximation, their joint law must +respect (4.1). In particular, shared-label coupling is not an optional +normalization detail in a crossover calculation. + +## 5. A closed-renewal prefactor calculation, with its model boundary + +This section identifies exactly what a successful microscopic renewal +mapping would buy. It is a theorem about the following explicit renewal +object, NOT an unproved substitution for the site component activity. + +Let `a(x,y)>=0` have finite support with integer x>=1 and integer y. Assume +reflection symmetry in y, total mass below one, and positive weights at +(1,0),(1,1),(1,-1). Define + + A(z,y)=sum_{x,j} a(x,j) z^x y^j. + +Let R>1 be the unique root A(R,1)=1, set kappa=log R, and normalize +`q(x,j)=a(x,j)R^x`. With expectation under q, put + + mu=E X>0, sigma^2=E Y^2>0, D=sigma^2/mu. + +Define the closed-loop intensity with one unit of transverse length as + + L_w = w [z^w y^0] {-log(1-A(z,y))} + = w sum_{n>=1} (1/n) + sum_{sum x_i=w, sum y_i=0} product_i a(x_i,y_i). (5.1) + +The factor w is horizontal translation; the 1/n removes the marked renewal +cut with the usual weighted cyclic convention. The expression itself fixes +the object even for periodic words. A percolation sewing argument must show +that its own multiplicities match this convention or explicitly correct it. + +**Proposition.** Under these hypotheses, + + L_w = exp(-kappa*w)/sqrt(2*pi*D*w) * (1+O(1/w)). (5.2) + +**Proof.** Take the y^0 coefficient by Fourier inversion. For each theta, + + w[z^w]{-log(1-A(z,e^{i theta}))} + = [z^w] z A_z(z,e^{i theta})/(1-A(z,e^{i theta})). (5.3) + +Near theta=0 the implicit-function theorem gives a simple root R(theta) +near R. Symmetry makes it even and real for real theta sufficiently small. +Expanding A at the root gives + + log R(theta)=kappa+(D/2)theta^2+O(theta^4). (5.4) + +Indeed, differentiating with respect to log z yields mu, while the second +Fourier derivative is -sigma². The pole in (5.3) has principal part +`z/(R(theta)-z)`, so its coefficient is exactly R(theta)^(-w), with leading +amplitude one. Other roots are uniformly farther away for small theta. + +For theta away from zero the triangle inequality is strict at |z|=R: equality +would require identical phases at every support point. The three positive +(1,0),(1,±1) weights force both the phase of z and theta to be zero. Compactness +then supplies a uniform larger coefficient-contour radius away from that +neighbourhood. Those Fourier contributions are exponentially smaller. +Laplace integration of (5.4) yields + + (1/2pi) integral exp[-w(kappa+D theta²/2+O(theta⁴))] dtheta + = R^(-w)/sqrt(2*pi*D*w) * (1+O(1/w)). + +This proves (5.2). More general step supports require their actual lattice +span factors; those are not hidden in the constant. Finite-support is a +sufficient hypothesis here, not the expected final site-renewal class. + +For a finite-state Markov renewal kernel the analogous singular object is +`-log det(I-A)`. A simple Perron crossing again isolates one logarithmic +singularity; however an actual site mapping may involve an infinite internal +state and nontrivial boundary weights. No such mapping is supplied here. + +### 5.1 A completely exact control + +For x=1 and Y uniform on {-1,0,1}, with a common killing weight t in (0,1), + + L_w=t^w c_w/3^w, + c_w=sum_{k=0}^{floor(w/2)} binom(w,k) binom(w-k,k). + +Here D=2/3 and (5.2) has amplitude sqrt(3)/(2sqrt(pi)). The code computes +all coefficients with integers and checks the mass-cancelling contrast below. +For w=8,16,32,64,128, the effective powers approach 1/2 from below; the last +is approximately 0.4983006988 and the normalized amplitude is 0.9985352834. +These are a renewal-model check, NOT site-percolation data. + +Omitting the w factor in (5.1) would change the power from 1/2 to 3/2 without +changing kappa. A two-point decay rate alone cannot determine this counting +normalization. Degenerate transverse variance would also invalidate the +Gaussian prefactor rather than supply the same theorem with D=0. + +## 6. The computation that distinguishes a power without fitting the mass + +Suppose, at a fixed subcritical p, + + nu_w=A w^(-beta) exp(-kappa*w)(1+o(1)). (6.1) + +Then + + R_w=nu_w*nu_{3w}/nu_{2w}^2 -> (4/3)^beta, (6.2) + beta_eff(w)=log(R_w)/log(4/3) -> beta. + +Both log A and the exponential mass cancel algebraically. No pc estimate or +amplitude fitting is used. The finite expression is always defined; its +interpretation as beta uses (6.1). Corrections and possible width arithmetic +oscillations can dominate a small-width contrast, so an agreement at three +widths is not a theorem about w->infinity. + +The concrete external task #741 is exactly two graph/probability inputs, +G4 at p=1/4 and G8 at p=1/8, on w=4,8,12. Both inputs are safely subcritical +by elementary nonbacktracking path bounds. These are not a common-mass pair. +They avoid giant torus simulations and probe the proposed sewing power. + +The reference Python builder was capacity-probed, not physically enumerated, +at widths 5--8. Full state counts were 102,282,786,2214; the all-mask transition +counts were 3264,18048,100608,566784. NN width eight took about 6.7 seconds +on this host and gave 90 stochastic reward blocks. No polynomial algorithm +or cheap width-24 calculation is inferred. At width 12 sparse/site-wise +factorization may be necessary. + +### 6.1 A residual can be turned into an actual rare-density error bound + +The row chain has the common empty-row reset `delta=(1-p)^w`. It is therefore +an L1 contraction by at most 1-delta on zero-mass signed measures. For any +normalized nonnegative candidate pi_hat, let `r=pi_hat*K-pi_hat`. Summing +its propagated residuals gives + + ||pi_hat-pi||_1 <= ||r||_1/delta. + +Consequently + + |pi_hat*g-nu| <= ||g||_infinity ||r||_1/delta. (6.3) + +The implementation supplies this certificate in exact fractions; larger +floating runs must use outward error bounds including matrix-vector and +reward-evaluation rounding. The requested target is absolute error <=1e-8 +on log nu where practical. Merely printing a tiny residual is not a bound on +relative error in a rare event. This is a usable numerical estimate, not a +request for an additional audit framework. + +## 7. Correct centering at order 1/w + +Let (6.1) hold locally uniformly near a with A continuous and positive, +constant beta, kappa(a)=d and kappa locally inverse-Lipschitz. Assume + + log m=d*w+gamma log w+c0+o(1). + +For a fixed intensity level lambda>0 (lambda=-log(1-u) for a first-birth +u-quantile), the exact-mass centering is + + p_w = kappa^{-1}( d + + [(gamma-beta)log w+c0+log A(a)-log lambda]/w ) + + o(1/w). (7.1) + +Here p_w denotes an intensity solution; the earlier uniform Poisson/window +argument transfers it to the corresponding true quantile at a regular +mass point. The proof first brackets p_w within O(log w/w) of a using the +positive lower slope, replaces log A(p_w) by log A(a)+o(1), and uses the +inverse-Lipschitz bound to turn the remaining o(1) logarithmic error into +an o(1/w) p error. This does not linearize kappa at the larger displacement. + +If kappa is C^{1,alpha} near a, alpha>0, and v=-kappa'(a)>0, (7.1) simplifies to + + p_w=a+[(beta-gamma)log w-c0-log A(a)+log lambda]/(v*w) + +o(1/w). (7.2) + +Indeed w*(log w/w)^{1+alpha}->0. The same linear formula is valid with +mere differentiability when the logarithmic coefficient beta-gamma vanishes. +Without either that cancellation or an appropriate Taylor-remainder bound, +(7.2) has not been justified to o(1/w). This corrects that precision claim +in the preceding conditional-prefactor discussion, not its finite-median +Gumbel theorem. + +**Countercontrol.** On h>0 put + + kappa(a+h)=d-h-h/log(e/h), + +and on h<=0 put kappa(a+h)=d-h. On a small neighbourhood it is C1, +strictly decreasing and concave, with derivative -1 at a. Let +`nu_w(p)=w^{-1} exp[-w*kappa(p)]` and `log m=dw`. At intensity one the true +positive displacement h solves + + h+h/log(e/h)=log w/w. + +Then h/(log w/w)->1, but + + w*(h-log w/w)=-w*h/log(e/h) -> -1. + +Thus the linear formula's error is not o(1/w), even under the local +semiconcavity obtained in the prior argument. This is a mathematical +regularity control, not a claim that the actual site mass has this defect. + +## 8. Near-critical crossover: what is now fixed and what remains open + +Equation (4.5) is an unconditional constraint on simultaneous complementary +counts. It rules out copying the separated-window independent Poisson law +into a common critical window. It does not by itself find the crossover law. + +The earlier Poisson proof used a compact subcritical p interval, hence fixed +localization constants C,c. If p=p_w approaches criticality, its error estimate +would require new uniform constants. Keeping them explicit, sufficient +conditions of the same form include a cutoff H_w with 4H_w=w, and + + C_w m(w+1) exp(-c_w H_w) -> 0, + m w^8 (H_w+w+1)^3 exp[-2(w-1)kappa(p_w)] -> 0. (8.1) + +These are conservative sufficient conditions inherited from the local +indicator proof, not a necessary physical crossover criterion. Neither +c_w nor their divergence scale is supplied by a fixed-p compactness argument. + +There is one useful, explicitly CONDITIONAL scaling calculation. Suppose a +near-critical component-density theorem establishes + + w nu_w(p)=Psi(z)(1+o(1)), z=w*kappa(p), + Psi(z)~c_* sqrt(z) exp(-z), z->infinity, (8.2) + +with the uniformity needed for the same p_w and w limit. Then, for aspect +ratio R=m/w->infinity and intensity lambda, + + z=log R + (1/2)log log R + log(c_*/lambda)+o(1). (8.3) + +This follows by taking logarithms of R*Psi(z)=lambda. It shows why a +near-critical analysis may depend on log(m/w), not simply log m. At a bounded +aspect ratio the large-z approximation itself fails. In fixed-p notation, +(8.2) would require A(p)~c_*sqrt(kappa(p)); a fixed subcritical amplitude cannot +be held constant all the way to pc. + +Neither (8.2) nor a site critical exponent is proved here. The literature +input most directly suggested by this question is [DM26], but it treats +BOND random-cluster configurations and its Theorem 1.1 gives two-sided +comparability of a TWO-POINT function, not the exact component amplitude. +The site model and the sewing/normalization remain genuine separate steps. +Do not label the heuristic `beta=1/2` or equation (8.3) as a demonstrated +near-critical square-site law. + +## 9. Execution and source record + +Commands from the repository root: + + python -m unittest discover -s tests -p 'test_cylinder_winding_intensity.py' -v + python scripts/cylinder_winding_intensity.py --output /tmp/intensity-new.json + +The report refuses to overwrite an existing path. Python stdlib suffices +for the transfer, Fraction stationary solves, controls and tests; report +regeneration also uses SymPy for the all-p rational functions. No existing +unmerged table or script is a dependency of this new calculation. + +Executed controls: 139,776 open-cylinder graph/configuration comparisons +on both adjacencies (2x4,3x4,4x4); 66,064 complementary torus pairs +(2x2,3x3,4x4); 18 Fraction/symbolic intensity checks and their complementary +variance checks; all-p symbolic density duality at the three widths. The +physical oracle is a separate raw-(dx,dy)-potential BFS, not the transfer's +one-dimensional gain DSU. Reduced and full transfer occupation/count +polynomials agree, without a false configurationwise lumping assertion. +Twelve local mathematical tests passed. Full Matching-One CI was not run. + +[MZ] S. Mertens and R. M. Ziff, *Percolation in Finite Matching Lattices*, +arXiv:1603.07289v2, section II: paragraphs on single/spiral counts and unique +cross-wrapping, before equations (6)--(11). Primary HTML read this round: +https://arxiv.org/html/1603.07289v2 . Its known classification supports section 4; +no prior-art absence is claimed for the count identity or ordinary transfer +methods. + +[DM26] L. D'Alimonte and I. Manolescu, *Near-critical Ornstein--Zernike theory +for the planar random-cluster model*, arXiv:2510.13648v3, 23 June 2026. +Primary PDF printed pp3--4 (model definition and Theorem 1.1) read AND rendered: +https://arxiv.org/pdf/2510.13648 . The configuration variables are edges and +the symbol in the theorem is asymp. Current abstract/version record checked: +https://arxiv.org/abs/2510.13648 . Only these targeted sections were read, +not the entire 44-page proof. Some HTML versions displayed mismatched dates +and incomplete formulas; the cited scope is tied to the v3 PDF. + +[CI/CIV leads] Campanino--Ioffe (2002), DOI 10.1214/AOP/1023481005; and +Campanino--Ioffe--Velenik, arXiv:math/0610100. Author/institution metadata or +abstracts checked in this round, not a full site-sewing theorem reading. +They are explicit starting points for #740, not imported site conclusions. + +The only external jobs opened are #740 (the actual site sewing/prefactor +mapping) and #741 (the six fixed-p intensity values and one cancelling +contrast per graph). Both return to #739. 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It is NOT the torus-rank automaton. +An independent graph-potential BFS supplies finite controls. + +Normal report: Python stdlib + SymPy (only for symbolic rational functions). +No simulation. Width 2--4 is exhaustively controlled. Larger widths are a +bounded capacity probe, not an asserted cheap asymptotic computation. +""" +from __future__ import annotations + +import argparse +from collections import Counter +from fractions import Fraction +import json +from math import comb, log, pi, sqrt +from pathlib import Path +from time import perf_counter +from typing import NamedTuple + + +class State(NamedTuple): + labels: tuple[int, ...] + gains: tuple[int, ...] + winding: tuple[int, ...] + + +class GainDSU: + """Potential(b)-potential(a)=gain on each oriented edge a->b.""" + def __init__(self, n: int) -> None: + self.parent = list(range(n)) + self.delta = [0] * n + self.winding = [False] * n + + def find(self, a: int) -> tuple[int, int]: + if self.parent[a] != a: + root, shift = self.find(self.parent[a]) + self.delta[a] += shift + self.parent[a] = root + return self.parent[a], self.delta[a] + + def join(self, a: int, b: int, gain: int) -> None: + ra, da = self.find(a) + rb, db = self.find(b) + if ra == rb: + self.winding[ra] |= db - da != gain + else: + self.parent[rb] = ra + self.delta[rb] = gain + da - db + self.winding[ra] |= self.winding[rb] + + +def empty_state(width: int) -> State: + if width < 2: + raise ValueError('width must be at least two; retain lifted parallel edges') + return State((-1,) * width, (0,) * width, ()) + + +def advance(state: State, mask: int, matching: bool = False) -> tuple[State, int]: + """Append one spatial row; reward every retired winding component once.""" + width = len(state.labels) + if not 0 <= mask < 1 << width: + raise ValueError('row mask outside width') + dsu = GainDSU(2 * width) + old = [i for i, k in enumerate(state.labels) if k >= 0] + new = [i for i in range(width) if mask >> i & 1] + representatives: dict[int, int] = {} + for i in old: + k = state.labels[i] + if k in representatives: + dsu.join(representatives[k], i, state.gains[i]) + else: + representatives[k] = i + for k, i in representatives.items(): + dsu.winding[dsu.find(i)[0]] = bool(state.winding[k]) + for i in new: + j = (i + 1) % width + if mask >> j & 1: + dsu.join(width + i, width + j, (i + 1) // width) + for dx in ((-1, 0, 1) if matching else (0,)): + j = (i + dx) % width + if state.labels[j] >= 0: + dsu.join(width + i, j, (i + dx) // width) + all_roots = {dsu.find(i)[0] for i in old + [width + i for i in new]} + kept_roots = {dsu.find(width + i)[0] for i in new} + reward = sum(dsu.winding[r] for r in all_roots - kept_roots) + labels = [-1] * width + gains = [0] * width + flags: list[int] = [] + root_map: dict[int, tuple[int, int]] = {} + for i in new: + root, potential = dsu.find(width + i) + if root not in root_map: + root_map[root] = (len(flags), potential) + flags.append(int(dsu.winding[root])) + k, origin = root_map[root] + labels[i] = k + gains[i] = 0 if dsu.winding[root] else potential - origin + return State(tuple(labels), tuple(gains), tuple(flags)), int(reward) + + +def build_transfer(width: int, matching: bool = False, + state_cap: int = 5000) -> tuple[list[State], list[list[tuple[int, int]]]]: + """BFS exhausts every row successor; fail explicitly at the chosen cap.""" + if width > 10: + raise ValueError('reference Python builder limited to width <=10') + states = [empty_state(width)] + index = {states[0]: 0} + transfer: list[list[tuple[int, int]]] = [] + for state in states: + row = [] + for mask in range(1 << width): + nxt, reward = advance(state, mask, matching) + if nxt not in index: + if len(states) >= state_cap: + raise RuntimeError('state cap reached; no incomplete closure returned') + index[nxt] = len(states) + states.append(nxt) + row.append((index[nxt], reward)) + transfer.append(row) + return states, transfer + + +def reward_lump(transfer: list[list[tuple[int, int]]]) -> tuple[list[list[tuple[int, int]]], list[int]]: + """Common all-p stochastic lumping retaining the joint next-class/reward law. + + This is not a claimed minimal positive/linear realization. Coefficients are + grouped by the new row's number of occupied sites, not numeric p samples. + """ + blocks = [0] * len(transfer) + while True: + classes: dict[tuple, int] = {} + refined = [] + for row in transfer: + counts = Counter((mask.bit_count(), reward, blocks[j]) + for mask, (j, reward) in enumerate(row)) + signature = tuple(sorted(counts.items())) + if signature not in classes: + classes[signature] = len(classes) + refined.append(classes[signature]) + if refined == blocks: + break + blocks = refined + reduced = [[(blocks[j], reward) for j, reward in transfer[blocks.index(k)]] + for k in range(max(blocks) + 1)] + return reduced, blocks + + +def row_weights(width: int, p: Fraction) -> list[Fraction]: + if not 0 <= p <= 1: + raise ValueError('p outside [0,1]') + return [p ** mask.bit_count() * (1 - p) ** (width - mask.bit_count()) + for mask in range(1 << width)] + + +def solve_fraction(a: list[list[Fraction]], b: list[Fraction]) -> list[Fraction]: + """Exact Gauss-Jordan solve with explicit singularity checking.""" + n = len(b) + if len(a) != n or any(len(row) != n for row in a): + raise ValueError('matrix must be square') + mat = [[Fraction(x) for x in row] + [Fraction(rhs)] for row, rhs in zip(a, b)] + for j in range(n): + pivot = next((i for i in range(j, n) if mat[i][j]), None) + if pivot is None: + raise ValueError('singular exact linear system') + mat[j], mat[pivot] = mat[pivot], mat[j] + v = mat[j][j] + mat[j] = [x / v for x in mat[j]] + for i in range(n): + if i != j and mat[i][j]: + v = mat[i][j] + mat[i] = [x - v * y for x, y in zip(mat[i], mat[j])] + return [row[-1] for row in mat] + + +def stationary_reward(transfer: list[list[tuple[int, int]]], p: Fraction) -> dict: + if not 0 < p < 1: + raise ValueError('stationary pressure calculation requires 0= 0 + return {'mean': mean, 'variance_rate': variance, 'stationary': stationary} + + +def stationary_certificate(transfer: list[list[tuple[int, int]]], p: Fraction, + candidate: list[Fraction]) -> dict: + """A rigorous forward-error bound, via the common empty-row reset. + + Candidate must be a probability vector. All arithmetic here is rational; + floating residuals supplied without outward rounding are not certificates. + """ + n = len(transfer) + if len(candidate) != n or min(candidate) < 0 or sum(candidate) != 1: + raise ValueError('candidate must be a normalized nonnegative vector') + if not 0 < p < 1: + raise ValueError('requires interior probability') + if any(row[0][0] != 0 for row in transfer): + raise ValueError('empty-row reset must lead to state zero') + width = (len(transfer[0])-1).bit_length() + weights = row_weights(width, p) + pushed = [Fraction(0)]*n + rewards = [Fraction(0)]*n + for i, row in enumerate(transfer): + for weight, (j, reward) in zip(weights, row): + pushed[j] += candidate[i]*weight + rewards[i] += weight*reward + residual = sum(abs(a-b) for a,b in zip(pushed, candidate)) + reset = (1-p)**width + estimate = sum(a*b for a,b in zip(candidate, rewards)) + error = max(rewards)*residual/reset + return {'estimate': estimate, 'absolute_error_bound': error, + 'lower': max(Fraction(0), estimate-error), 'upper': estimate+error, + 'stationarity_l1_residual': residual, 'empty_row_reset': reset} + + +def symbolic_intensity(transfer: list[list[tuple[int, int]]]) -> dict: + import sympy as sp + from sympy.polys.matrices import DomainMatrix + p = sp.Symbol('p') + width = (len(transfer[0]) - 1).bit_length() + n = len(transfer) + k = sp.zeros(n) + g = sp.zeros(n, 1) + for i, row in enumerate(transfer): + for mask, (j, reward) in enumerate(row): + probability = p ** mask.bit_count() * (1-p) ** (width-mask.bit_count()) + k[i, j] += probability + g[i] += probability * reward + eq = (k.T - sp.eye(n)).applyfunc(sp.expand) + eq[-1, :] = sp.ones(1, n) + rhs = sp.zeros(n, 1) + rhs[-1] = 1 + dm = DomainMatrix.from_Matrix(eq).to_field() + solution = dm.inv().matmul(DomainMatrix.from_Matrix(rhs).convert_to(dm.domain)).to_Matrix() + expression = sp.factor((solution.T * g)[0]) + num, den = sp.fraction(expression) + return {'expression': str(expression), + 'numerator_descending': [int(x) for x in sp.Poly(num, p).all_coeffs()], + 'denominator_descending': [int(x) for x in sp.Poly(den, p).all_coeffs()], + 'low_p_through_8': str(sp.series(expression, p, 0, 9))} + + +def evaluate_polynomial(coefficients: list[int], p: Fraction) -> Fraction: + out = Fraction(0) + for x in coefficients: + out = out * p + x + return out + + +def graph_components(mask: int, width: int, height: int, + matching: bool = False, torus: bool = False) -> tuple[int, int]: + """Independent BFS, raw (dx,dy) potentials. Return rank and essential count. + + No production DSU, frontier state, reward, or reduction is used here. + In a free cylinder only horizontal winding can occur. Parallel lifted + edges on width two are deliberately retained. + """ + steps = [(1, 0), (-1, 0), (0, 1), (0, -1)] + if matching: + steps += [(1, 1), (1, -1), (-1, 1), (-1, -1)] + potential: dict[int, tuple[int, int]] = {} + generators: list[tuple[int, int]] = [] + count = 0 + for start in range(width * height): + if not (mask >> start & 1) or start in potential: + continue + potential[start] = (0, 0) + todo = [start] + essential = False + while todo: + v = todo.pop() + x, y = v % width, v // width + vx, vy = potential[v] + for dx, dy in steps: + ny = y + dy + if not torus and not 0 <= ny < height: + continue + j = (ny % height) * width + (x + dx) % width + if not (mask >> j & 1): + continue + proposed = (vx + dx, vy + dy) + if j not in potential: + potential[j] = proposed + todo.append(j) + else: + gx, gy = proposed[0] - potential[j][0], proposed[1] - potential[j][1] + if gx or gy: + essential = True + generators.append((gx, gy)) + count += essential + if not generators: + return 0, int(count) + x, y = generators[0] + rank = 2 if any(x*b-y*a for a, b in generators) else 1 + return rank, int(count) + + +def reward_count(mask: int, width: int, height: int, + transfer: list[list[tuple[int, int]]]) -> int: + state = 0 + count = 0 + for y in range(height): + row_mask = mask >> (width*y) & ((1 << width)-1) + state, reward = transfer[state][row_mask] + count += reward + _, final_reward = transfer[state][0] # a deliberate EMPTY closing row + return count + final_reward + + +def reward_histogram(width: int, height: int, transfer: list[list[tuple[int, int]]]) -> Counter: + """Count law by total occupancy, not a false configurationwise lumping claim.""" + current = Counter({(0, 0, 0): 1}) + for _ in range(height): + nxt = Counter() + for (state, occupied, count), coefficient in current.items(): + for mask, (j, reward) in enumerate(transfer[state]): + nxt[j, occupied+mask.bit_count(), count+reward] += coefficient + current = nxt + histogram = Counter() + for (state, occupied, count), coefficient in current.items(): + reward = transfer[state][0][1] + histogram[occupied, count+reward] += coefficient + return histogram + + +def central_trinomial(n: int) -> int: + return sum(comb(n, k) * comb(n-k, k) for k in range(n//2+1)) + + +def renewal_controls() -> dict: + """Exactly solvable directed renewal LOOP; not a site-cluster theorem.""" + controls = [] + for w in (8, 16, 32, 64, 128): + c = [central_trinomial(k*w) for k in (1, 2, 3)] + ratio = Fraction(c[0]*c[2], c[1]**2) # 3 powers and exponential fugacity cancel + beta_eff = log(float(ratio)) / log(4/3) + amplitude_ratio = float(Fraction(c[0], 3**w)) * sqrt(w) / sqrt(3/(4*pi)) + controls.append({'width': w, 'exact_tripling_ratio': str(ratio), + 'effective_beta': beta_eff, + 'normalized_amplitude': amplitude_ratio}) + return {'model': 'X=1, Y=-1,0,1 with equal probabilities; distinct from site percolation', + 'predicted_beta': '1/2', 'transverse_diffusion': '2/3', + 'predicted_amplitude_squared_times_pi': '3/4', 'controls': controls} + + +def centering_counterexample() -> list[dict]: + """C1/semiconcave mass is insufficient for an o(1/w) linearized centre. + + Exact model on h>0: kappa(a+h)=d-h-h/log(e/h). The intensity is + w^{-1} exp(-w*kappa) and log m=dw. At intensity level one, solve + h+h/log(e/h)=log(w)/w. Then w*(h-log(w)/w) -> -1, not zero. + Numbers below are Decimal diagnostics; the limit has a direct proof. + """ + from decimal import Decimal, localcontext + result=[] + with localcontext() as ctx: + ctx.prec=80 + for exponent in (4,8,16,32): + w=Decimal(10)**exponent + linear=w.ln()/w + lo,hi=Decimal(0),linear + for _ in range(400): + h=(lo+hi)/2 + if h+h/(1-h.ln()) dict: + import sympy as sp + start = perf_counter() + tables = {} + generators = [] + physical_checks = 0 + symbolic_checks = 0 + for width in (2, 3, 4): + for matching in (False, True): + states, transfer = build_transfer(width, matching) + reduced, blocks = reward_lump(transfer) + tables[width, matching] = transfer + expression = symbolic_intensity(reduced) + point_controls = [] + for p in (Fraction(1, 4), Fraction(1, 2), Fraction(3, 4)): + exact = stationary_reward(reduced, p) + rational = evaluate_polynomial(expression['numerator_descending'], p) / evaluate_polynomial(expression['denominator_descending'], p) + assert exact['mean'] == rational + point_controls.append({'p': str(p), 'intensity': str(exact['mean']), + 'variance_per_row': str(exact['variance_rate']), + 'fano_rate': str(exact['variance_rate']/exact['mean'])}) + truncated = [Fraction(int(x*(1 << 20)), 1 << 20) for x in exact['stationary']] + truncated[0] += 1-sum(truncated) + certificate = stationary_certificate(reduced, p, truncated) + assert certificate['lower'] <= exact['mean'] <= certificate['upper'] + point_controls[-1]['rounded_stationary_certificate'] = {k: str(v) for k,v in certificate.items()} + symbolic_checks += 1 + height = 4 + histogram = Counter() + for mask in range(1 << (width*height)): + _, observed = graph_components(mask, width, height, matching, torus=False) + assert reward_count(mask, width, height, transfer) == observed + histogram[mask.bit_count(), observed] += 1 + physical_checks += 1 + assert reward_histogram(width, height, reduced) == histogram + generators.append({'width': width, 'graph': 'matching' if matching else 'NN', + 'frontier_states': len(states), 'reward_lumps': len(reduced), + 'max_absolute_gain': max(abs(v) for state in states for v in state.gains), + 'transition_entries': len(states)*(1 << width), + 'symbolic_intensity': expression, + 'point_controls': point_controls, + 'states': [list(state) for state in states], + 'transition_table': transfer, 'reward_lump_map': blocks, + 'reduced_table': reduced}) + p = sp.Symbol('p') + for w in (2, 3, 4): + g4 = next(g for g in generators if g['width']==w and g['graph']=='NN') + g8 = next(g for g in generators if g['width']==w and g['graph']=='matching') + e4 = sp.sympify(g4['symbolic_intensity']['expression']) + e8 = sp.sympify(g8['symbolic_intensity']['expression']) + assert sp.cancel(e4-e8.subs(p,1-p)) == 0 + for point in g4['point_controls']: + paired = next(v for v in g8['point_controls'] if Fraction(v['p'])==1-Fraction(point['p'])) + assert paired['intensity'] == point['intensity'] + assert paired['variance_per_row'] == point['variance_per_row'] + duality_checks = 0 + joint_support = set() + for width, height in ((2,2), (3,3), (4,4)): + full = (1 << (width*height)) - 1 + for mask in range(full+1): + rank4, n4 = graph_components(mask, width, height, False, torus=True) + rank8, n8 = graph_components(full^mask, width, height, True, torus=True) + assert rank4+rank8 == 2 + assert n4-n8 == rank4-1 + assert (n4,n8) in ((1,0),(0,1)) or n4==n8>=1 + joint_support.add((n4,n8)) + duality_checks += 1 + return {'schema': 'matching-one.cylinder-winding-intensity.v1', + 'scope': 'two graphs; exact fixed-width component rewards, not a large-width prefactor proof', + 'models': generators, + 'checks': {'open_cylinder_graph_configurations': physical_checks, + 'torus_complement_pairs': duality_checks, + 'symbolic_point_controls': symbolic_checks, + 'all_p_duality_identities': 3, + 'joint_torus_support_observed': sorted(joint_support)}, + 'renewal': renewal_controls(), + 'centering_counterexample': centering_counterexample(), + 'elapsed_seconds': perf_counter()-start} + + +def main() -> None: + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output', type=Path) + parser.add_argument('--probe-width', type=int) + parser.add_argument('--state-cap', type=int, default=5000) + args = parser.parse_args() + if args.probe_width is not None: + start = perf_counter() + states, transfer = build_transfer(args.probe_width, state_cap=args.state_cap) + print(json.dumps({'width': args.probe_width, 'states': len(states), + 'transition_entries': sum(map(len,transfer)), + 'elapsed_seconds': perf_counter()-start}, indent=2)) + return + if args.output is None: + parser.error('--output is required unless --probe-width is used') + if args.output.exists(): + raise FileExistsError(f'refusing to overwrite {args.output}') + result = run_report() + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(result, indent=2)+'\n', encoding='utf-8') + print(json.dumps(result['checks'], indent=2)) + print(f'elapsed_seconds={result["elapsed_seconds"]:.3f}') + + +if __name__ == '__main__': + main() diff --git a/tests/test_cylinder_winding_intensity.py b/tests/test_cylinder_winding_intensity.py new file mode 100644 index 00000000..af67fa70 --- /dev/null +++ b/tests/test_cylinder_winding_intensity.py @@ -0,0 +1,109 @@ +import sys +from pathlib import Path +from fractions import Fraction as F +from collections import Counter +import unittest +sys.path.insert(0, str(Path(__file__).resolve().parents[1]/'scripts')) +import cylinder_winding_intensity as c + +class CylinderIntensityTests(unittest.TestCase): + def test_complete_closure_counts(self): + for w,n,k in ((2,6,3),(3,14,4),(4,38,7)): + for matching in (False,True): + states,table=c.build_transfer(w,matching) + red,blocks=c.reward_lump(table) + self.assertEqual((len(states),len(red)),(n,k)) + self.assertTrue(all(len(row)==1< Date: Sun, 13 Sep 2026 14:00:50 +0800 Subject: [PATCH 07/17] Compute the winding-prefactor contrast at widths 4, 8 and 12 (issue #741) Numerical counterpart to the prefactor ticket. One result JSON, the engine that made it reachable, and a regression lock. Result. With nu_w the once-per-COMPLETE-component cylinder density, R_4 = nu_4*nu_12/nu_8^2, beta_eff(4) = log(R_4)/log(4/3) at the two fixed subcritical inputs the ticket names: NN site, p = 1/4 (3p<1): beta_eff = 0.792584457 +- 1.9e-9 matching NN+NNN, p = 1/8 (7p<1): beta_eff = 0.508452577 +- 5.1e-10 The 1/2 sewing hypothesis is numerically close on the matching graph and not on NN, where the identical construction returns 0.79. An identical construction cannot have two different true exponents, so at least one is not asymptotic. Two checks make that concrete rather than rhetorical, and they are recorded in the result JSON: 1. The same construction on the (2,4,8) window returns beta_eff = -7.137 (NN) and -7.070 (matching). The assumed form does not describe those widths at all, so the (4,8,12) values are finite-window effective exponents. 2. The adjacent-window effective kappa is still moving at w = 12: NN 1.151104 then 1.094101, matching 1.109358 then 1.072790. Neither value should be quoted as beta, and this does not refute the sewing hypothesis. Deciding it needs the A, beta derivation of #740, not a fourth width. Engine. `scripts/cylinder_winding_intensity.py` caps the builder at width <= 10 and solves densely in Fraction, so `scripts/cylinder_winding_intensity_fast.cpp` ports advance()/empty_state()/reward_lump() allocation-free and caches the transition table after BFS. Validation is exact: all 18 published controls of results/geometric-consistency/cylinder-winding-intensity.json reproduce as equal rationals, and the frontier-state and reward-lump counts match (6/3, 14/4, 38/7), as do the w=5..8 capacity counts 102/282/786/2214. One defect found by that validation and worth carrying: the port first disagreed on the MATCHING graph alone. Cause was integer division -- C++ '/' truncates, Python '//' floors, and the only affected call is the dx=-1 diagonal step at i=0, where (i+dx)//w = -1 but (i+dx)/w = 0. NN never exercises it. Cost, measured: w=12 gives 147 578 frontier states and 2 105 reward lumps, 604 479 488 transitions, 504 s (NN) and 561 s (matching) wall, 2.4 GB peak RSS, 2.25 GB of which is the cached table. w=8 is 0.5 s here against 6.7 s in Python; the w<=10 Python cap is what forced the port. Error control. w=4 and w=8 are exact rationals. w=12 is a float64 solve followed by one correction step driven by the EXACT rational stationary residual, then certified exactly: |pi_hat.g - nu| <= ||g||_inf * ||pi_hat K - pi_hat||_1 / delta with delta=(1-p)^w. That step takes the bound from 1.6e-14 to 3.1e-16 (NN) and 7.5e-17 (matching), i.e. log nu to 5.5e-10 and 1.5e-10, inside the 1e-8 the ticket asks for. tests/test_winding_prefactor_contrast.py locks the solver against the committed control grid without needing the C++ builder. No exponent determined, no asymptote claimed, no new p_c, no Monte Carlo, no GPU. Full Matching-One repository CI has not been run for this commit. --- .../winding-prefactor-contrast-20260913.md | 130 ++++++++ .../winding-prefactor-contrast.json | 174 +++++++++++ scripts/cylinder_winding_intensity_fast.cpp | 292 ++++++++++++++++++ scripts/winding_prefactor_contrast.py | 135 ++++++++ tests/test_winding_prefactor_contrast.py | 95 ++++++ 5 files changed, 826 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/winding-prefactor-contrast-20260913.md create mode 100644 results/geometric-consistency/winding-prefactor-contrast.json create mode 100644 scripts/cylinder_winding_intensity_fast.cpp create mode 100644 scripts/winding_prefactor_contrast.py create mode 100644 tests/test_winding_prefactor_contrast.py diff --git a/docs/manuscripts/geometric-balance/winding-prefactor-contrast-20260913.md b/docs/manuscripts/geometric-balance/winding-prefactor-contrast-20260913.md new file mode 100644 index 00000000..0e4d8a2a --- /dev/null +++ b/docs/manuscripts/geometric-balance/winding-prefactor-contrast-20260913.md @@ -0,0 +1,130 @@ +# The winding-prefactor contrast at widths 4, 8 and 12 + +2026-09-13. Computed for issue #741, returning to the #739 manuscript. This is a +numerical counterpart to #740, not a new mechanism programme and not a width scan. + +## What was computed + +The once-per-COMPLETE-component cylinder density + + nu_w^G(p) = expected number of horizontally winding components retired per vertical row + +for two fixed inputs, both rigorously subcritical by elementary path bounds: + +| graph | p | why subcritical | +|---|---|---| +| NN square site | 1/4 | `3p < 1` | +| matching NN+NNN site | 1/8 | `7p < 1` | + +at `w = 4, 8, 12`. The contrast is + + R_4 = nu_4 * nu_12 / nu_8^2, beta_eff(4) = log(R_4) / log(4/3). + +If `nu_w = A w^-beta exp(-kappa w)(1+o(1))` then the exponential mass and the +amplitude cancel in `R_4` and `R_4 -> (4/3)^beta`. This is therefore a direct +diagnostic of the `w^{-1/2}` sewing hypothesis of #740 — one number, not a fit of +three unknown parameters. + +## Result + +| graph | p | log nu_4 | log nu_8 | log nu_12 | beta_eff | deviation from 1/2 | +|---|---|---|---|---|---|---| +| NN | 1/4 | −5.419513505388 | −10.023930900251 | −14.400335955877 | **0.792584457** ± 1.9e−9 | +0.292584 | +| matching NN+NNN | 1/8 | −5.742669143987 | −10.180101524507 | −14.471261214034 | **0.508452577** ± 5.1e−10 | +0.008453 | + +`w = 4` and `w = 8` are exact rationals with a zero certificate bound; `w = 12` +carries the certificate below, all below 6e−10 on `log nu`. + +### The three-width agreement is not an asymptotic proof, and here it is not even self-consistent + +Two checks make that concrete rather than rhetorical. + +1. **A second, equally admissible window disagrees wildly.** The same construction on + `(2,4,8)` returns `beta_eff = −7.137` (NN) and `−7.070` (matching). The assumed + form does not describe those widths at all, which says the `(4,8,12)` values are + finite-window effective exponents rather than a settled amplitude. +2. **The effective kappa is still moving.** Adjacent differences give + + NN (4,8) 1.151104 (8,12) 1.094101 matching (4,8) 1.109358 (8,12) 1.072790 + + both still decreasing in magnitude at `w = 12`, so the amplitude has not settled either. + +An identical construction cannot have two different true exponents, so at least one +of the two `beta_eff` values above is not asymptotic. The honest reading is: + +- the `1/2` sewing hypothesis is **numerically close** on the matching graph at `p = 1/8`; +- it is **not** close on NN at `p = 1/4`, where the same formula returns `0.79`; +- and the `(2,4,8)` window shows the whole two-parameter description is not yet valid, + so neither number should be quoted as `beta`. + +Reported as required even though inconsistent with `1/2`. This neither confirms nor +refutes the sewing hypothesis; deciding that needs the `A, beta` derivation of #740, +not a fourth width. + +## Engine, validation and cost + +The supplied `scripts/cylinder_winding_intensity.py` caps the builder at width ≤ 10 and +solves densely in `Fraction`. `scripts/cylinder_winding_intensity_fast.cpp` is an +allocation-free C++ port of the same `advance()` / `empty_state()` / `reward_lump()` +semantics, with the transition table cached after BFS so that the reward-lumping +refinement does not recompute 6e8 successors per iteration. + +**Validation is exact, not indicative.** All 18 published controls of +`results/geometric-consistency/cylinder-winding-intensity.json` — both graphs, widths +2/3/4, `p = 1/4, 1/2, 3/4` — reproduce as **equal rationals**, and the frontier-state +and reward-lump counts match (`6/3`, `14/4`, `38/7`). The capacity-probe counts +`102/282/786/2214` at `w = 5..8` and `90` reward lumps at `w = 8` also reproduce. + +One caveat worth recording, found by this validation: the C++ port first disagreed on +the matching graph alone. The cause was integer division — C++ `/` truncates toward +zero while Python `//` floors, and the only affected call is the `dx = −1` diagonal step +at `i = 0`, where `(i+dx)//w = −1` but `(i+dx)/w = 0`. NN was unaffected, which is +exactly why a matching-side control matters. Fixed by an explicit floor division. + +Cost, measured rather than extrapolated, single process on a 16 vCPU aarch64 container: + +| | w = 4 | w = 8 | w = 12 | +|---|---:|---:|---:| +| frontier states | 38 | 2 214 | 147 578 | +| reward lumps | 7 | 90 | 2 105 | +| transitions | 152 | 566 784 | 604 479 488 | +| wall time | <1 s | 0.5 s | 504 s (NN) / 561 s (matching) | +| peak RSS | — | — | 2.4 GB (2.25 GB cached table) | + +`w = 12` broke down as BFS 234 s + table cache 238 s + refinement ≈ 30 s. The same +counts are reachable on a laptop at `w = 8` (0.5 s) but not at `w = 12`. + +## Error control + +For a normalised nonnegative `pi_hat` and the uniform empty-row reset +`delta = (1-p)^w` (the empty row always maps to the empty state, so the chain is +uniformly ergodic with gap at least `delta`), + + |pi_hat . g - nu| <= ||g||_inf * ||pi_hat K - pi_hat||_1 / delta. + +`w = 4, 8` are solved in exact rational arithmetic. `w = 12` is solved in float64 +(2 105 states), then corrected once using the **exact rational** stationary residual and +certified exactly; that step is what brings the bound from 1.6e−14 down to 3e−16 (NN) +and 7.5e−17 (matching), i.e. `log nu` to 5.5e−10 and 1.5e−10. + +## What this does not establish + +No exponent is determined. No asymptote is claimed. The `w^-beta` form is not verified; +the `(2,4,8)` window shows it fails at small width. `kappa_G` is not measured as a +limit. No new `p_c`, no Monte Carlo, no GPU, no continuum identification. The residual +certificate bounds arithmetic only; it says nothing about the model assumptions or the +imported inputs of #739. + +## Reproducing + +```sh +g++ -O3 -std=c++17 -o winding_build scripts/cylinder_winding_intensity_fast.cpp +for w in 4 8 12; do ./winding_build $w 0 out_nn_$w.json; done # NN, 0 = site NN +for w in 4 8 12; do ./winding_build $w 1 out_m_$w.json; done # matching +python3 scripts/winding_prefactor_contrast.py \ + '[["out_nn_4.json","1/4",true],["out_nn_8.json","1/4",true],["out_nn_12.json","1/4",false],'\ +'"["out_m_4.json","1/8",true],["out_m_8.json","1/8",true],["out_m_12.json","1/8",false]]' \ + results/geometric-consistency/winding-prefactor-contrast.json +``` + +The `w = 12` builds need ~2.4 GB and ~9 minutes each; the `w ≤ 8` builds are seconds. diff --git a/results/geometric-consistency/winding-prefactor-contrast.json b/results/geometric-consistency/winding-prefactor-contrast.json new file mode 100644 index 00000000..920c56b7 --- /dev/null +++ b/results/geometric-consistency/winding-prefactor-contrast.json @@ -0,0 +1,174 @@ +{ + "schema": "matching-one.winding-prefactor-contrast.v1", + "issue": 741, + "pr": 739, + "date": "2026-09-13", + "definition": "nu_w = expected number of complete horizontally winding components retired per vertical row, pi.g on the reward-preserving lumped chain", + "graph_inputs": [ + { + "graph": "NN", + "p": "1/4", + "why_subcritical": "3p<1" + }, + { + "graph": "matching NN+NNN", + "p": "1/8", + "why_subcritical": "7p<1" + } + ], + "engine": { + "python_reference": "scripts/cylinder_winding_intensity.py", + "fast_builder": "scripts/cylinder_winding_intensity_fast.cpp", + "note": "allocation-free C++ port of advance()/empty_state()/reward_lump(); validated against every published control of the Python reference" + }, + "validation": { + "points": 18, + "source": "results/geometric-consistency/cylinder-winding-intensity.json", + "agreement": "exact rational equality at every (graph,width,p) control, plus identical frontier-state and reward-lump counts" + }, + "runs": [ + { + "graph": "NN", + "p": "1/4", + "widths": [ + { + "w": 4, + "frontier_states": 38, + "reward_lumps": 7, + "log_nu": -5.419513505387646, + "nu": "52135149017187/11770514026725376", + "certificate_bound": 0.0, + "log_nu_abs_error_bound": 0.0, + "mode": "exact-rational solve" + }, + { + "w": 8, + "frontier_states": 2214, + "reward_lumps": 90, + "log_nu": -10.023930900250674, + "nu": 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3.0907075957690434e-16, + "log_nu_abs_error_bound": 5.546825221358631e-10, + "mode": "float64 solve + exact-rational residual correction + exact certification" + } + ] + }, + { + "graph": "matching NN+NNN", + "p": "1/8", + "widths": [ + { + "w": 4, + "frontier_states": 38, + "reward_lumps": 7, + "log_nu": -5.742669143987384, + "nu": "578542977420046503529/180445122308089296453632", + "certificate_bound": 0.0, + "log_nu_abs_error_bound": 0.0, + "mode": "exact-rational solve" + }, + { + "w": 8, + "frontier_states": 2214, + "reward_lumps": 90, + "log_nu": -10.180101524506828, + "nu": "4304066353276814600044997473461749946733998654996906665848710094050668526018463750118116035133786833693570766188431654980451251563150021628926696864281471364809534986496434452171916677495079889807512646873142689361000629991021992202254034107483405752782584490865423182129099195425092500784726730027782992682182888628256311224842606882093651369807185556594705994887406409039631339042921350991283756541994077559051452642113341699201143846379431566869183683434604000069810038992473728867102962226989648218217/113511764657387757743385852913961444340846792248681594418514915909829354582886141640860978491305768062425209800158664229180983934165690093259309006582490727717774659821138193397179076192301603758529671081392965053326462178884774141814282691798376964456718728193496138901518192555819255671364890966589998223829885349719255811426060081926113431586436483919311190561771297363987867516713176200921755746267736239849245504325297603519015385553972106891313881673522269765109458789284250226055406632424069191379714048", + "certificate_bound": 0.0, + "log_nu_abs_error_bound": 0.0, + "mode": "exact-rational solve" + }, + { + "w": 12, + "frontier_states": 147578, + "reward_lumps": 2105, + "log_nu": -14.471261214034028, + "nu": "2825989450032982531722048546838021/5444517870735033276005206910959493898816", + "certificate_bound": 7.541392155565239e-17, + "log_nu_abs_error_bound": 1.4529177860822529e-10, + "mode": "float64 solve + exact-rational residual correction + exact certification" + } + ] + } + ], + "contrast": { + "NN": { + "log_nu_by_width": { + "2": -2.868379786881457, + "4": -5.419513505387646, + "8": -10.023930900250674, + "12": -14.400335955876589 + }, + "effective_kappa_4_8": 1.151104348715757, + "effective_kappa_8_12": 1.0941012639064787, + "R_window_4_8_12": 1.2561008246626348, + "log_R": 0.2280123392371145, + "log_R_error_bound": 5.546825221358631e-10, + "beta_eff_4_8_12": 0.7925844571886983, + "beta_eff_error_bound": 1.9281094487694742e-09, + "R_window_2_4_8": 0.128312873113551, + "beta_eff_2_4_8": -7.137336222788077, + "deviation_from_one_half": 0.2925844571886983 + }, + "matching": { + "log_nu_by_width": { + "2": -3.3390984215201938, + "4": -5.742669143987384, + "8": -10.180101524506828, + "12": -14.471261214034028 + }, + "effective_kappa_4_8": 1.109358095129861, + "effective_kappa_8_12": 1.0727899223818, + "R_window_4_8_12": 1.157511788056221, + "log_R": 0.1462726909922445, + "log_R_error_bound": 1.4529177860822529e-10, + "beta_eff_4_8_12": 0.5084525766434808, + "beta_eff_error_bound": 5.050428668354995e-10, + "R_window_2_4_8": 0.13082932628049532, + "beta_eff_2_4_8": -7.069824131613746, + "deviation_from_one_half": 0.008452576643480758 + } + }, + "resource_report": { + "w12_NN": { + "frontier_states": 147578, + "reward_lumps": 2105, + "transitions": 604479488, + "bfs_seconds": 234.2, + "table_cache_seconds": 237.8, + "table_gigabytes": 2.25, + "total_seconds": 504.1, + "peak_rss_gigabytes": 2.4, + "exit": "0" + }, + "w12_matching": { + "frontier_states": 147578, + "reward_lumps": 2105, + "transitions": 604479488, + "bfs_seconds": 261.4, + "table_cache_seconds": 263.3, + "total_seconds": 560.9, + "peak_rss_gigabytes": 2.4, + "exit": "0" + }, + "host": "Huawei Cloud EulerOS 2.0 aarch64 container, 16 vCPU, 30 GiB visible, g++ 10.3.1 -O2/-O3, single-process single-thread", + "naive_python_estimate": "the Python builder is capped at width<=10 and its builder took 6.7 s at w=8 against 0.5 s here; a w=12 build in pure Python was not attempted" + }, + "interpretation": [ + "3-width window (4,8,12) gives beta_eff = 0.792584457 +- 1.9e-9 for NN at p=1/4 and 0.508452577 +- 5.1e-10 for matching at p=1/8.", + "The 1/2 sewing hypothesis is therefore numerically close on the matching graph at p=1/8 (deviation +0.00845) but not on NN at p=1/4 (deviation +0.29258). An identical construction cannot have two different true exponents, so at least one of these is not asymptotic.", + "The (2,4,8) window returns beta_eff = -7.137 (NN) and -7.070 (matching), i.e. the assumed form nu_w = A w^-beta exp(-kappa w) does not hold at those widths at all. This is direct evidence that the (4,8,12) values are finite-window effective exponents.", + "The adjacent-window effective kappa is still decreasing in magnitude over 4->8->12 for both graphs (-1.2756,-1.1511,-1.0941 for NN; -1.2018,-1.1094,-1.0728 for matching), so the amplitude has not settled either.", + "Three widths cannot establish an asymptote. The contrast is reported as an effective finite-width diagnostic, not as a determination of beta, and not as a refutation of the sewing hypothesis.", + "Exact rational values are available at w=4 and w=8 (certificate bound zero); the w=12 values carry the certificate bounds listed above, all below 6e-10 on log nu." + ] +} \ No newline at end of file diff --git a/scripts/cylinder_winding_intensity_fast.cpp b/scripts/cylinder_winding_intensity_fast.cpp new file mode 100644 index 00000000..1fa02227 --- /dev/null +++ b/scripts/cylinder_winding_intensity_fast.cpp @@ -0,0 +1,292 @@ +// Faithful, allocation-free C++ port of the supplied one-frontier winding-component +// transfer (scripts/cylinder_winding_intensity.py). advance()/empty_state()/ +// reward_lump() semantics are reproduced exactly; only the width limit and the +// dense Fraction linear algebra of the Python reference are changed. +// +// Output: the reward-preserving lumped chain aggregated by row popcount, i.e. all +// that is needed to evaluate nu_w(p) as an exact rational downstream. + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +using namespace std; + +static inline int floordiv(int a, int b) { // Python // semantics; C++ / truncates + return (a >= 0) ? (a / b) : -(((-a) + b - 1) / b); +} + +static const int MAXW = 16; +static const int MAXN = 4 * MAXW; // 2*width DSU nodes + +struct DSU { + int parent[MAXN], delta[MAXN]; + uint8_t wind[MAXN]; + void init(int n) { + for (int i = 0; i < n; ++i) { parent[i] = i; delta[i] = 0; wind[i] = 0; } + } + inline int find(int a, int &pot) { + // iterative path compression + int root = a, acc = 0; + while (parent[root] != root) { acc += delta[root]; root = parent[root]; } + // second pass: compress, accumulating potential relative to root + int cur = a, run = 0; + while (parent[cur] != cur) { + int nxt = parent[cur]; + int d = delta[cur]; + parent[cur] = root; + delta[cur] = acc - run; + run += d; + cur = nxt; + } + pot = (a == root) ? 0 : delta[a]; + return root; + } + inline void join(int a, int b, int gain) { + int da, db; + int ra = find(a, da), rb = find(b, db); + if (ra == rb) { wind[ra] |= (uint8_t)(db - da != gain); } + else { parent[rb] = ra; delta[rb] = gain + da - db; wind[ra] |= wind[rb]; } + } +}; + +struct State { + int16_t labels[MAXW]; // -1 when unoccupied + int16_t gains[MAXW]; + uint8_t flags[MAXW]; + int nflags; +}; + +static inline void pack_state(const State &s, int W, string &out) { + out.resize((size_t)W * 2 + s.nflags); + size_t p = 0; + for (int i = 0; i < W; ++i) out[p++] = (char)(s.labels[i] & 0xFF); + for (int i = 0; i < W; ++i) out[p++] = (char)(s.gains[i] & 0xFF); + for (int i = 0; i < s.nflags; ++i) out[p++] = (char)s.flags[i]; +} + +struct KeyHash { + inline size_t operator()(const string &k) const { + size_t h = 1469598103934665603ULL; + const unsigned char *p = (const unsigned char *)k.data(); + size_t n = k.size(); + for (size_t i = 0; i < n; ++i) { h ^= p[i]; h *= 1099511628211ULL; } + return h; + } +}; + +// Returns reward; writes the successor into `nx`. +static inline int advance(const State &st, int mask, int W, bool matching, State &nx) { + static DSU dsu; + static int oldv[MAXW], newv[MAXW]; + static int rep_of[MAXW]; // label -> first site + static int root_of[MAXW]; // label -> root + static int allroots[MAXN], keptroots[MAXN]; + int nold = 0, nnew = 0; + for (int i = 0; i < W; ++i) if (st.labels[i] >= 0) oldv[nold++] = i; + for (int i = 0; i < W; ++i) if ((mask >> i) & 1) newv[nnew++] = i; + + dsu.init(2 * W); + for (int t = 0; t < W; ++t) rep_of[t] = -1; + for (int t = 0; t < nold; ++t) { + int i = oldv[t], k = st.labels[i]; + if (rep_of[k] < 0) rep_of[k] = i; + else dsu.join(rep_of[k], i, st.gains[i]); + } + for (int k = 0; k < W; ++k) if (rep_of[k] >= 0) { + int pot; int r = dsu.find(rep_of[k], pot); + dsu.wind[r] = (uint8_t)(st.flags[k] != 0); + root_of[k] = r; + } + (void)root_of; + for (int t = 0; t < nnew; ++t) { + int i = newv[t]; + int j = i + 1; if (j == W) j = 0; + if ((mask >> j) & 1) dsu.join(W + i, W + j, (i + 1) / W); + int lo = matching ? -1 : 0, hi = matching ? 1 : 0; + for (int dx = lo; dx <= hi; ++dx) { + int jj = (i + dx) % W; if (jj < 0) jj += W; + if (st.labels[jj] >= 0) dsu.join(W + i, jj, floordiv(i + dx, W)); + } + } + int na = 0, nk = 0; + for (int t = 0; t < nold; ++t) { int p2; allroots[na++] = dsu.find(oldv[t], p2); } + for (int t = 0; t < nnew; ++t) { int p2; keptroots[nk++] = dsu.find(W + newv[t], p2); } + sort(allroots, allroots + na); na = (int)(unique(allroots, allroots + na) - allroots); + sort(keptroots, keptroots + nk); nk = (int)(unique(keptroots, keptroots + nk) - keptroots); + int reward = 0; + for (int a = 0, b = 0; a < na; ++a) { + while (b < nk && keptroots[b] < allroots[a]) ++b; + if (b >= nk || keptroots[b] != allroots[a]) reward += dsu.wind[allroots[a]] ? 1 : 0; + } + // build successor + nx.nflags = 0; + for (int i = 0; i < W; ++i) { nx.labels[i] = -1; nx.gains[i] = 0; } + static int tag[MAXN], origin[MAXN]; + for (int i = 0; i < 2 * W; ++i) tag[i] = -1; + for (int t = 0; t < nnew; ++t) { + int i = newv[t]; + int pot; int r = dsu.find(W + i, pot); + if (tag[r] < 0) { tag[r] = nx.nflags; origin[r] = pot; nx.flags[nx.nflags++] = dsu.wind[r] ? 1 : 0; } + nx.labels[i] = (int16_t)tag[r]; + nx.gains[i] = (int16_t)(dsu.wind[r] ? 0 : pot - origin[r]); + } + return reward; +} + +int main(int argc, char **argv) { + if (argc < 3) { fprintf(stderr, "usage: %s WIDTH 0|1(matching) [out.json]\n", argv[0]); return 2; } + int W = atoi(argv[1]); + bool matching = atoi(argv[2]) != 0; + const char *outpath = (argc > 3) ? argv[3] : nullptr; + if (W < 2 || W > MAXW) { fprintf(stderr, "width out of range\n"); return 2; } + auto t0 = chrono::steady_clock::now(); + + State empty; for (int i = 0; i < W; ++i) { empty.labels[i] = -1; empty.gains[i] = 0; } + empty.nflags = 0; + + vector states; states.push_back(empty); + unordered_map index; + index.reserve(1 << 20); + { string k; pack_state(empty, W, k); index[k] = 0; } + + string kb; + State nxt, cur; + size_t cursor = 0; + while (cursor < states.size()) { + cur = states[cursor]; + for (int mask = 0; mask < (1 << W); ++mask) { + advance(cur, mask, W, matching, nxt); + pack_state(nxt, W, kb); + auto it = index.find(kb); + if (it == index.end()) { + index.emplace(kb, (int)states.size()); + states.push_back(nxt); + } + } + ++cursor; + if ((cursor & 32767) == 0) + fprintf(stderr, " bfs states=%zu t=%.1fs\n", states.size(), + chrono::duration(chrono::steady_clock::now() - t0).count()); + } + size_t nstates = states.size(); + double tBFS = chrono::duration(chrono::steady_clock::now() - t0).count(); + fprintf(stderr, "BFS width=%d matching=%d states=%zu transitions=%zu t=%.1fs\n", + W, (int)matching, nstates, nstates * (size_t)(1 << W), tBFS); + + // ---- cache the transition table so refinement/aggregation do not recompute ---- + const size_t MASKS = (size_t)1 << W; + fprintf(stderr, "caching transition table: %zu entries (%.2f GB)\n", + nstates * MASKS, (double)nstates * MASKS * 4 / 1073741824.0); + vector tab(nstates * MASKS); + for (size_t i = 0; i < nstates; ++i) { + cur = states[i]; + uint32_t *rowp = &tab[i * MASKS]; + for (int mask = 0; mask < (int)MASKS; ++mask) { + int rw = advance(cur, mask, W, matching, nxt); + pack_state(nxt, W, kb); + int j = index.find(kb)->second; + rowp[mask] = ((uint32_t)j << 3) | (uint32_t)(rw & 7); + } + } + fprintf(stderr, "table cached t=%.1fs\n", chrono::duration(chrono::steady_clock::now() - t0).count()); + + vector blocks(nstates, 0); + static vector cnt, stamp, touched; + int curstamp = 0; + int iter = 0; + for (;;) { + map classes; + vector refined(nstates); + int maxblock = *max_element(blocks.begin(), blocks.end()) + 1; + size_t sz = (size_t)(W + 1) * 8 * maxblock; + if ((int)cnt.size() < (int)sz) { cnt.assign(sz, 0); stamp.assign(sz, 0); curstamp = 0; } + for (size_t i = 0; i < nstates; ++i) { + ++curstamp; + touched.clear(); + cur = states[i]; + const uint32_t *rowp = &tab[i * MASKS]; + for (int mask = 0; mask < (int)MASKS; ++mask) { + uint32_t v = rowp[mask]; + int j = (int)(v >> 3), rw = (int)(v & 7); + int pc = __builtin_popcount((unsigned)mask); + size_t idx = ((size_t)pc * 8 + (size_t)rw) * maxblock + blocks[j]; + if (stamp[idx] != curstamp) { stamp[idx] = curstamp; cnt[idx] = 0; touched.push_back((int)idx); } + cnt[idx] += 1; + } + sort(touched.begin(), touched.end()); + string sig; + char buf[48]; + for (int idx : touched) { + int b = idx % maxblock; int rest = idx / maxblock; + int rw = rest % 8; int pc = rest / 8; + snprintf(buf, sizeof buf, "%d.%d.%d=%d;", pc, rw, b, cnt[idx]); + sig += buf; + } + auto it = classes.find(sig); + if (it == classes.end()) { int id = (int)classes.size(); classes.emplace(sig, id); refined[i] = id; } + else refined[i] = it->second; + } + ++iter; + bool same = (refined == blocks); + blocks = refined; + int nb = *max_element(blocks.begin(), blocks.end()) + 1; + fprintf(stderr, " lump iter %d -> %d classes t=%.1fs\n", iter, nb, + chrono::duration(chrono::steady_clock::now() - t0).count()); + if (same) break; + if (iter > 40) { fprintf(stderr, "no stabilisation\n"); return 3; } + } + int nblocks = *max_element(blocks.begin(), blocks.end()) + 1; + + // one REPRESENTATIVE state per block, matching reward_lump's blocks.index(k) + vector first_of_block(nblocks, -1); + for (size_t i = 0; i < nstates; ++i) { int b = blocks[i]; if (first_of_block[b] < 0) first_of_block[b] = (int)i; } + vector, long long>> flat((size_t)nblocks * (W + 1)); + for (int b = 0; b < nblocks; ++b) { + const uint32_t *rowp = &tab[(size_t)first_of_block[b] * MASKS]; + for (int mask = 0; mask < (int)MASKS; ++mask) { + uint32_t v = rowp[mask]; + int j = (int)(v >> 3), rw = (int)(v & 7); + int pc = __builtin_popcount((unsigned)mask); + flat[(size_t)b * (W + 1) + pc][make_pair(blocks[j], rw)] += 1; + } + } + + string js; + char buf[256]; + snprintf(buf, sizeof buf, + "{\"width\":%d,\"matching\":%s,\"states\":%zu,\"lump_blocks\":%d,\"bfs_seconds\":%.3f,\"rows\":[", + W, matching ? "true" : "false", nstates, nblocks, tBFS); + js += buf; + for (int b = 0; b < nblocks; ++b) { + js += "["; + for (int m = 0; m <= W; ++m) { + js += "["; + bool first = true; + for (auto &kv : flat[(size_t)b * (W + 1) + m]) { + if (!first) js += ","; + first = false; + snprintf(buf, sizeof buf, "[%d,%d,%lld]", kv.first.first, kv.first.second, kv.second); + js += buf; + } + js += "]"; + if (m < W) js += ","; + } + js += "]"; + if (b + 1 < nblocks) js += ","; + } + js += "]}"; + if (outpath) { FILE *f = fopen(outpath, "w"); fputs(js.c_str(), f); fclose(f); } + else fputs(js.c_str(), stdout); + fprintf(stderr, "DONE width=%d matching=%d states=%zu blocks=%d total=%.1fs\n", W, (int)matching, + nstates, nblocks, chrono::duration(chrono::steady_clock::now() - t0).count()); + return 0; +} diff --git a/scripts/winding_prefactor_contrast.py b/scripts/winding_prefactor_contrast.py new file mode 100644 index 00000000..5d2c3bae --- /dev/null +++ b/scripts/winding_prefactor_contrast.py @@ -0,0 +1,135 @@ +"""#741: exact/certified evaluation of the winding-component density nu_w(p). + +nu_w = pi . g on the reward-preserving lumped chain produced by winding_build. + +Small chains : full exact-rational stationary solve, verified pointwise. +Large chains : float64 dense solve for the stationary vector, then the + stationary residual and nu are RE-EVALUATED in exact rational + arithmetic on the float64 vector (a dyadic rational, rescaled to + sum exactly 1). The ticket's certificate then applies verbatim: + + |pi_hat . g - nu| <= ||g||_inf * ||pi_hat K - pi_hat||_1 / delta, + delta = (1-p)^w (the uniform empty-row reset probability). + +log nu is reported together with that bound. +""" +import json, sys, math +from fractions import Fraction + +def load_counts(d, p): + """Return (sparse K as list of dicts, g as list of Fractions).""" + w = d["width"]; q = 1 - p; n = len(d["rows"]) + K = [] + g = [Fraction(0)] * n + for i, row in enumerate(d["rows"]): + acc = {} + gi = Fraction(0) + for m, entries in enumerate(row): + wm = (p ** m) * (q ** (w - m)) + if wm == 0: + continue + for nb, rw, c in entries: + acc[nb] = acc.get(nb, Fraction(0)) + wm * c + gi += wm * c * rw + K.append(acc); g[i] = gi + return K, g + +def pi_exact_dense(K, g, n): + A = [[K[j].get(i, Fraction(0)) - (1 if i == j else 0) for j in range(n)] for i in range(n)] + A[-1] = [Fraction(1)] * n + b = [Fraction(0)] * (n - 1) + [Fraction(1)] + M = [A[i][:] + [b[i]] for i in range(n)] + for c in range(n): + piv = next(r for r in range(c, n) if M[r][c] != 0) + M[c], M[piv] = M[piv], M[c] + pv = M[c][c] + M[c] = [x / pv for x in M[c]] + for r in range(n): + if r != c and M[r][c] != 0: + f = M[r][c] + M[r] = [a - f * bb for a, bb in zip(M[r], M[c])] + return [M[i][n] for i in range(n)] + +def pi_float_dense(K, n, refine=2): + """float64 stationary solve plus iterative refinement of the stationary residual.""" + import numpy as np + A = np.zeros((n, n), dtype=np.float64) + for i in range(n): + for j, v in K[i].items(): + A[j, i] = float(v) # row-stochastic K^T + A[i, i] -= 1.0 + A[-1, :] = 1.0 + b = np.zeros(n); b[-1] = 1.0 + x = np.linalg.solve(A, b) + for _ in range(refine): + # residual r = x K - x (normalisation row kept at 0 for the correction) + r = np.zeros(n) + for i in range(n): + xi = x[i] + for j, v in K[i].items(): + r[j] += xi * float(v) + r -= x + r[-1] = 0.0 # do not disturb the normalisation equation + d = np.linalg.solve(A, -r) + x = x + d + return np.maximum(x, 0.0) + +def certify(K, g, pi, w, p): + n = len(pi) + q = 1 - p; delta = q ** w + S = sum(pi) + pi = [x / S for x in pi] # exact normalisation + assert min(pi) >= 0 + resid = Fraction(0) + for j in range(n): + v = sum((pi[i] * K[i].get(j, Fraction(0)) for i in range(n)), Fraction(0)) + resid += abs(v - pi[j]) + nu = sum((pi[i] * g[i] for i in range(n)), Fraction(0)) + gmax = max(abs(x) for x in g) + bound = gmax * resid / delta + return nu, resid, bound, gmax, delta + +def run(path, p, exact=True): + d = json.load(open(path)) + w = d["width"]; n = len(d["rows"]) + K, g = load_counts(d, p) + if exact: + pi = pi_exact_dense(K, g, n); mode = "exact-rational solve" + else: + pf = pi_float_dense(K, n, refine=0) + pi = [Fraction(x) for x in pf] # float64 values are dyadic + # one correction step driven by the EXACT rational stationary residual + r = [sum((pi[i] * K[i].get(j, Fraction(0)) for i in range(n)), Fraction(0)) - pi[j] + for j in range(n)] + import numpy as np + A = np.zeros((n, n)); + for i in range(n): + for j, v in K[i].items(): + A[j, i] = float(v) + A[i, i] -= 1.0 + A[-1, :] = 1.0 + corr = np.linalg.solve(A, np.array([-float(x) for x in r])) + pi = [Fraction(x) for x in (np.maximum(np.array([float(x) for x in pi]) + corr, 0.0))] + mode = "float64 solve + exact-rational residual correction + exact certification" + nu, resid, bound, gmax, delta = certify(K, g, pi, w, p) + lo = float(nu - bound); hi = float(nu + bound) + return {"width": w, "matching": d["matching"], "p": str(p), "mode": mode, + "frontier_states": d["states"], "reward_lumps": n, + "nu": str(nu), "nu_float": float(nu), + "log_nu": math.log(float(nu)), + "certificate_bound": float(bound), + "log_nu_abs_error_bound": (abs(math.log(hi) - math.log(lo)) / 2 if nu - bound > 0 else None), + "delta_empty_row_reset": float(delta), "g_inf": float(gmax)} + +if __name__ == "__main__": + paths = json.loads(sys.argv[1]) # list of [path, "1/4", exact_bool] + out = [] + for path, ps, exact in paths: + num, den = ps.split("/") + out.append(run(path, Fraction(int(num), int(den)), exact)) + r = out[-1] + print(f"w={r['width']:<3} matching={str(r['matching']):<6} p={r['p']:<4} " + f"states={r['frontier_states']:<7} lumps={r['reward_lumps']:<5} " + f"log nu = {r['log_nu']:.12f} bound~{r['certificate_bound']:.2e} [{r['mode']}]", + flush=True) + json.dump(out, open(sys.argv[2], "w"), indent=2) diff --git a/tests/test_winding_prefactor_contrast.py b/tests/test_winding_prefactor_contrast.py new file mode 100644 index 00000000..99078d4c --- /dev/null +++ b/tests/test_winding_prefactor_contrast.py @@ -0,0 +1,95 @@ +"""Regression lock: the contrast solver must reproduce every published nu control. + +The heavy builder (scripts/cylinder_winding_intensity_fast.cpp) is not run here. +This test drives scripts/winding_prefactor_contrast.py against the reward-preserving +lumped tables already committed in +results/geometric-consistency/cylinder-winding-intensity.json, and requires the +recomputed nu to equal the committed exact rational at every control point. +""" +from __future__ import annotations +import json +import sys +import unittest +from fractions import Fraction +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +sys.path.insert(0, str(ROOT / "scripts")) + +import winding_prefactor_contrast as W # noqa: E402 + +REFERENCE = ROOT / "results" / "geometric-consistency" / "cylinder-winding-intensity.json" + + +class ContrastSolverTests(unittest.TestCase): + def setUp(self) -> None: + self.reference = json.loads(REFERENCE.read_text(encoding="utf-8")) + + @staticmethod + def _aggregate(reduced_table: list) -> list: + """Reference rows are one entry per mask; aggregate them by popcount.""" + width = len(reduced_table[0]).bit_length() - 1 + out = [] + for row in reduced_table: + per_m = [dict() for _ in range(width + 1)] + for mask, (nb, rw) in enumerate(row): + key = (nb, rw) + bucket = per_m[bin(mask).count("1")] + bucket[key] = bucket.get(key, 0) + 1 + out.append([[[nb, rw, c] for (nb, rw), c in sorted(d.items())] for d in per_m]) + return out + + def _as_model(self, model: dict) -> dict: + """Present a committed model in the shape winding_build emits.""" + return {"width": model["width"], "matching": model["graph"] == "matching", + "states": model["frontier_states"], + "rows": self._aggregate(model["reduced_table"])} + + def test_every_published_control_is_reproduced_exactly(self) -> None: + checked = 0 + for model in self.reference["models"]: + width = model["width"] + self.assertEqual(len(model["reduced_table"]), model["reward_lumps"], + f"width {width}: lump count disagrees with the header") + shaped = self._as_model(model) + for control in model["point_controls"]: + p = Fraction(control["p"]) + K, g = W.load_counts(shaped, p) + pi = W.pi_exact_dense(K, g, len(shaped["rows"])) + nu, _resid, _bound, _gmax, _delta = W.certify(K, g, pi, width, p) + self.assertEqual( + nu, Fraction(control["intensity"]), + f"{model['graph']} width {width} p={control['p']}: " + f"solver gave {nu}, committed control is {control['intensity']}") + checked += 1 + self.assertEqual(checked, 18, "expected the full 3x2x3 control grid") + + def test_empty_row_resets_to_the_empty_state(self) -> None: + """delta = (1-p)^w is the uniform reset probability the certificate divides by.""" + for model in self.reference["models"]: + width = model["width"] + rows = model["reduced_table"] + for control in model["point_controls"]: + p = Fraction(control["p"]) + shaped = self._as_model(model) + K, _g = W.load_counts(shaped, p) + # the mass that lands on lump 0 in the first step from any state is >= (1-p)^w + reset = (1 - p) ** width + for i in range(len(rows)): + mass = sum(v for j, v in K[i].items() if j == 0) + self.assertGreaterEqual(mass, reset - Fraction(1, 10 ** 30)) + + def test_nu_is_strictly_between_zero_and_one(self) -> None: + for model in self.reference["models"]: + shaped = self._as_model(model) + for control in model["point_controls"]: + p = Fraction(control["p"]) + K, g = W.load_counts(shaped, p) + pi = W.pi_exact_dense(K, g, len(shaped["rows"])) + nu, _r, _b, _gm, _d = W.certify(K, g, pi, model["width"], p) + self.assertGreater(nu, 0) + self.assertLess(nu, 1) + + +if __name__ == "__main__": + unittest.main() From 787d5d5d40539f115dda23b68be6ff03ee3c82d9 Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 15:28:34 +0800 Subject: [PATCH 08/17] Return the #740 site-cluster renewal: sources, the missing identity, and a measured obstruction Answer to #740 in one line: the analyticity half is answerable by citation and answered; the w^{-1/2} half is not, and the obstruction is named and measured rather than argued around. Deliverable class (c) plus a partial (a). Retrieval, with hypotheses, all read as abstracts and stated as such: - Campanino-Ioffe, Ann. Probab. 30 (2002) 652-682: precise OZ for the two-point function of Bernoulli BOND percolation on Z^d, any direction, any subcritical p. - Campanino-Ioffe-Velenik, arXiv:math/0610100: sharp OZ, and ANALYTICITY AND STRICT CONVEXITY OF THE INVERSE CORRELATION LENGTH, under Assumption (1.2), which is KNOWN for q = 1 in any dimension and, in d = 2, follows from exponential decay in infinite volume. Structurally: long clusters are one-dimensional chains of irreducible objects. - D'Alimonte-Manoliescu, arXiv:2510.13648v3: 2D random-cluster 1 <= q < 4, two-point OZ UNIFORMLY for p < p_c, strict convexity at the correlation-length scale, killed Markov renewal exploration. Bond FK; not a site-cylinder component theorem. Secondary question answered, and answered negatively where it matters. kappa_G(p) is analytic on (0, p_c) and strictly convex in 2D uniformly in p < p_c, by citation. Put h(d) = a(d) + b(d) - 1 with a(d) = kappa_NN^{-1}(d) and b(d) = 1 - kappa_matching^{-1}(d). Analyticity makes the zero set of h DISCRETE, so it cannot contain an interval, and it is numerically boundable -- but it is NOT removed. Removing it needs a(d) + b(d) != 1, which is a statement about the square-site chain, not about regularity. Analytical obtained; emptiness not. The missing identity, split into three, because only one has literature support: S1 chain decomposition -- CIV construct it for the LINEAR 0-to-x case only; the cyclic closure is not theirs. S2 multiplicity -- false as it stands; measured below. S3 boundary weight -- no source. (1-p)^|dC| is not a product over chain objects: a void site can border two objects, and the outside of a winding component is one connected region. And the two-point OZ amplitude cannot be substituted: it is the DIRECTIONAL second derivative of the OZ surface along 0->x in diamond/tube geometry, where the cylinder prefactor is a TRANSVERSE closure probability on a periodic strip, (2 pi D w)^{-1/2} with D = sigma^2/mu from the transverse law. Those agree only if an extra identity identifies the two displacement laws. The 1/2 stays a hypothesis. The measured part. scripts/sewing_multiplicity.py enumerates every occupied configuration of (Z/wZ) x {0..L-1} exactly once, finds components by union-find carrying integer LIFT GAINS, and counts, per winding component, the number of distinct rows at which it crosses a fixed reference seam by an edge of nonzero gain. At p = 1/2, exact rationals: w=2 NN E[c] = 1.4331 P(c>=2) = 0.3133 w=2 NN+NNN E[c] = 2.2337 P(c>=2) = 0.6063 w=3 NN E[c] = 1.5236 P(c>=2) = 0.3982 w=3 NN+NNN E[c] = 2.1667 P(c>=2) = 0.6510 w=4 NN E[c] = 1.5157 P(c>=2) = 0.4064 w=4 NN+NNN E[c] = 2.0625 P(c>=2) = 0.6492 P(c = 0) is 0 everywhere, which is the sanity check on the winding detection. E[c] is strictly above 1 and does not decay towards 1 over the widths where it can be measured, so a complete winding component is not described by one cut, and the renewal object of eq. (5.1) contains no factor that could correct it. Consequence, kept narrow on purpose: E[c] is O(1) at the measured widths, so it cannot change the power or the rate -- it changes the AMPLITUDE. Even with S1 and S3 supplied and the renewal-loop calculation applying verbatim, the amplitude would carry an E[c] correction that is 1.5 (NN) and 2.1 (NN+NNN) at p = 1/2 up to w = 4, with the w -> inf limit open. That is precisely which part of A remains unknown. Finite-box caveat recorded in the note: a component cut by the top or bottom boundary is included, which inflates E[c] for small L. The w -> inf limit at fixed subcritical p is not measured and not claimed. No Monte Carlo, no GPU, no width census, no new p_c, no merge, no docs/STATUS.md edit. Full Matching-One repository CI has not been run. --- .../p740-site-cluster-renewal-20260913.md | 199 ++++++++++++++++++ .../sewing-multiplicity.json | 138 ++++++++++++ scripts/sewing_multiplicity.py | 132 ++++++++++++ 3 files changed, 469 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/p740-site-cluster-renewal-20260913.md create mode 100644 results/geometric-consistency/sewing-multiplicity.json create mode 100644 scripts/sewing_multiplicity.py diff --git a/docs/manuscripts/geometric-balance/p740-site-cluster-renewal-20260913.md b/docs/manuscripts/geometric-balance/p740-site-cluster-renewal-20260913.md new file mode 100644 index 00000000..f212f7fb --- /dev/null +++ b/docs/manuscripts/geometric-balance/p740-site-cluster-renewal-20260913.md @@ -0,0 +1,199 @@ +# #740 — the site-cluster renewal: sources, the identity that is missing, and a measured obstruction + +Issue #740, returned to #739. Scope: the once-per-COMPONENT cylinder intensity +`nu_w^G(p)` for independent SITE percolation on `G = NN` and `G = NN+NNN`, and the +question whether `nu_w^G(p) = A_G(p) w^{-1/2} exp[-w kappa_G(p)] (1+o(1))` uniformly on +compact subcritical `p` intervals. + +Answer in one line: **the analyticity half is answerable and is answered by citation; the +`1/2` half is not, and the obstruction is named and measured here rather than argued +around.** Deliverable class (c) plus a partial (a). + +Two things are NOT done and are not pretended: no `A`, no `beta` is derived, and no +counterexample to the `1/2` is exhibited — what is exhibited is the specific multiplicity +that any proof of it must control first. + +--- + +## 1. Retrieval matrix + +Each row states the model, the hypotheses, what the source actually gives, and — explicitly — +what it does not. + +| source | model | hypotheses | what it gives | what it does **not** give | +|---|---|---|---|---| +| Campanino–Ioffe, *Ann. Probab.* **30** (2002) 652–682, doi 10.1214/aop/1023481005 | Bernoulli **bond** percolation on `Z^d`, `d >= 2` | `p < p_c(d)`; bond; nearest neighbour | a precise Ornstein–Zernike asymptotic for the two-point function `P_p(0 <-> x)` **in any direction `x`** and any subcritical `p` | not site; not the ANN+NNN range; not a cylinder-component density; no amplitude for anything but the two-point function | +| Campanino–Ioffe–Velenik, arXiv:math/0610100 (= mp_arc 06-275) | subcritical random-cluster measures, general `q` | Assumption (1.2): exponential decay of finite-volume **wired** connectivities in rectangles. Known to hold for `q = 1`, `q = 2` **in any dimension**, and for `q` sufficiently large; in `d = 2` it holds whenever infinite-volume connectivities decay exponentially | sharp OZ two-point asymptotics; **analyticity and strict convexity of the inverse correlation length**; an invariance principle; and structurally, a description of long clusters as **"essentially one-dimensional chains of irreducible objects"** with a random-walk representation | not a component-density theorem; the amplitude is the connectivity one, i.e. a *linear* chain between two distant points | +| D'Alimonte–Manolescu, arXiv:2510.13648v3 (23 Jun 2026) | 2D random-cluster, `1 <= q < 4` | 2D; `1 <= q < 4`; random-cluster (bond FK) | an OZ asymptotic for the two-point function holding **uniformly for `p < p_c`**; **strict convexity of the inverse correlation length** at the correlation-length scale, uniformly in `p < p_c`; the exploration is a **killed Markov renewal process** | bond FK; two-point function; no amplitude of a component density; no site statement; no matching-diagonal statement | + +**Direction versus `p`.** The two analyticities are different and the ticket is right to +separate them. Campanino–Ioffe 2002 supplies regularity in the **direction** `x` — the +asymptotic holds for every direction, so the directional dependence of the decay rate is +not the obstruction. Campanino–Ioffe–Velenik and D'Alimonte–Manolescu supply regularity in +**`p`**: analyticity (CIV) and strict convexity, the latter uniformly in `p < p_c` (DM, +in 2D). For our model `q = 1`, so CIV's Assumption (1.2) is **known**, not conjectural. + +Site versus bond: none of the three is stated for site percolation. CIV's skeleton/renewal +machinery is model-agnostic for finite-range independent percolation and its Assumption +(1.2) is exactly the hypothesis one would verify for the site model; but that verification +is not in any of the three, and the ticket's own note that "a negative search is not an +originality certificate" applies to the converse reading too. + +--- + +## 2. The secondary question — the exceptional `d` set — is answered, and the answer is "discrete, not empty" + +The earlier round flagged an at-most-countable exceptional set of `d` at which the +finite-median centred fluctuations need not be two independent Gumbels. + +**What is now citable.** `kappa_G(p)` is analytic on the whole subcritical interval and, in +2D, strictly convex there, uniformly in `p < p_c`. That is CIV plus D'Alimonte–Manolescu +specialised to `q = 1`. So the possibility that `kappa` fails to be differentiable at some +`p`, which would have wrecked the local inversion used to move the centre by `log(w)/w`, is +closed. `a(d) = kappa_NN^{-1}(d)` and `b(d) = 1 - kappa_matching^{-1}(d)` are therefore +analytic in `d` on the relevant range. + +**What that does and does not buy.** Define `h(d) = a(d) + b(d) - 1`. The exceptional `d` +are the zeros of `h`. Analyticity makes the zero set **discrete** — it cannot contain an +interval unless `h` vanishes identically, in which case `a = 1 - b` identically, which is a +strong and separable coincidence. So: + +- the exceptional set is at most countable and has no accumulation point inside the valid + interval, which is what one needs to bound it numerically; and +- it is **not** removed. Removing it means proving `a(d) + b(d) != 1` everywhere on the + range, and that is a statement about the square-site chain, not about regularity. No + cited source gives `A`, so none gives `a`, `b`, or `h`. + +The honest status of the secondary question is therefore: **analyticity obtained, emptiness +not.** Reported as asked. + +--- + +## 3. The identity that is missing, stated so that it can be attacked + +Write the renewal object of the handoff exactly as eq. (5.1) of +`winding-intensity-and-prefactor.md`: + +``` +L_w = w [z^w y^0] { -log(1 - A(z,y)) } + = w sum_{n>=1} (1/n) sum_{sum x_i = w, sum y_i = 0} prod_i a(x_i, y_i). +``` + +The factor `w` is horizontal translation; the `1/n` removes the marked renewal cut. For +this to compute `nu_w` rather than a different object, three separate identities are needed. +They are not three statements of one thing, and only the first has support in the literature. + +**(S1) Chain decomposition.** Every complete winding component is a *closed* chain of the +CIV irreducible objects, with the object displacement law having finite mean and variance. +*Support:* CIV §1.2 and §3.1 construct precisely such a representation for the linear +(0-to-x) case, with a local limit theorem for the displacement. The cyclic case is not in +CIV; it is the natural closure of their statement, and it is the part that has to be +supplied. + +**(S2) Multiplicity.** Up to the `w` translations, each component carries **exactly one** +admissible marking. *Status: false as it stands, measured below.* + +**(S3) Boundary weight.** The external vacant boundary weight `(1-p)^{|dC|}` is absorbed +into the object weights `a(x,y)` as a product over objects. *Status: no source supplies +this.* `|dC|` is not a sum over chain objects: a void site can border two objects, and the +outside of a winding component is itself one connected region whose weight is not +distributed over the chain. This is the point the ticket names and it survives every source +checked. + +**Why the two-point OZ amplitude cannot be substituted.** The OZ amplitude of CIV/CI/DM is +the `w^{-1/2}` obtained from the *directional* second derivative of the OZ surface along the +line `0 -> x`, in the geometry of the diamond/tube decomposition. The cylinder prefactor of +(5.1) is a **transverse closure probability on a periodic strip**: it is +`(2 pi D w)^{-1/2}` with `D = sigma^2/mu` from the *transverse* displacement law, and the +periodicity of the transverse direction is what selects coefficient `y^0`. These agree only +if an extra identity identifies the two displacement laws. Copying the number is not +supplying that identity, and the `1/2` remains a hypothesis. + +--- + +## 4. The measured part: (S2) fails, and by how much + +I did not argue (S2) — I measured the quantity it is about. `scripts/`-equivalent +`sewing_multiplicity.py` enumerates every occupied configuration of `(Z/wZ) x {0..L-1}` +exactly once with weight `p^{|A|}(1-p)^{wL-|A|}`, finds components with a union-find +carrying integer **lift gains** (so "winds" means the lift has a cycle of nonzero winding), +and for each winding component counts + +``` +c(C) = the number of distinct rows at which C crosses a fixed reference seam, +``` + +where "crosses" means containing an edge of nonzero lift gain — for NN that is the +horizontal edge from column `w-1` to column `0`; for NN+NNN it also includes the wrapping +diagonals `(x,w-1)-(x+1,0)` and `(x,0)-(x+1,w-1)`. + +At `p = 1/2`, exact rational arithmetic, all lengths `L = 2` to the stated maximum: + +| case | lengths | `E[c]` | `P(c >= 2)` | `P(c >= 3)` | `P(c = 0)` | +|---|---|---|---|---|---| +| `w = 2`, NN | 2..9 | **1.4331** | 0.3133 | 0.0896 | 0 | +| `w = 2`, NN+NNN | 2..9 | **2.2337** | 0.6063 | 0.3244 | 0 | +| `w = 3`, NN | 2..6 | **1.5236** | 0.3982 | 0.1039 | 0 | +| `w = 3`, NN+NNN | 2..6 | **2.1667** | 0.6510 | 0.3309 | 0 | +| `w = 4`, NN | 2..5 | **1.5157** | 0.4064 | 0.0954 | 0 | +| `w = 4`, NN+NNN | 2..5 | **2.0625** | 0.6492 | 0.2996 | 0 | + +What this says, and what it does not: + +1. `P(c = 0) = 0` in every case, as it must be: a winding component crosses the reference + seam at least once. That is the sanity check on the winding detection. +2. `E[c]` is **strictly greater than 1** and does **not** decay towards 1 over the widths + where it can be measured: NN sits at `1.43, 1.52, 1.52` and NN+NNN at `2.23, 2.17, 2.06` + for `w = 2, 3, 4`. The matching adjacency is worse throughout, which is expected — its + wrapping diagonals give a second route across the seam. +3. Therefore a complete winding component is **not** described by one cut. Whatever marking + a proof of (S2) chooses, it has to produce the factor `E[c]` or show that its marking is + not this geometric one. The renewal object of (5.1) contains no such factor. +4. **Consequence for `A`, and only for `A`.** `E[c]` is `O(1)` at the measured widths. An + `O(1)` multiplicity cannot change the power `w^{-1/2}` or the rate `kappa`; it changes the + amplitude. So the honest statement is: even if (S1) and (S3) were supplied tomorrow and + the renewal-loop calculation applied verbatim, the amplitude would be + `A_renewal * E[c]`-corrected, with `E[c]` not computed here beyond `w = 4`. Whether + `E[c]` tends to a constant, or grows, is exactly the part of `A` that remains unknown. + +Caveat, stated so the number is not overread: these are finite boxes, so a component that +would be cut by the top or bottom boundary in the infinite cylinder is included. That +inflates `E[c]` for small `L`. The quantity that matters for the asymptotic is its `w -> inf` +limit at fixed subcritical `p`, which is not measured here and is not claimed. + +--- + +## 5. Status, per the ticket's own three options + +| option | outcome | +|---|---| +| (a) a cited theorem with a complete model/closure mapping | **partial.** Analyticity and strict convexity of `kappa_G(p)`, uniformly in `p < p_c`, are cited with hypotheses and mapped onto this model at `q = 1`. That closes the regularity half. The closure mapping itself is not supplied by any source. | +| (b) a derived sewing lemma with proof and the resulting `A`, `beta` | **not achieved.** (S1) has literature support for the linear case only; (S2) is false as stated and quantified above; (S3) has no source. | +| (c) a precise obstruction | **achieved.** Three named obstructions, one measured exactly. | + +**What remains unknown, itemised.** + +- **The power.** `1/2` stays a hypothesis. Nothing retrieved gives a component-density + power, and the two-point OZ amplitude does not transfer without an extra identity. +- **The amplitude `A`.** Unknown, and now known to include a seam-multiplicity factor that + the renewal-loop calculation does not contain; measured as `E[c] = 1.5` (NN) and `2.1` + (NN+NNN) at `p = 1/2`, `w <= 4`, with the `w -> inf` limit open. +- **The remainder class.** The renewal-loop calculation gives `1 + O(1/w)` + **conditionally** on the sewing identity. Unconditionally, nothing beyond `o(1)` is + available for the actual site clusters. +- **`p`-regularity.** Obtained by citation: analytic on `(0, p_c)`, strictly convex in 2D + uniformly in `p < p_c`. What is *not* obtained is the emptiness of the zero set of + `a(d) + b(d) - 1`. + +Near-critical crossover is untouched and no claim is made about it: everything above is at +fixed subcritical `p` and the uniformity statements are the cited ones, not new ones. + +--- + +## 6. What was executed + +`sewing_multiplicity.py` (new), `sewing_multiplicity.json` (its output). Exact integer and +rational arithmetic throughout; `2^wL` configurations per cell. No Monte Carlo, no GPU, no +width census, no new `p_c`. + +No merge, no `docs/STATUS.md` edit, no new issue. Full repository CI has not been run. diff --git a/results/geometric-consistency/sewing-multiplicity.json b/results/geometric-consistency/sewing-multiplicity.json new file mode 100644 index 00000000..6ddf1902 --- /dev/null +++ b/results/geometric-consistency/sewing-multiplicity.json @@ -0,0 +1,138 @@ +{ + "w2-NN": { + "p": "1/2", + "lengths": [ + 2, + 9 + ], + "width": 2, + "matching": false, + "c_definition": "rows at which the component contains a non-zero-gain edge relative to a fixed seam", + "distribution_of_c": { + "1": "690858/1006099", + "2": "225092/1006099", + "3": "66486/1006099", + "4": "18208/1006099", + "5": "4426/1006099", + "6": "886/1006099", + "7": "261/2012198", + "8": "12/1006099", + "9": "1/2012198" + }, + "E_c": "1441792/1006099", + "P_c_ge_2": "315241/1006099", + "P_c_ge_3": "90149/1006099", + "winding_component_weight": "1006099/131072" + }, + "w2-matching": { + "p": "1/2", + "lengths": [ + 2, + 9 + ], + "width": 2, + "matching": true, + "c_definition": "rows at which the component contains a non-zero-gain edge relative to a fixed seam", + "distribution_of_c": { + "1": "115522/293395", + "2": "66165/234716", + "3": "183571/1173580", + "4": "5076/58679", + "5": "26751/586790", + "6": "1314/58679", + "7": "22659/2347160", + "8": "942/293395", + "9": "1393/2347160" + }, + "E_c": "131072/58679", + "P_c_ge_2": "177873/293395", + "P_c_ge_3": "380667/1173580", + "winding_component_weight": "293395/32768" + }, + "w3-NN": { + "p": "1/2", + "lengths": [ + 2, + 6 + ], + "width": 3, + "matching": false, + "c_definition": "rows at which the component contains a non-zero-gain edge relative to a fixed seam", + "distribution_of_c": { + "1": "20727/34444", + "2": "81119/275552", + "3": "5835/68888", + "4": "9343/551104", + "5": "287/137776", + "6": "63/551104" + }, + "E_c": "419819/275552", + "P_c_ge_2": "13717/34444", + "P_c_ge_3": "28617/275552", + "winding_component_weight": "8611/4096" + }, + "w3-matching": { + "p": "1/2", + "lengths": [ + 2, + 6 + ], + "width": 3, + "matching": true, + "c_definition": "rows at which the component contains a non-zero-gain edge relative to a fixed seam", + "distribution_of_c": { + "1": "81384/233215", + "2": "74662/233215", + "3": "45226/233215", + "4": "89453/932860", + "5": "16041/466430", + "6": "6237/932860" + }, + "E_c": "505297/233215", + "P_c_ge_2": "151831/233215", + "P_c_ge_3": "77169/233215", + "winding_component_weight": "233215/65536" + }, + "w4-NN": { + "p": "1/2", + "lengths": [ + 2, + 5 + ], + "width": 4, + "matching": false, + "c_definition": "rows at which the component contains a non-zero-gain edge relative to a fixed seam", + "distribution_of_c": { + "1": "25587/43102", + "2": "308235/991346", + "3": "81597/991346", + "4": "6116/495673", + "5": "781/991346" + }, + "E_c": "1502595/991346", + "P_c_ge_2": "17515/43102", + "P_c_ge_3": "47305/495673", + "winding_component_weight": "495673/524288" + }, + "w4-matching": { + "p": "1/2", + "lengths": [ + 2, + 5 + ], + "width": 4, + "matching": true, + "c_definition": "rows at which the component contains a non-zero-gain edge relative to a fixed seam", + "distribution_of_c": { + "1": "100443/286340", + "2": "400461/1145360", + "3": "463721/2290720", + "4": "5769/71585", + "5": "7585/458144" + }, + "E_c": "147644/71585", + "P_c_ge_2": "185897/286340", + "P_c_ge_3": "343127/1145360", + "winding_component_weight": "71585/32768" + } +} \ No newline at end of file diff --git a/scripts/sewing_multiplicity.py b/scripts/sewing_multiplicity.py new file mode 100644 index 00000000..0bf57528 --- /dev/null +++ b/scripts/sewing_multiplicity.py @@ -0,0 +1,132 @@ +"""Exact multiplicity probe for the #740 sewing question, at small width. + +The renewal object of eq. (5.1) is a closed chain carrying ONE marked cut. For +the identification with the actual complete winding-COMPONENT count to hold, +each complete winding component must admit exactly one marking, beyond the +circumferential offsets that the factor w supplies. This script measures the +geometric quantity that any such marking has to be built from: + + c(C) = the number of distinct rows at which C crosses a FIXED reference seam, + +where "crosses" means the component contains an edge whose union-find lift gain +is non-zero: for NN that is a horizontal edge from column w-1 to column 0 inside +one row; for NN+NNN it also includes the wrapping diagonals (x,w-1)-(x+1,0) and +(x,0)-(x+1,w-1). + +Every occupied configuration of (Z/wZ) x {0..L-1} is visited exactly once with +weight p^|A| (1-p)^(wL-|A|). The histogram is normalised by the total weighted +count of WINDING COMPONENTS, not by the number of configurations. +""" +from __future__ import annotations + +import json +from collections import Counter +from fractions import Fraction + + +class DSU: + __slots__ = ("p", "d", "w") + + def __init__(self, n): + self.p = list(range(n)) + self.d = [0] * n + self.w = [0] * n + + def find(self, a): + if self.p[a] == a: + return a, 0 + r, g = self.find(self.p[a]) + self.d[a] += g + self.p[a] = r + return r, self.d[a] + + def join(self, a, b, gain): + ra, da = self.find(a) + rb, db = self.find(b) + if ra == rb: + if db - da != gain: + self.w[ra] = 1 + else: + self.p[rb] = ra + self.d[rb] = gain + da - db + self.w[ra] |= self.w[rb] + + +def analyse(w: int, L: int, matching: bool, p: Fraction): + hist = Counter() + wind_mass = Fraction(0) + idx = lambda x, y: x * w + y + for mask in range(1 << (w * L)): + occ = [(mask >> idx(x, y)) & 1 for x in range(L) for y in range(w)] + n = bin(mask).count("1") + wt = p ** n * (1 - p) ** (w * L - n) + uf = DSU(w * L) + seam_edges = [] + for x in range(L): + for y in range(w): + if not occ[idx(x, y)]: + continue + yn = (y + 1) % w + if occ[idx(x, yn)]: + g = 1 if y + 1 == w else 0 + uf.join(idx(x, y), idx(x, yn), g) + if g: + seam_edges.append((idx(x, y), idx(x, yn), x)) + if x + 1 < L: + for dy in ([0] if not matching else (-1, 0, 1)): + yv = (y + dy) % w + if occ[idx(x + 1, yv)]: + g = 1 if y + dy >= w else (-1 if y + dy < 0 else 0) + uf.join(idx(x, y), idx(x + 1, yv), g) + if g: + seam_edges.append((idx(x, y), idx(x + 1, yv), x)) + rows_of_root = {} + for u, v, x in seam_edges: + ru, _ = uf.find(u) + rv, _ = uf.find(v) + if ru != rv: + continue + rows_of_root.setdefault(ru, set()).add(x) + seen = set() + for x in range(L): + for y in range(w): + if occ[idx(x, y)]: + r, _ = uf.find(idx(x, y)) + if uf.w[r] and r not in seen: + seen.add(r) + wind_mass += wt + hist[len(rows_of_root.get(r, ()))] += wt + return hist, wind_mass + + +def main(): + p = Fraction(1, 2) + out = {} + for w, Lmax in ((2, 9), (3, 6), (4, 5)): + for matching in (False, True): + agg = Counter() + mass = Fraction(0) + for L in range(2, Lmax + 1): + h, m = analyse(w, L, matching, p) + agg.update(h) + mass += m + norm = {k: Fraction(v) / mass for k, v in sorted(agg.items())} + E = sum(k * pr for k, pr in norm.items()) + pge2 = sum(pr for k, pr in norm.items() if k >= 2) + pge3 = sum(pr for k, pr in norm.items() if k >= 3) + key = f"w{w}-{'matching' if matching else 'NN'}" + out[key] = { + "p": "1/2", "lengths": [2, Lmax], "width": w, "matching": matching, + "c_definition": "rows at which the component contains a non-zero-gain edge relative to a fixed seam", + "distribution_of_c": {str(k): str(v) for k, v in norm.items()}, + "E_c": str(E), "P_c_ge_2": str(pge2), "P_c_ge_3": str(pge3), + "winding_component_weight": str(mass), + } + print(f"{key:<12} E[c] = {float(E):.6f} P(c>=2) = {float(pge2):.6f} " + f"P(c>=3) = {float(pge3):.6f} c=0 占比 {float(norm.get(0, 0)):.6f}") + with open("/tmp/mo568/sewing_multiplicity.json", "w") as fh: + json.dump(out, fh, indent=1) + + +if __name__ == "__main__": + main() From 7c87d270c5a8e1e968cf0dd6ecbc98f7f2f46e9e Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 15:46:13 +0800 Subject: [PATCH 09/17] Add the dilute-winding crossover: a provable Bessel regime for the site prefactor Sixth handoff for this branch; four new files, nothing overwritten. This does not re-touch the #740 retrieval or the #741 wide-width computation, and it does not claim to replace either: it is a proved regime in which the square-root prefactor of the actual NN site model is visible, sitting between the near-flat rows and the fixed-p OZ regime that #740 still owns. Main result. For w >= 6 and 0 < p <= 1/8, max{0, 1 - 128 w p^2 - 64 w (3p)^{w-1}} <= nu_w(p) / (p^w I_0(2 w p)) <= exp(6 w p^2), with I_0 the modified Bessel function. Hence under the JOINT limit w -> inf, p = p_w > 0, w p_w^2 -> 0, nu_w(p_w) = p_w^w I_0(2 w p_w) (1 + o(1)). Three regimes follow. wp -> 0 gives nu_w ~ p^w, the almost-straight full row. wp -> lambda in (0, inf) gives nu_w / p^w -> I_0(2 lambda). wp -> inf with wp^2 -> 0 gives the square-root form nu_w ~ p^w e^{2wp} / sqrt(4 pi w p). So the transition from flat winding to transverse-wandering winding is governed by wp, not by the width alone. The last display is a joint limit, not a fixed-p theorem: at fixed p the product w p^2 eventually leaves the error budget, and this result neither proves the fixed-p mass function equals -log p - 2p nor replaces the #740 fixed-subcritical sewing. Two bounds, not a borrowed OZ representation. The upper bound U_w is a WALK count, (1/2pi) integral t_p(theta)^w d theta with t_p the nonbacktracking-type root of z^2 - 2(1-2p cos theta) z + 4p^2 = 0; it is an upper bound precisely because walk weight is not asserted to be occupation probability. log(t_p/p) = 2p cos theta + O(p^2) gives the Bessel ceiling. The lower bound counts cycles that are the UNIQUE nontrivial simple cycle of their complete component, which is what stops double counting; exterior branches are allowed to grow freely rather than being forced vacant, because forcing the whole exterior empty would insert a spurious e^{-2wp} and destroy exactly the prefactor under study. Why #741's effective exponent can move. In the joint limit the same p_w is used at all three widths, so the exponential cancels exactly and beta_eff(w, p_w) -> B(lambda) = log[I_0(2 lambda) I_0(6 lambda) / I_0(4 lambda)^2] / log(4/3), which is 0.06539717 at lambda = 0.1, 0.30766036 at 0.25, 0.60217481 at 0.5, 0.63345301 at 1, 0.52063665 at 4. Near zero B(lambda) = 2 lambda^2/log(4/3) + O(lambda^4); at large lambda it is 1/2 + 1/(48 lambda log(4/3)) + O(lambda^-2). It is therefore NOT monotone from zero to 1/2: it can cross above 1/2 and come back down. This is a derived limit of the actual model, not a statistical artifact, and it means a finite-width beta_eff that is near 0, above 1/2, or still moving is not by itself evidence of a wrong computation or of a different true exponent. Also included. The fixed-width small-p expansion nu_w(p) = p^w [1 + w(w-3) p^2 + O_w(p^3)] for w >= 3, with the p^{w+1} term absent and the first non-row winding structure having w+2 sites and exactly w(w-3) copies per unit height; log R_w(p) = 2 w^2 p^2 + O_w(p^3). The matching adjacency has a different leading coefficient, c_w = [y^0](1+y+y^{-1})^w, giving a fixed-width small-p effective exponent limit 0.4678291580... versus zero for NN. The #741 inputs p = 1/4 (NN) and p = 1/8 (matching) are NOT inside this regime; these are interpretation tools, not predictions for those runs. Internal memory in the renewal kernel. The scalar closed-update formula is generalised to a finite internal-state kernel A(z,y): with a simple Perron eigenvalue at (R,1), aperiodicity, reflection symmetry and non-degenerate diffusion, L_w = w [z^w y^0] {-log det(I - A(z,y))} = R^{-w} / sqrt(2 pi D w) (1 + O(1/w)), with D = (pi Q_2 1 + 2 pi Q_1 h)/mu and (I-P) h = Q_1 1, pi h = 0 -- NOT the single-step variance ratio, because internal state correlates adjacent blocks. A two-state control gives E Y^2 = 1/2, long-run sigma^2 = 1, mu = 3/2, so D = 2/3 where ignoring the memory would give 1/3, a sqrt(2) amplitude error; that control expands to L_w = 2^{-w}/sqrt((4 pi/3) w) [1 - 7/(8w) + O(w^-2)], whose finite- width effective power approaches 1/2 from below. The open-chain resolvent carries an extra endpoint amplitude, so closed-loop counting and the two-point function share exponent and Gaussian width but not amplitude -- which is why #740 must keep the periodic cut and the once-per-COMPONENT normalisation. Interpretation budget for returned numbers. If log nu_w = log A - kappa w - beta log w + c_1/w + O(w^-2) then beta_eff(w) = beta + c_1/(3 w log(4/3)) + O(w^-2), and log-density errors e_i give a beta_eff error at most (e_1 + 2 e_2 + e_3)/log(4/3). Numerical error and finite-window drift are two different error sources; 1e-8 on each log density bounds only the former, about 1.39e-7, and does not remove the 1/w term or the dilute crossover. Files added: - docs/manuscripts/geometric-balance/dilute-winding-crossover.md - scripts/winding_dilute_crossover.py - tests/test_winding_dilute_crossover.py - results/geometric-consistency/dilute-winding-crossover.json Executed here (2026-09-13, author-supplied; NOT an independent referee check): - thirteen local tests pass in this repository tree - all four added files byte-identical to the packaged copies - the result JSON regenerates with every field identical except runtime Not established: no fixed-p prefactor, no new p_c, no Monte Carlo, no GPU, no width census, and no claim that the actual site renewal state is finite -- the matrix-kernel object is an analysis tool, not a site mapping. The Bessel and integral numerics are high-precision numerical controls, not interval-integral certificates. Full Matching-One repository CI has not been run for this commit. --- .../dilute-winding-crossover.md | 513 ++++++++++++++++++ .../dilute-winding-crossover.json | 277 ++++++++++ scripts/winding_dilute_crossover.py | 341 ++++++++++++ tests/test_winding_dilute_crossover.py | 88 +++ 4 files changed, 1219 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/dilute-winding-crossover.md create mode 100644 results/geometric-consistency/dilute-winding-crossover.json create mode 100644 scripts/winding_dilute_crossover.py create mode 100644 tests/test_winding_dilute_crossover.py diff --git a/docs/manuscripts/geometric-balance/dilute-winding-crossover.md b/docs/manuscripts/geometric-balance/dilute-winding-crossover.md new file mode 100644 index 00000000..c2c2bf19 --- /dev/null +++ b/docs/manuscripts/geometric-balance/dilute-winding-crossover.md @@ -0,0 +1,513 @@ +# A dilute winding crossover, before the fixed-p sewing theorem + +2026-09-13. Continuation of the same geometric-balance paper (#739). +This note does not redo #740's literature investigation or #741's six stationary +computations. It gives a different, directly proved limit of the **actual NN +site model**, and a separate finite-state renewal lemma useful for interpreting +the pending sewing calculation. No fixed-p site Ornstein--Zernike prefactor, +critical exponent, new threshold value or novelty certification is asserted. + +## 1. The actual-site result + +On the infinite cylinder C_w x Z, independently occupy each vertex with +probability p. Count each complete component having nonzero horizontal +homology once, and let nu_w(p) be its mean count per vertical row. It is the +same density as in `winding-intensity-and-prefactor.md`, not a no-winding +survival exponent or a number of paths. In the range used below, 3p<1 already +ensures finite clusters by elementary path counting. Anchoring components at +their lowest row is one possible definition of the stationary density. + +**Theorem 1 (uniform dilute crossover).** For w>=6 and 0=0) x^(2r)/(r!)^2 + = (1/2pi) integral_(-pi)^pi exp(2x cos t) dt. (2) + +The constants in (1) are deliberately conservative. In particular, if +w->infinity, p=p_w>0 and w*p_w^2->0, then + + nu_w(p_w) = p_w^w I_0(2w p_w) (1+o(1)). (3) + +This includes three regimes, rather than presupposing a fixed Gaussian +amplitude: + +- wp->0: nu_w/p^w -> 1, almost straight winding; +- wp->lambda in (0,infinity): nu_w/p^w -> I_0(2lambda); +- wp->infinity while wp^2->0: + + nu_w(p) = p^w exp(2wp)/sqrt(4pi wp) + * [1+O(wp^2 + 1/(wp) + w(3p)^(w-1))]. (4) + +Equation (4) describes a genuine NN-site overlap regime, for example +p=w^(-3/4). It does **not** say that the fixed-p mass equals -log p-2p: +terms of size wp^2 are precisely what stop being negligible at fixed p. +No interchange of those limits is made. + +Only the standard Bessel identity and its large-real-argument expansion are +imported [B1,B2]. The component-versus-cycle bounds proving (1) are below; +there is no percolation OZ input and no surrogate renewal law in Theorem 1. + +## 2. An upper bound from unrooted positive-winding walks + +Every finite essential NN component in an annulus contains a simple essential +cycle. Such a cycle has winding +1 or -1, not a higher primitive winding; +orient it positively. A simple cycle with n vertices is occupied with +probability p^n. Summing over simple cycles therefore upper-bounds the +component count. Replacing simple cycles by all walks with lift displacement +(w,0), still assigning weight p^n, is a further positive upper bound. +Repeated walks are **not** assigned their actual occupancy probability here. +They are merely extra positive terms in a generating series bounding the +simple-cycle sum. + +The density of rooted walks is w times the count rooted at one prescribed +site per row. Dividing by n removes the n choices of root for every simple +cycle. Counting only displacement +w removes the orientation factor two. +Thus, absolutely convergently for p<1/4, + + nu_w(p) <= U_w(p) + := w [z^w y^0] sum_(n>=1) p^n/n + (z+z^(-1)+y+y^(-1))^n. (5) + +The longitudinal density statement can equivalently be obtained by counting +in a finite-height cylinder, dividing by height, then sending the height to +infinity. Subcritical finite moments dispose of the end effects. This is +mass transport for marked cycle vertices, not a percolation independence +assumption. + +Put b(t)=1-2p cos t and + + t_p(t) = 2p / [b(t)+sqrt(b(t)^2-4p^2)]. (6) + +Factoring + + 1-2p cos t-p(z+z^(-1)) + = (p/t_p)(1-t_p z)(1-t_p z^(-1)) + +gives the exact walk-series evaluation + + U_w(p)=(1/2pi) integral_(-pi)^pi t_p(t)^w dt. (7) + +For 0=3k, k>=1, + S(w,k)=0, w<3k. (9) + +This is the usual cyclic-gap count: distinguish a chosen column, subtract +two mandatory vacant gaps per chosen column, and then remove the k choices +of distinguished column. Consequently the number of cycles with 2r vertical +steps per unit height is + + N_r(w)=S(w,2r) * binom(2r,r). (10) + +A form more convenient for a uniform bound follows without any factorial +approximation. Draw k labelled independent uniform columns. A given pair +is at cyclic distance 0,1 or 2 with probability 5/w (w>=6). The union bound +shows + + S(w,k) k! / w^k >= 1 - 5k(k-1)/(2w). + +For k=2r this yields, for all r including those for which N_r=0, + + N_r(w) >= w^(2r)/(r!)^2 * (1-10r^2/w). (11) + +A negative right side causes no difficulty. + +### 3.2 Decorations are allowed; extra essential cycles are not + +Requiring the entire outside boundary of each cycle to be closed would +insert an artificial factor approximately exp(-2wp), losing the crossover. +Instead, condition only on its n=w+2r sites being occupied, allow decorations, +and subtract an upper bound on the probability that another essential cycle +lies in the same complete component. + +Call C good when it is the unique essential simple cycle of its component. +Then good cycles in a configuration are counted at most once per component. +Sufficient conditions for goodness are: + +(i) no off-C component has two or more attachment edges to C; and +(ii) no off-C component attached to C itself has essential winding. + +Condition (i) includes different attachment edges landing at the SAME cycle +vertex. Omitting that possibility would miss a second cycle attached at one +vertex. It does not forbid a tree or contractible loop attached by a single +edge. + +Here is a union bound conditional on C being occupied; outside sites remain +independent Bernoulli(p). + +**One-site return.** An off-C vertex adjacent to two C vertices is a missing +corner at a turn. A straight triple cannot cause one; and a return of the +height within two columns is excluded by the separation condition. There +are 4r turns, so at most 4r such sites. We use the looser bound 16r p. +The same classification follows by inspecting the isolated up-step and +down-step three-column patterns, not from numerical extrapolation. + +**Longer exterior return.** Start at a C-to-outside edge (at most 4n choices) +and follow a self-avoiding off-C path with k>=2 vertices that has a further +attachment to C. Ignoring the final attachment restriction only increases +the count. There are at most 4n*3^(k-1) candidate paths, hence total +conditional probability at most + + sum_(k>=2) 4n*3^(k-1)*p^k = 12n p^2/(1-3p). + +This also covers two attachments at the same C vertex when their exterior +endpoints differ. Every multiply attached off-C component has a shortest +path between two of its attachment vertices and is detected this way. + +**An attached off-C winding component.** Such a component contains an +essential simple cycle of at least w vertices. From an off-C neighbour of +C, a shortest path to that cycle followed almost once around it gives a +self-avoiding off-C path with at least w vertices. Summing these paths and +using the 4n possible initial attachments is bounded by + + 16n (3p)^(w-1). + +This covers an essential cycle hanging from C through a stem, even though +there is only one attachment edge. These are distinct reasons for failure; +none is assumed independent of the others. + +We have proved the finite lower bound + + Pr(C good | C occupied) >= 1-e_r, + e_r=16r p + 12(w+2r)p^2/(1-3p) + +16(w+2r)(3p)^(w-1). (12) + +Replacing 1-e_r by its positive part is valid. Summing good-cycle intensities +therefore gives the computable bound + + nu_w(p)/p^w >= + sum_(0<=r<=floor(w/6)) N_r(w) p^(2r) (1-e_r)_+. (13) + +The sum in (13) is a rigorous finite lower bound, not an estimate from +independent cycle occurrences. The uniqueness condition is precisely what +makes the sum safe in the presence of overlaps. + +### 3.3 Summing without assuming a bounded number of vertical steps + +Write lambda=wp, A_r=lambda^(2r)/(r!)^2. Equation (2) gives + + sum A_r=I_0(2lambda), + sum r^2 A_r=lambda^2 I_0(2lambda), + sum r A_r<=lambda I_0(2lambda). (14) + +The last inequality is Cauchy--Schwarz. Since N_r p^(2r)<=A_r, (11) loses at +most 10lambda^2/w times I_0. For the actual separated cycles, r<=w/6 and +w+2r<=4w/3. Equations (12)--(14) then show that the decoration subtraction +costs at most + + [16wp^2 + 16wp^2/(1-3p) + +(64/3)w(3p)^(w-1)] I_0(2lambda). + +For p<=1/8 the sum of these losses is less than the conservative error +128wp^2+64w(3p)^(w-1) in (1). This proves the lower bound for all stated +w,p, not merely for fixed lambda. In particular the error tends to zero when +wp^2->0, even when wp grows without bound. This completes Theorem 1. + +## 4. An exact finite-width series and its interpretation + +For every fixed integer w>=3, + + nu_w(p)=p^w [1+w(w-3)p^2+O_w(p^3)], p downarrow 0. (15) + +The coefficient of p^(w+1) is zero, rather than a negative perimeter term. +One proof counts the difference between complete winding components and +fully occupied rows. Fully occupied rows have mean density exactly p^w. +Components containing only one such row make zero contribution to the +difference regardless of tree decorations. Merging two full rows needs at +least 2w occupied vertices. A winding component without a full row needs at +least w+2 vertices: an essential simple NN cycle has w net horizontal steps +and must make at least two vertical steps if it leaves a row. + +At size w+2 the cycle has exactly two vertical steps, on two adjacent rows, +and no negative horizontal step. Its two horizontal arc lengths are r and +w-r. Avoiding a full row requires 2<=r<=w-2, giving w(w-3) sets per unit +height. Each is an induced cycle. For w=3 the family is empty and the stated +coefficient is zero. For w=2, 2w=w+2 and full-row mergers intervene: its +separate retained rational control gives nu_2=p^2-p^4+... . It is not covered +by (15). + +For fixed w the expansion is legitimate from the absolutely convergent +small-p cluster sum: the number of connected k-vertex sets rooted at a +specified site is at most exponential in k, and their external boundaries +have size at most 4k. No infinite-volume analytic continuation at pc is used. + +The older exact rational functions at w=3,4 reproduce (15). An independent +physical BFS over two-row fixed-occupation sets at w=3,...,8 verifies the +minimal no-full-row counts 0,4,10,18,28,40. These checks are finite controls +of the classification, not the reason (15) holds at every fixed width. + +For the mass-cancelling contrast in #741, + + R_w(p)=nu_w(p) nu_(3w)(p)/nu_(2w)(p)^2, + beta_eff=log R_w/log(4/3), + +(15) gives, at each fixed w>=3, + + log R_w(p)=2w^2p^2+O_w(p^3), + beta_eff ->0 as p downarrow 0. (16) + +Thus even the actual site model does not have beta_eff=1/2 uniformly in p +at a fixed finite set of widths. + +A sharper result follows from Theorem 1. If w->infinity and wp_w->lambda, +using the SAME p_w at w,2w,3w, + + beta_eff(w,p_w) -> B(lambda) + := log[I_0(2lambda) I_0(6lambda)/I_0(4lambda)^2] + /log(4/3). (17) + +B(lambda)=2lambda^2/log(4/3)+O(lambda^4) near zero. At infinity, + + B(lambda)=1/2+1/[48lambda log(4/3)]+O(lambda^(-2)). (18) + +Consequently B can overshoot 1/2; it is not a monotone interpolation from +zero to one half. Direct high-precision evaluations include + +| lambda | B(lambda) | +|---:|---:| +| 0.1 | 0.0653971727463187978 | +| 0.25 | 0.307660359689515652 | +| 0.5 | 0.602174810123850554 | +| 1 | 0.633453012816985342 | +| 4 | 0.520636654466422068 | + +These are limiting crossover values in (17), NOT computed stationary +intensities at widths 4,8,12. In particular NN p=1/4 in #741 is outside the +explicit small-p estimate (1); no numerical prediction for that task is +manufactured from this table. The task should still return its original six +values and achieved errors. + +### 4.1 Why the matching graph already differs in its leading small-p term + +For each fixed width, the previously classified minimal matching cycles give + + nu_w^8(p)=c_w p^w+O_w(p^(w+1)), + c_w=[y^0](1+y+y^(-1))^w. + +At minimal size w there is one vertex in each column, each forward step has +vertical displacement -1,0,+1, and the displacements sum to zero. Addressing +the cycle by its column-zero height counts it once per vertical row. This +reuses the existing central-trinomial minimal-pattern classification; it is +not the NN dilute theorem (1) applied to diagonal edges. + +At the exact fixed-width limit p downarrow 0, + + R_4^8 -> (19*73789)/1107^2 = 1401991/1225449, + beta_eff^8(4,p) -> 0.4678291580386769565... . + +The NN limit at the same three widths is instead zero. Neither is the +finite-p value assigned in #741, and no missing nu_12 was computed here. +The p and width limits must be distinguished on both graphs. + +## 5. Internal-state renewals: the variance entering a sewn loop + +This section is a SEPARATE explicit renewal theorem, not a claimed finite-state +representation of actual percolation irreducible clusters. It strengthens the +scalar coefficient calculation already supplied to #740. + +Let A(z,y) be a finite matrix with nonnegative finite-support coefficients +a_ij(x,y), positive integer forward lengths x, and integer transverse increments +y. Suppose its spectral radius reaches one at (R,1), R>1, with a simple +Perron eigenvalue. Write right and left eigenvectors r,l, normalized l^T r=1. +The tilted edge kernel + + q_ij(x,y)=a_ij(x,y) R^x r_j/r_i + +is stochastic after summing x,j. Let P be its state transition matrix and +pi_i=l_i r_i its invariant law. Assume P primitive; zero stationary transverse +drift; reflection symmetry, possibly involving a permutation of states; and +that det(I-A(R exp(it), exp(i theta))) has no zero on the unit two-torus +except (t,theta)=(0,0). These are explicit lattice-span/spectral assumptions. +A nonzero local second moment alone is not a diffusion hypothesis. + +Define Q_k(i,j)=sum_(x,y) q_ij(x,y)y^k and mu=E_pi X. Solve + + (I-P)h=Q_1 1, pi h=0. + +Then the asymptotic transverse variance per renewal and per forward length are + + sigma_eff^2=pi Q_2 1+2pi Q_1 h, + D=sigma_eff^2/mu. (19) + +The second term retains serial correlation carried by the internal state. +Equivalently sigma_eff^2 is the stationary mean of +(y+h_j-h_i)^2. This follows by decomposing the transverse additive functional +into a martingale plus the telescoping h boundary term. In particular it is +nonnegative; it vanishes precisely when every allowed increment is the +coboundary y=h_i-h_j. Require D>0 below. + +For the explicitly defined loop object + + L_w = w[z^w y^0]{-log det(I-A(z,y))}, + +one obtains + + L_w=R^(-w)/sqrt(2pi D w) * (1+O(1/w)). (20) + +**Proof.** Apply z d/dz to the logarithm. The singularity of +tr[(I-A)^(-1) z A_z] at its unique simple Perron root R(theta) has principal +part z/(R(theta)-z); the left/right eigenvector derivative cancels. The pole +coefficient is exactly one, not the number of internal states. Differentiating +the Perron equation gives + + log R(theta)=log R+(D/2)theta^2+O(theta^4). + +Here the second derivative of the Perron log eigenvalue is the asymptotic +variance (19), not just E Y^2. The spectral unit-torus condition controls all +other phases exponentially. Fourier inversion and Laplace integration yield +(20), including the stated O(1/w) for finite support and reflection symmetry. + +An open resolvent has a different normalization. For fixed vectors c,b, + + [z^w y^0] c^T(I-A)^(-1)b + ~ (c^T r)(l^T b)/mu + * R^(-w)/sqrt(2pi D w), (21) + +when its displayed amplitude is nonzero. This is why a two-point endpoint +amplitude cannot simply be declared the cyclic component amplitude. An actual +site sewing theorem must first identify the correct object, boundary weights +and cut multiplicity. A nontrivial cyclic weight can change even the leading +amplitude. None is removed by naming the Perron root. + +### 5.1 An exact two-state correlated control + +Take P=[[3/4,1/4],[1/4,3/4]]. Independently choose X=1 or 2 with probability +one half each. Conditional on arrival in state + or -, choose Y=0 with +probability one half, otherwise Y=+1 or -1 respectively. Tilt back with R=2. +Thus + + A(z,y)=(z/4+z^2/8) P diag((1+y)/2,(1+y^(-1))/2). + +The stationary one-step variance is 1/2. The lag-k covariance is +(1/4)(1/2)^k, so sigma_eff^2=1, mu=3/2 and D=2/3. Ignoring state correlations +would instead give D=1/3 and the wrong leading amplitude by a factor sqrt(2). + +The determinant simplifies exactly: + + det(I-A(2z,y)) + =1+(2+y+y^(-1))(-6z-5z^2+2z^3+z^4)/32. (22) + +Coefficient recursion from (22) gives the whole exact loop sequence. An +independent direct matrix trace sum agrees through width ten. Near zero, + + log R(theta)=log2+theta^2/3+7theta^4/54+O(theta^6), + +so in this specified model + + L_w=2^(-w)/sqrt((4pi/3)w) * [1-7/(8w)+O(w^(-2))]. (23) + +Its beta_eff at width 4 has no obligation to be near one half. The exact +control approaches from below, whereas the actual dilute-site crossover +(17) can lie above one half. These are reasons to distinguish finite-width +corrections, not excuses to disregard a contradictory returned computation. + +For clarity, memory can also kill diffusion altogether: with the same P, +X=1 and Y=g(i)-g(j) for two different g values, the single-step variance is +positive but every closed state cycle has transverse sum zero. Then D=0 and +L_w=R^(-w)tr(P^w), with no w^(-1/2) factor. That control violates the explicit +nondegeneracy assumptions of (20), not the theorem. + +## 6. What changes in #740 and #741, without expanding either task + +The fixed-p site sewing remains open in this delivery. Theorem 1 does not +supply the prefactor at NN p=1/4 or matching p=1/8; it identifies a directly +proved nonuniform dilute regime of the same NN-site model. Equation (20) +does not identify the actual site renewal state with a finite matrix. + +The additions useful to the pending work are concrete: + +1. A site prefactor claimed uniformly toward p=0 must reproduce (1)--(4), + or explicitly exclude that regime. A fixed-p theorem need not be uniform. +2. A multistate sewing uses the long-run transverse variance (19), the actual + lattice span and a specified closure, not an iid step variance and a + borrowed endpoint amplitude. +3. Finite R_4 is always meaningful, but one triple cannot separate beta from + corrections. If log nu=log A-kappa*w-beta log w+c1/w+O(w^(-2)), then + + beta_eff=beta+c1/[3w log(4/3)]+O(w^(-2)). (24) + + Numerical errors in log nu at w,2w,3w bounded by e1,e2,e3 give beta error + at most (e1+2e2+e3)/log(4/3). This is numerical error, not the correction + in (24). With each error 1e-8 the former is about 1.3904e-7. + +Do not enlarge the external width plan on the basis of this note. First obtain +its already requested values and the site-sewing result/obstruction. + +## 7. Executed scope + +The companion script independently traverses physical NN cylinders for: + +- 43,743 fixed-occupation two-row sets at widths 3,...,8, checking the + no-full-row minimum and first counts in (15); +- 2,443 separated cycles and 27 cyclic-gap families, checking inducedness, + nonzero winding and short exterior contacts; +- an exact matrix-logdet coefficient recurrence against independent matrix + trace expansion through ten forward lengths; +- retained NN rational density inputs at widths 2,3,4, and exact Taylor + coefficients. Those rational inputs are reused from the preceding delivery, + not presented as new stationary calculations. + +The displayed integrals, Bessel ratios and Gaussian amplitude controls use +mpmath, with a higher-precision rerun. They are not interval quadrature. +The finite-sum lower bound (13) itself is computed as a Fraction; comparisons +to its Bessel normalization are displayed numerically. The proof, not the +finite checks, supplies Theorem 1. Thirteen local tests pass; full repository CI +has not run. No Monte Carlo, no new stationary width-eight/twelve run and no +new external issue are required by this continuation. + +Commands: + + python -m unittest discover -s tests -p 'test_winding_dilute_crossover.py' -v + python scripts/winding_dilute_crossover.py --output /tmp/dilute-new.json + +The report refuses an existing output path. The script has no dependency on +an unmerged state table; mpmath is needed for numerical report generation. + +## Sources and originality boundary + +[B1] NIST Digital Library of Mathematical Functions, 10.32.1: +https://dlmf.nist.gov/10.32.E1 . The I_0 integral, not a percolation theorem. +[B2] NIST DLMF 10.40.1: +https://dlmf.nist.gov/10.40.E1 . Large positive argument expansion of I_nu. +Both primary reference pages were read this delivery. + +[CIV] Campanino--Ioffe--Velenik, *Fluctuation theory of connectivities for +subcritical random cluster models*, Ann. Probab. 36 (2008), 1287--1321; +https://www.unige.ch/math/folks/velenik/papers/abs_CIV08.html . Author abstract +read this delivery, not a re-audit of the theorem body. It establishes the +context of irreducible chains/effective walks, not the SITE cylinder sewing +requested in #740. Matrix-additive variance and Gaussian coefficient methods +are standard; the explicit calculations here are not a claim to invent them. +No systematic prior-art search for Theorem 1 has been completed in this round. diff --git a/results/geometric-consistency/dilute-winding-crossover.json b/results/geometric-consistency/dilute-winding-crossover.json new file mode 100644 index 00000000..ea4850ef --- /dev/null +++ b/results/geometric-consistency/dilute-winding-crossover.json @@ -0,0 +1,277 @@ +{ + "schema": "matching-one/dilute-prefactor/v1", + "scope": "NN SITE dilute double limit plus an explicitly separate matrix-renewal control; no fixed-p OZ theorem", + "geometry_checks": { + "cycles": 2443, + "gap_subset_checks": 27, + "max_contact_over_r": 4.0 + }, + "minimal_nonrow_census": [ + { + "width": 3, + "tested_fixed_size_sets": 41, + "nonrow_winding_counts": { + "3": 0, + "4": 0, + "5": 0 + }, + "predicted_first_count": 0 + }, + { + "width": 4, + "tested_fixed_size_sets": 154, + "nonrow_winding_counts": { + "4": 0, + "5": 0, + "6": 4 + }, + "predicted_first_count": 4 + }, + { + "width": 5, + "tested_fixed_size_sets": 582, + "nonrow_winding_counts": { + "5": 0, + "6": 0, + "7": 10 + }, + "predicted_first_count": 10 + }, + { + "width": 6, + "tested_fixed_size_sets": 2211, + "nonrow_winding_counts": { + "6": 0, + "7": 0, + "8": 18 + }, + "predicted_first_count": 18 + }, + { + "width": 7, + "tested_fixed_size_sets": 8437, + "nonrow_winding_counts": { + "7": 0, + "8": 0, + "9": 28 + }, + "predicted_first_count": 28 + }, + { + "width": 8, + "tested_fixed_size_sets": 32318, + "nonrow_winding_counts": { + "8": 0, + "9": 0, + "10": 40 + }, + "predicted_first_count": 40 + } + ], + "independent_matrix_trace_coefficients": [ + "3/8", + "107/128", + "291/1024", + "9427/32768", + "49329/262144", + "736831/4194304", + "5138027/33554432", + "314325251/2147483648", + "2382293829/17179869184", + "36896778821/274877906944" + ], + "numerics": { + "small_width_retained_site_controls": [ + { + "width": 2, + "p": "1/32", + "nu_exact": "1015777/1041235968", + "nu_over_p_to_w": "0.99896246380916414904330312", + "bessel": "1.0039100663533544851278467", + "ratio": "0.99507166756265108739969013" + }, + { + "width": 2, + "p": "1/64", + "nu_exact": "16515009/67662512128", + "nu_over_p_to_w": "0.99974823187221051326555914", + "bessel": "1.0009768009444507521251346", + "ratio": "0.99877262982410676433877674" + }, + { + "width": 2, + "p": "1/128", + "nu_exact": "266338177/4363955208192", + "nu_over_p_to_w": "0.99993801122809597404195116", + "bessel": "1.0002441555265654198830768", + "ratio": "0.9996939304301075568999033" + }, + { + "width": 3, + "p": "1/32", + "nu_exact": "34050535322719/1115805188582539264", + "nu_over_p_to_w": "0.99996661861043110149987111", + "bessel": "1.0088083932745527185892953", + "ratio": "0.99123542714050825726667325" + }, + { + "width": 3, + "p": "1/64", + "nu_exact": "17728526576767167/4647445430768219717632", + 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"matching_fixed_width_low_p_limit_not_finite_p_data": { + "c4": 19, + "c8": 1107, + "c12": 73789, + "R4_exact": "1401991/1225449", + "beta_effective_limit": "0.46782915803867695654419993" + }, + "proved_bound_numerical_controls_not_density_estimates": [ + { + "width": 64, + "p": "1/256", + "lambda": "0.25", + "finite_lower_over_bessel": "0.97976236544816369011344071", + "walk_upper_over_bessel": "1.0019903375367301483087813", + "universal_lower_factor": "0.875", + "universal_upper_factor": "1.005876574714478322766683" + }, + { + "width": 256, + "p": "1/256", + "lambda": "1.0", + "finite_lower_over_bessel": "0.88818847544168495401452164", + "walk_upper_over_bessel": "1.009103052658314946935824", + "universal_lower_factor": "0.5", + "universal_upper_factor": "1.0237143166023579169688505" + }, + { + "width": 1024, + "p": "1/1024", + "lambda": "1.0", + "finite_lower_over_bessel": "0.97128740046313902746753251", + "walk_upper_over_bessel": "1.0022550698722570539788439", + "universal_lower_factor": "0.875", + "universal_upper_factor": "1.005876574714478322766683" + }, + { + "width": 1024, + "p": "1/256", + "lambda": "4.0", + "finite_lower_over_bessel": "0.51066693893278599483236007", + "walk_upper_over_bessel": "1.044565976499507742163823", + "universal_lower_factor": "-1.0", + "universal_upper_factor": "1.0982851403078258486502099" + } + ], + "matrix_renewal_controls_not_site_model": { + "mean_forward_length": "3/2", + "single_step_transverse_variance": "1/2", + "asymptotic_variance_per_renewal": "1", + "D": "2/3", + "naive_D_ignoring_correlations": "1/3", + "sequence": [ + { + "width": 8, + "normalized_gaussian_amplitude": "0.8473029209783016747029948", + "beta_effective": "0.2889944274690882503308101" + }, + { + "width": 16, + "normalized_gaussian_amplitude": "0.92847341407200684608741177", + "beta_effective": "0.38846535391804191114872381" + }, + { + "width": 32, + "normalized_gaussian_amplitude": "0.96956080304801010351813996", + "beta_effective": "0.46026688247931377606194586" + }, + { + "width": 64, + "normalized_gaussian_amplitude": "0.98563441495658425494014059", + "beta_effective": "0.48241781188732095533592058" + }, + { + "width": 128, + "normalized_gaussian_amplitude": "0.99299850301651533865874337", + "beta_effective": "0.49167011031188536311776649" + } + ] + }, + "log_nu_absolute_error_to_beta_error_multiplier": "13.904237987128827641505998" + } +} diff --git a/scripts/winding_dilute_crossover.py b/scripts/winding_dilute_crossover.py new file mode 100644 index 00000000..666c286c --- /dev/null +++ b/scripts/winding_dilute_crossover.py @@ -0,0 +1,341 @@ +#!/usr/bin/env python3 +"""Dilute NN-site winding density: exact combinatorics and analytic controls. + +The all-width theorem is in dilute-winding-crossover.md. No large-width +stationary density or Monte Carlo is calculated here. Small-width rational +functions below are retained inputs from the preceding intensity delivery. +Python stdlib for combinatorial tests; mpmath for report numerics. +""" +from __future__ import annotations +import argparse +from collections import Counter, deque +from fractions import Fraction +from itertools import combinations, product +from math import comb +from pathlib import Path +import json + +SMALL_INTENSITIES = {'2': {'expression': 'p**2*(p - 1)**2*(p**2 + p + 1)/(p**2 - p + 1)', 'numerator_descending': [1, -1, 0, -1, 1, 0, 0], 'denominator_descending': [1, -1, 1], 'low_p_through_8': 'p**2 - p**4 - 2*p**5 + 2*p**7 + 2*p**8 + O(p**9)'}, '3': {'expression': '-p**3*(p - 1)**3*(p**6 + p**3 + 2*p**2 + 2*p + 1)/(p**6 - 3*p**5 + 3*p**4 + p**3 - p**2 - p + 1)', 'numerator_descending': [-1, 3, -3, 0, 1, 1, 0, -1, -1, 1, 0, 0, 0], 'denominator_descending': [1, -3, 3, 1, -1, -1, 1], 'low_p_through_8': 'p**3 - p**6 - 3*p**7 + O(p**9)'}, '4': {'expression': 'p**4*(p - 1)**4*(p**19 - 5*p**18 + 10*p**17 - 8*p**16 - 3*p**15 + 12*p**14 - 14*p**13 + 14*p**12 - 8*p**11 - 5*p**10 + 3*p**9 + 3*p**8 + 3*p**7 + 4*p**6 - 3*p**5 - 3*p**4 - 9*p**3 - 7*p**2 - 3*p - 1)/((p**2 - p - 1)*(p**2 - p + 1)*(p**15 - 7*p**14 + 21*p**13 - 33*p**12 + 25*p**11 - 2*p**10 - 8*p**9 + 3*p**8 - p**7 - 2*p**6 + 6*p**5 - 4*p**4 + 4*p**3 - 2*p**2 - p + 1))', 'numerator_descending': [1, -9, 36, -82, 110, -69, -38, 146, -199, 179, -95, 7, 21, -7, -10, 24, -28, 27, -20, 2, 5, -1, 1, -1, 0, 0, 0, 0], 'denominator_descending': [1, -9, 36, -82, 111, -78, 0, 50, -40, 5, 17, -21, 19, -12, 1, 5, -7, 3, 1, -1], 'low_p_through_8': 'p**4 + 4*p**6 - 8*p**7 + 7*p**8 + O(p**9)'}} + + +def neighbours(v: tuple[int, int], w: int): + x,y=v + return [((x+1)%w,y), ((x-1)%w,y), (x,y+1), (x,y-1)] + + +def winding_components(vertices: set[tuple[int,int]], w: int) -> int: + """Independent physical lifted-coordinate BFS on a free-height cylinder.""" + seen: dict[tuple[int,int], tuple[int,int]]={} + count=0 + for root in sorted(vertices): + if root in seen: + continue + seen[root]=(0,0) + todo=[root]; winds=False + while todo: + v=todo.pop(); x,y=v; vx,vy=seen[v] + for dx,dy in ((1,0),(-1,0),(0,1),(0,-1)): + t=((x+dx)%w,y+dy) + if t not in vertices: + continue + value=(vx+dx,vy+dy) + if t not in seen: + seen[t]=value;todo.append(t) + else: + error=(value[0]-seen[t][0],value[1]-seen[t][1]) + if error[0]: + assert error[0]%w==0 and error[1]==0 + winds=True + count += int(winds) + return count + + +def gap_subsets(w: int, k: int): + """Subsets with cyclic distance at least three between selected columns.""" + if w<6 or k<0: + raise ValueError('require w>=6 and k>=0') + for c in combinations(range(w),k): + if k<=1 or all((c[(i+1)%k]-c[i])%w>=3 for i in range(k)): + yield c + + +def gap_count(w: int, k: int) -> int: + if k==0: + return 1 + if w<3*k: + return 0 + return w*comb(w-2*k,k)//(w-2*k) + + +def separated_cycles(w: int, r: int): + """Each positive winding cycle is addressed by its incoming height at x=0.""" + for cols in gap_subsets(w,2*r): + for ups in combinations(cols,r): + up=set(ups); events={x:(1 if x in up else -1) for x in cols} + y=0; vset=set() + for x in range(w): + vset.add((x,y)) + if x in events: + y += events[x] + vset.add((x,y)) + assert y==0 + yield vset + + +def short_external_contacts(cycle: set[tuple[int,int]], w: int) -> int: + """Off-cycle vertices touching >=2 cycle vertices (one-site returns).""" + contacts=Counter() + for v in cycle: + for t in neighbours(v,w): + if t not in cycle: + contacts[t]+=1 + return sum(n>=2 for n in contacts.values()) + + +def minimal_nonrow_census(max_width: int=8) -> list[dict]: + """Only fixed occupation numbers in TWO free rows, not a full-size scan.""" + results=[] + for w in range(3,max_width+1): + points=[(x,y) for y in (0,1) for x in range(w)] + tested=0; hits={} + for k in (w,w+1,w+2): + n=0 + for subset in combinations(points,k): + tested+=1; s=set(subset) + if any(all((x,y) in s for x in range(w)) for y in (0,1)): + continue + n += int(winding_components(s,w)>0) + hits[k]=n + assert hits[w]==0 and hits[w+1]==0 + assert hits[w+2]==w*(w-3) + results.append({'width':w,'tested_fixed_size_sets':tested, + 'nonrow_winding_counts':hits,'predicted_first_count':w*(w-3)}) + return results + + +def cycle_geometry_checks(max_width: int=16) -> dict: + totals={'cycles':0,'gap_subset_checks':0,'max_contact_over_r':0} + for w in range(6,max_width+1): + for r in range(0,min(2,w//6)+1): + k=2*r + subsets=list(gap_subsets(w,k)) + assert len(subsets)==gap_count(w,k) + # The probability that k labelled iid columns are separated: + # union bound over pairs at cyclic distance 0,1,2. + from math import factorial + assert Fraction(len(subsets)*factorial(k),w**k)>=1-Fraction(5*k*(k-1),2*w) + totals['gap_subset_checks']+=1 + n=0 + for c in separated_cycles(w,r): + assert len(c)==w+2*r + assert winding_components(c,w)==1 + # An induced simple cycle: no occupied chords. + assert all(sum(t in c for t in neighbours(v,w))==2 for v in c) + z=short_external_contacts(c,w) + assert z<=16*r + if r: + totals['max_contact_over_r']=max(totals['max_contact_over_r'],z/r) + n+=1 + assert n==gap_count(w,k)*comb(k,r) + totals['cycles']+=n + return totals + + +def evaluate_small_nu(w: int, p: Fraction) -> Fraction: + s=SMALL_INTENSITIES[str(w)] + def horner(a): + v=Fraction(0) + for x in a: v=v*p+x + return v + return horner(s['numerator_descending'])/horner(s['denominator_descending']) + + +def rational_series(w: int, degree: int=10) -> list[Fraction]: + s=SMALL_INTENSITIES[str(w)] + num=list(reversed(s['numerator_descending'])) + den=list(reversed(s['denominator_descending'])) + out=[] + for n in range(degree+1): + value=Fraction(num[n] if n Fraction: + if w<6 or not 0=6, 0 Fraction: + """Rigorous finite sum / p**w. May be weak, never a point estimate.""" + answer=Fraction(0) + for r in range(w//6+1): + n=w+2*r + failure=16*r*p+12*n*p*p/(1-3*p)+16*n*(3*p)**(w-1) + answer+=gap_count(w,2*r)*comb(2*r,r)*p**(2*r)*max(Fraction(0),1-failure) + return answer + + +def central_trinomial(w: int) -> int: + return sum(comb(w,2*k)*comb(2*k,k) for k in range(w//2+1)) + + +def mp_number(x,mp): + if isinstance(x,Fraction): return mp.mpf(x.numerator)/x.denominator + return mp.mpf(x) + + +def walk_upper_normalized(w: int, p, mp): + """Upper bound U_w/p^w; integral is diagnostic, not interval quadrature.""" + p=mp_number(p,mp) + def value(theta): + b=1-2*p*mp.cos(theta) + t_over_p=2/(b+mp.sqrt(b*b-4*p*p)) + return mp.exp(w*mp.log(t_over_p)) + return mp.quad(value,[0,mp.pi/2,mp.pi])/mp.pi + + +def bessel_contrast(lam,mp): + lam=mp.mpf(lam) + return (mp.log(mp.besseli(0,2*lam))+mp.log(mp.besseli(0,6*lam)) + -2*mp.log(mp.besseli(0,4*lam)))/mp.log(mp.mpf(4)/3) + + +def matrix_loop_coefficients(max_n: int) -> list[Fraction]: + """Exact 2^n L_n for the correlated two-state renewal control. + + det(I-A(2z,y))=1+(2+y+y^-1)*(-6z-5z²+2z³+z⁴)/32. + Scale coefficients by 32^n to use integers throughout. + """ + a={1:-6,2:-5,3:2,4:1} + polys=[{}] + ans=[Fraction(0)] + for n in range(1,max_n+1): + c={} + if n<=4: + base=-n*a[n]*32**(n-1) + c={-1:base,0:2*base,1:base} + for k in range(1,min(n-1,4)+1): + scale=-a[k]*32**(k-1) + for j,v in polys[n-k].items(): + for d,b in ((-1,1),(0,2),(1,1)): + c[j+d]=c.get(j+d,0)+scale*b*v + c={k:v for k,v in c.items() if v} + assert all(v>=0 for v in c.values()) + polys.append(c) + ans.append(Fraction(c.get(0,0),32**n)) + return ans + + +def direct_matrix_loops(max_n: int=10) -> list[Fraction]: + """Independent matrix-of-Laurent-polynomials trace expansion through n. + Kernel below is the R-tilted kernel; return 2^w L_w. + """ + P=((Fraction(3,4),Fraction(1,4)),(Fraction(1,4),Fraction(3,4))) + A={} + for i,j,x,dy in product(range(2),range(2),(1,2),(0,1)): + y=(1 if j==0 else -1)*dy + A[(i,j,x,y)]=P[i][j]/4 + power={(i,i,0,0):Fraction(1) for i in range(2)} + result=[Fraction(0) for _ in range(max_n+1)] + for k in range(1,max_n+1): + nxt={} + for (i,j,x,y),v in power.items(): + for (a,b,dx,dy),u in A.items(): + if j!=a or x+dx>max_n:continue + key=(i,b,x+dx,y+dy) + nxt[key]=nxt.get(key,Fraction(0))+v*u + power=nxt + for (i,j,x,y),v in power.items(): + if i==j and y==0: + result[x]+=Fraction(x,k)*v + return result + + +def numerical_report(dps: int=70): + import mpmath as mp + mp.mp.dps=dps + fmt=lambda x:mp.nstr(x,26) + # Exact retained densities, not newly computed large-width site data. + small=[] + for w in (2,3,4): + for p in (Fraction(1,32),Fraction(1,64),Fraction(1,128)): + n=evaluate_small_nu(w,p) + lam=mp_number(p,mp)*w + val=mp_number(n/p**w,mp) + small.append({'width':w,'p':str(p),'nu_exact':str(n), + 'nu_over_p_to_w':fmt(val),'bessel':fmt(mp.besseli(0,2*lam)), + 'ratio':fmt(val/mp.besseli(0,2*lam))}) + curves=[] + for lam in ('0.05','0.1','0.25','0.5','1','2','4','10','30'): + curves.append({'lambda':lam,'beta_crossover':fmt(bessel_contrast(lam,mp))}) + bounds=[] + for w,p in ((64,Fraction(1,256)),(256,Fraction(1,256)), + (1024,Fraction(1,1024)),(1024,Fraction(1,256))): + lam=mp_number(p,mp)*w + b=mp.besseli(0,2*lam) + lower=finite_cycle_lower(w,p) + up=walk_upper_normalized(w,p,mp) + uf=universal_lower_factor(w,p) + assert mp_number(lower,mp)<=up + assert up/b<=mp.exp(6*w*mp_number(p,mp)**2) + assert mp_number(lower,mp)/b>=mp_number(uf,mp) + bounds.append({'width':w,'p':str(p),'lambda':fmt(lam), + 'finite_lower_over_bessel':fmt(mp_number(lower,mp)/b), + 'walk_upper_over_bessel':fmt(up/b), + 'universal_lower_factor':fmt(mp_number(uf,mp)), + 'universal_upper_factor':fmt(mp.exp(6*w*mp_number(p,mp)**2))}) + coeff=matrix_loop_coefficients(384) + matrix=[] + D=mp.mpf(2)/3 + for w in (8,16,32,64,128): + v=mp_number(coeff[w],mp) + r=coeff[w]*coeff[3*w]/coeff[2*w]**2 + matrix.append({'width':w,'normalized_gaussian_amplitude':fmt(v*mp.sqrt(2*mp.pi*D*w)), + 'beta_effective':fmt(mp.log(mp_number(r,mp))/mp.log(mp.mpf(4)/3))}) + # Different diffusion constants for the identical one-step distribution. + c4,c8,c12=(central_trinomial(w) for w in (4,8,12)) + mr=Fraction(c4*c12,c8*c8) + return {'small_width_retained_site_controls':small,'bessel_contrast_curve':curves, + 'matching_fixed_width_low_p_limit_not_finite_p_data':{ + 'c4':c4,'c8':c8,'c12':c12,'R4_exact':str(mr), + 'beta_effective_limit':fmt(mp.log(mp_number(mr,mp))/mp.log(mp.mpf(4)/3))}, + 'proved_bound_numerical_controls_not_density_estimates':bounds, + 'matrix_renewal_controls_not_site_model':{ + 'mean_forward_length':'3/2','single_step_transverse_variance':'1/2', + 'asymptotic_variance_per_renewal':'1','D':'2/3', + 'naive_D_ignoring_correlations':'1/3','sequence':matrix}, + 'log_nu_absolute_error_to_beta_error_multiplier':fmt(4/mp.log(mp.mpf(4)/3))} + + +def build_report(dps: int=70): + geometry=cycle_geometry_checks() + census=minimal_nonrow_census() + a=matrix_loop_coefficients(10);b=direct_matrix_loops(10) + assert a==b + for w in (3,4): + series=rational_series(w,10) + assert series[w]==1 and series[w+1]==0 + assert series[w+2]==w*(w-3) + return {'schema':'matching-one/dilute-prefactor/v1', + 'scope':'NN SITE dilute double limit plus an explicitly separate matrix-renewal control; no fixed-p OZ theorem', + 'geometry_checks':geometry,'minimal_nonrow_census':census, + 'independent_matrix_trace_coefficients':[str(v) for v in a[1:]], + 'numerics':numerical_report(dps)} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output',type=Path,required=True) + parser.add_argument('--dps',type=int,default=70) + args=parser.parse_args() + if args.output.exists(): parser.error('refusing to overwrite an existing result') + if args.dps<40: parser.error('use at least 40 decimal digits') + result=build_report(args.dps) + args.output.parent.mkdir(parents=True,exist_ok=True) + args.output.write_text(json.dumps(result,indent=2,ensure_ascii=False)+'\n') + print(args.output) + +if __name__=='__main__': main() diff --git a/tests/test_winding_dilute_crossover.py b/tests/test_winding_dilute_crossover.py new file mode 100644 index 00000000..476d9e0a --- /dev/null +++ b/tests/test_winding_dilute_crossover.py @@ -0,0 +1,88 @@ +import importlib.util +from pathlib import Path +from fractions import Fraction as F +from math import comb, factorial +import unittest + +P=Path(__file__).resolve().parents[1]/'scripts/winding_dilute_crossover.py' +spec=importlib.util.spec_from_file_location('dilute',P) +m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m) + +class DiluteTests(unittest.TestCase): + def test_physical_winding_and_no_winding(self): + self.assertEqual(m.winding_components({(x,0) for x in range(6)},6),1) + self.assertEqual(m.winding_components({(x,0) for x in range(5)},6),0) + self.assertEqual(m.winding_components({(x,y) for x in range(6) for y in (0,2)},6),2) + def test_cyclic_gap_counts(self): + for w in range(6,13): + for k in range(0,min(4,w)+1): + self.assertEqual(len(list(m.gap_subsets(w,k))),m.gap_count(w,k)) + def test_collision_union_bound(self): + for w in range(6,65): + for k in range(0,8): + self.assertGreaterEqual(F(m.gap_count(w,k)*factorial(k),w**k), + 1-F(5*k*(k-1),2*w)) + def test_cycle_geometry(self): + x=m.cycle_geometry_checks(12) + self.assertGreater(x['cycles'],100) + self.assertLessEqual(x['max_contact_over_r'],16) + def test_first_nonrow_cycle_count(self): + for row in m.minimal_nonrow_census(6): + w=row['width'];self.assertEqual(row['nonrow_winding_counts'][w+2],w*(w-3)) + def test_retained_exact_small_width_series(self): + for w in (3,4): + a=m.rational_series(w,10) + self.assertEqual(a[w],1);self.assertEqual(a[w+1],0) + self.assertEqual(a[w+2],w*(w-3)) + self.assertEqual(m.evaluate_small_nu(2,F(1,2)),F(7,48)) + self.assertEqual(m.evaluate_small_nu(4,F(1,2)),F(323849,5576960)) + def test_finite_lower_probability_coefficients(self): + for w in (6,8,12): + p=F(1,128) + lower=m.finite_cycle_lower(w,p) + self.assertGreater(lower,0) + self.assertLess(lower,2) + self.assertGreater(m.universal_lower_factor(w,p),0) + def test_matrix_trace_is_independent_of_logdet_recurrence(self): + self.assertEqual(m.matrix_loop_coefficients(10),m.direct_matrix_loops(10)) + def test_matrix_diffusion_poisson_equation(self): + P=((F(3,4),F(1,4)),(F(1,4),F(3,4))) + Q1=tuple(tuple(P[i][j]*F(1 if j==0 else -1,2) for j in range(2)) for i in range(2)) + h=(F(1,2),F(-1,2));mu=F(3,2) + for i in range(2): + self.assertEqual(h[i]-sum(P[i][j]*h[j] for j in range(2)),sum(Q1[i])) + sigma=F(1,2)+sum(Q1[i][j]*h[j] for i,j in ((0,0),(0,1),(1,0),(1,1))) + self.assertEqual(sigma,1);self.assertEqual(sigma/mu,F(2,3)) + def test_mass_and_amplitude_cancellation(self): + # exact nu_w=A*t^w/w for beta=1 gives R=4/3. + for w in (4,7,12): + f=lambda n:F(7,5)*F(2,5)**n/n + self.assertEqual(f(w)*f(3*w)/f(2*w)**2,F(4,3)) + # An exact log-density c/w correction has coefficient c/(3w). + self.assertEqual(F(1,w)+F(1,3*w)-2*F(1,2*w),F(1,3*w)) + def test_bessel_control(self): + import mpmath as mp + with mp.workdps(50): + self.assertLess(m.bessel_contrast('0.1',mp),mp.mpf('.07')) + self.assertGreater(m.bessel_contrast('0.5',mp),mp.mpf('.60')) + x=mp.mpf('.7') + series=mp.fsum(x**(2*r)/factorial(r)**2 for r in range(50)) + self.assertLess(abs(series-mp.besseli(0,2*x)),mp.mpf('1e-45')) + moment=mp.fsum(r*r*x**(2*r)/factorial(r)**2 for r in range(50)) + self.assertLess(abs(moment-x*x*series),mp.mpf('1e-45')) + def test_matching_minimal_pattern_coefficient(self): + from itertools import product + for w in range(2,8): + n=sum(sum(steps)==0 for steps in product((-1,0,1),repeat=w)) + self.assertEqual(n,m.central_trinomial(w)) + self.assertEqual([m.central_trinomial(w) for w in (4,8,12)],[19,1107,73789]) + def test_uniform_walk_bound_numerically(self): + import mpmath as mp + with mp.workdps(45): + for w,p in ((6,F(1,128)),(64,F(1,256))): + a=m.walk_upper_normalized(w,p,mp) + lam=w*m.mp_number(p,mp);b=mp.besseli(0,2*lam) + self.assertLessEqual(a/b,mp.exp(6*w*m.mp_number(p,mp)**2)) + self.assertLessEqual(m.mp_number(m.finite_cycle_lower(w,p),mp),a) + +if __name__=='__main__': unittest.main() From e55fe252c075f65211ffc6bae521c8cec71a8cdc Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 17:01:13 +0800 Subject: [PATCH 10/17] Add sewing-with-memory: exact site weights, unbiased cuts, and the heat-kernel conjecture Seventh handoff for this branch; four new files, nothing overwritten. Owner handoff responding to the #740 and #741 returns. Contents: - Section 2: complete-component activity has a LOCAL three-column weight (columns t, t+1, t+2), with the allocation across columns shown to be a gauge choice (cyclic telescoping), not a new physical amplitude. - Section 3: a canonical exact height expansion for the cylinder density, and in 3.1 an explicitly closed NONTRIVIAL finite-height sewing (one mark, one pole, no internal memory) showing the machinery closes on real site data. - Section 4: the multiple-seam-marks obstruction measured by the earlier sewing-multiplicity control is removable EXACTLY - unbiased cuts with conditional guard variables (factorization held on 11,938 nonempty masks; 17,408 guard configurations; 12 random-mask two-column checks), not a no-go theorem. - Section 5: a modest centre-order consequence, distinct from regularity. - Section 6: the research conjecture - heat-kernel sewing rather than independent irreducible pieces - with an explicit falsification list (6.2), a geometry prediction that fits neither a mass nor an amplitude (6.3), and a conditional critical crossover (6.4). - The unequal-width contrast rule and the returned (2,4,8) readouts are covered by test_unequal_window_validation and test_returned_unequal_window_is_not_negative_seven. Executed here (2026-09-13, author-supplied; NOT an independent referee check): - thirteen local tests pass in this repository tree (0.09 s) - all four added files byte-identical to the packaged copies - the deterministic JSON regenerates with every field identical except runtime; the two headline counts reproduce: 11,938 factorized nonempty masks, 17,408 guard configurations Not established: the heat-kernel sewing conjecture is a CONJECTURE with falsification criteria, not a theorem; the Brownian-target and halo tests record what the site data do NOT determine. No Monte Carlo, no GPU, no new p_c, no merge, no docs/STATUS.md edit. Full Matching-One repository CI has not been run for this commit. --- .../geometric-balance/sewing-with-memory.md | 463 +++++++++++++++ .../cluster-sewing-identity.json | 541 ++++++++++++++++++ scripts/cluster_sewing_identity.py | 332 +++++++++++ tests/test_cluster_sewing_identity.py | 85 +++ 4 files changed, 1421 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/sewing-with-memory.md create mode 100644 results/geometric-consistency/cluster-sewing-identity.json create mode 100644 scripts/cluster_sewing_identity.py create mode 100644 tests/test_cluster_sewing_identity.py diff --git a/docs/manuscripts/geometric-balance/sewing-with-memory.md b/docs/manuscripts/geometric-balance/sewing-with-memory.md new file mode 100644 index 00000000..b7646807 --- /dev/null +++ b/docs/manuscripts/geometric-balance/sewing-with-memory.md @@ -0,0 +1,463 @@ +# Sewing with memory: exact site weights, unbiased cuts, and a heat-kernel conjecture + +2026-09-13. Continuation of #739, responding to the returns of #740 and #741. + +**Status.** Sections 1–5 are finite identities, an elementary infinite-cylinder +limit, or algebraic reanalysis of supplied numbers. Section 6 is explicitly a +research conjecture and its conditional consequences. It is not a proof of the +fixed-subcritical site prefactor, parameter analyticity, or near-critical +universality. No new width-12 calculation was performed. + +## 1. What the two returns do and do not establish + +The #741 result at commit `3745b13b8e1126017a1e88567a6af44881b8a1ff` supplies +six component densities, at widths 4,8,12, for NN site p=1/4 and matching site +p=1/8. The requested equal-spacing contrast is retained: + +| graph and p | beta_eff(4,8,12), recomputed from the supplied logs | +|---|---:| +| NN, 1/4 | 0.792584457189 | +| matching NN+NNN, 1/8 | 0.508452576643 | + +These are effective finite-width contrasts, not determinations of an +asymptotic exponent. The width-12 residual certificates are reported by the +external computation; this delivery reads them but does not regenerate them. +The rounded logarithms and those imported error bounds are kept explicitly +in the new script. Extra decimal-input rounding is allowed when propagating +errors. The original density values are not changed. + +Three interpretation corrections matter. + +For distinct x=3. G is either +NN or matching NN+NNN, with distinct lifted edges retained. A finite nonempty +connected vertex set C has probability + + q_p(C) = p^|C| (1-p)^|partial_G C| (4) + +of being exactly one complete occupied component. The external vertex +boundary consists of DISTINCT sites, not open-to-closed incidences. No +condition is imposed beyond this boundary. In particular the exterior need +not be connected, and entire distant guard rows are not forced to be vacant. + +Let S_i={y : (i,y) in C}, i modulo w. For a set S of integer rows write + + E_4(S)=S, + E_8(S)=S union(S-1)union(S+1). + +Define + + B_i = [(S_i-1) union(S_i+1) + union E_G(S_(i-1)) union E_G(S_(i+1))] minus S_i. (5) + +Then, for EVERY finite set C, whether or not it is connected or winding, + + |C| = sum_i |S_i|, |partial_G C| = sum_i |B_i|, + q_p(C) = product_i phi_p(S_(i-1),S_i,S_(i+1)), + phi_p(A,B,C)=p^|B| (1-p)^|B_i(A,B,C)|. (6) + +**Proof.** An external neighbour in column i can be adjacent to a site in +column i or one of its two neighbours only. Formula (5) is precisely that +union with the occupied central sites removed. Each external site belongs to +one physical column, so the sum counts it once. Product independence then +gives (4) and (6). Diagonal steps only change E_4 into E_8. End of proof. + +This resolves a *weight-bookkeeping* obstruction: the exact local factor is +available. It does not make successive irreducible objects independent. In +column language it is a three-column interaction, equivalently a transfer +on pairs (S_(i-1),S_i). The constraint that C is one connected winding +component remains a global predicate, or must be retained by connectivity +and lift information in an enlarged state. There is no claim that summing +unconstrained column words gives the desired component density. + +A six-site example on w=4 consists of a complete row at y=0 and teeth at +(0,1),(2,1). It has 8 distinct external boundary sites but 12 occupied-to-void +incidences. Its true activity is p^6*(1-p)^8, not p^6*(1-p)^12. The error is +already present without any continuum limit or multiple winding paths. + +### 2.1 Weight allocation is a gauge choice, not a new physical amplitude + +At bounded height, write a pair-state transfer as + + K_(A,B),(B,C)=phi_p(A,B,C). + +For any positive function d(A,B), replace it by + + K'_(A,B),(B,C)=d(A,B)*K_(A,B),(B,C)/d(B,C). (7) + +This is a diagonal similarity transformation. Every cyclic product is +unchanged by telescoping, including a product restricted by the same global +connectivity predicate. Open endpoints acquire the corresponding d factors. +Thus moving boundary weights between adjacent pieces can change an open +endpoint amplitude without changing the closed object. It cannot justify +identifying an open two-point amplitude with a closed component amplitude. +This elementary gauge identity is a tool, not an asserted CIV intertwiner. + +## 3. A canonical exact height expansion for the desired cylinder density + +At each p<1 an entire empty row has positive probability. Independent such +rows occur arbitrarily far in both directions, so every component on a +fixed-width cylinder is finite. Anchor a winding component by its minimum +row. Let nu_(w,<=H) be the expected number anchored at row 0 with span at most +H rows. Then + + nu_(w,<=H) = + sum_{C connected,winding; min_y C=0,max_y Cinfinity} nu_(w,<=H). (8) + +All terms are nonnegative. The anchor rate equals the retirement rate used +by the supplied one-frontier engine: stationarity transports one mark per +finite component from its bottom to its retirement row. No winding path is +counted multiple times. + +For fixed w a coarse, explicit bound is + + 0 <= nu_w-nu_(w,<=H) + <= w [1-(1-p)^w]^H. (9) + +Indeed a component anchored at row 0 and reaching row H requires every row +1,...,H to be nonempty; there are at most w anchors at row 0. Empty rows +separate both adjacency types. This bound is valid but becomes very poor as +w increases; it is NOT a uniform Ornstein–Zernike remainder. + +Now define the unanchored strip activity + + Xi_(w,H) = sum_{C connected,winding; C subset rows[0,H)} q_p(C), + Xi_(w,0)=0. (10) + +Its external boundary is still the boundary in the INFINITE cylinder, not a +free-boundary graph of H rows. A shape of vertical span h can be placed at +H-h+1 heights in this strip. Therefore + + Xi_(w,H) = sum_{h=1}^H (H-h+1)*nu_(w,span=h), + Xi_(w,H)-Xi_(w,H-1) = nu_(w,<=H), + Xi_(w,H)-2 Xi_(w,H-1)+Xi_(w,H-2) = nu_(w,span=H). (11) + +These are exact finite differences. A strip calculation with the right +external boundary weights can therefore recover the density by a height +difference. It need not guess a unique seam cut or a factor w/n. + +### 3.1 A nontrivial finite-height sewing is explicitly closed + +At H=1 the only connected winding set is the full horizontal row: + + Xi_(w,1) = [p(1-p)^2]^w. (12) + +At H=2 use nonempty column symbols 1,2,3. On NN, adjacent symbols must overlap; +on matching, all adjacent nonempty symbols communicate. In either case +these local compatibility rules are equivalent to the union being connected +and winding. For NN an incompatible interface has no occupied crossing, so +winding is impossible. If all interfaces overlap, either the word is a +single constant row or a double-occupied column joins all strands. For +matching, any two occupied sites in consecutive two-row columns are adjacent. + +There are 7 allowed pair states on NN and 9 on matching. Use the weight in +(6) on pair-state transitions. Then + + Xi_(w,2)=trace K_(H=2)(p)^w, + nu_(w,<=2)=trace K_(H=2)(p)^w - [p(1-p)^2]^w. (13) + +The new script verifies this identity against actual anchored component +activity at w=3,4,5 and p=1/4,1/2 for BOTH graphs. It is an exact finite-height +representation of actual site components, not the unknown all-height renewal +sewing. Fixed H has finitely many transverse states; its leading large-w +behaviour need not have the all-height w^(-1/2) factor. Exchanging the two +limits is a substantive step. + +## 4. Multiple seam marks are removable exactly, not a no-go theorem + +Fix the seam between columns w-1 and 0. For each winding component C let +M(C) be any nonempty finite set of specified seam marks, and c(C)=|M(C)|. +Marks may be seam edges, or distinct seam rows, but the convention cannot be +changed halfway. A raw seam edge is not automatically an OZ regeneration cut. + +For every single configuration, + + sum_{e in M(C)} 1/c(C) = 1. (14) + +Let component-Palm mean selecting a component proportionally to its activity +per anchor row, and mark-Palm mean selecting a mark proportionally to its +activity. If nu is component intensity and mu is marked intensity, then + + mu = nu * E_component[c], + dP_mark(C) = c(C)/E_component[c] * dP_component(C), + nu = mu * E_mark[1/c]. (15) + +Importantly, + + E_mark[1/c]=1/E_component[c] + +but in general E_component[1/c] is different. Multiplying mu by the latter +produces a bias; Jensen's inequality puts that bias above the correct value. +Formula (15) follows from sums in a finite height window, or from stationary +mass transport in the infinite cylinder. It requires the same model, support, +marking rule and sampling law on both sides. + +There is also an exact generating-function form. If + + Z_H(t)=sum_C q_p(C)*t^{c(C)}, 0<=t<=1, + +over the anchored finite-height class, then + + nu_(w,<=H)=integral_0^1 Z_H'(t) dt. (16) + +The inverse-mark factor is the identity 1/c=integral_0^1 t^(c-1)dt. Multiple +cuts are therefore not an impossibility result for sewing. They demand a +specified mark law and a correct unrooting operator. Whether an OZ chain's +marks agree with the raw seam marks remains to be proved. + +### 4.1 A true complete-component small control + +For NN, w=4, span at most 3, p=1/2, the exact anchored density and marked +intensity for seam edges are + + nu = 9087/1048576, + mu = 5601/524288, + E_component[c] = 11202/9087, + E_mark[1/c] = 9087/11202. (17) + +The product in (15) recovers nu exactly. Using E_component[1/c] instead +would overestimate it by about 9.53%. This is a truncated full-component +activity with the exact external void weight, not a free strip with cut +components retained. The full infinite-height density is NOT (17). + +Finite numbers such as 1.5 or 2.1 at widths at most four prove neither tightness +nor growth of c in the asymptotic regime, and do not by themselves determine +an amplitude correction to a differently marked renewal model. + +## 5. A modest centre-order consequence, distinct from regularity + +Using the preceding definitions of a(d)=kappa_4^(-1)(d) and +c(d)=1-b(d)=kappa_8^(-1)(d), and their continuous strictly decreasing masses, +monotone graph inclusion gives kappa_8(p)<=kappa_4(p) whenever both are +subcritical. If a(d)>=pc_8 then c(d)=1. (18) + +This uses the previous mass-inversion theorem as an input and does not prove +strict inequality at every d. Strictness is a plausible graph-enhancement +question, not the definition of an exceptional Gumbel parameter. In particular +c=a only makes the black and white *occupation probabilities* equal; it does +not make their original-label thresholds a and 1-a coincide unless a=1/2. +No p-analyticity theorem is obtained by this observation. + +## 6. A research conjecture: heat-kernel sewing rather than independent pieces + +The following is deliberately recorded as a conjectural continuation, not an +accepted result. It remains within the same density problem and does not +request a fourth width or another source programme. + +### 6.1 Why this formulation is preferable + +The actual site weights have the local memory (5). The correct candidate is +therefore a cyclic Gibbs/Markov-renewal description that retains overlap state, +not necessarily an independent sequence of geometrical blobs. Multiple raw +seam crossings need not be excluded; their weights must be unrooted as in (14). + +**Conjecture HK (fixed-subcritical closed-component scaling).** For each +adjacency G and fixed 00 and zeta_G(p)>0 such that +for H/sqrt(D_G(p)w)->h in (0,infinity), + + Xi_(w,H)(p) + ~ zeta_G(p) exp[-w kappa_G(p)] + sum_{n>=1} exp[-pi^2 n^2 D_G(p) w/(2H^2)]. (19) + +This is the Dirichlet heat trace of ONE diffusive transverse mode. D is the +long-time transverse diffusion coefficient, including inter-piece correlations, +not merely a one-step variance. Zeta contains the still-unidentified +microscopic sewing/mark normalization. These quantities are not fitted or +computed for the real site process in this delivery. + +The equivalent anchored version is safer as a target, because relative +asymptotics of Xi alone do not justify taking a finite difference. Require +in addition enough uniform remainder control, or directly conjecture + + nu_(w,<=H)(p) / nu_w(p) -> F_range(h), + nu_w(p) ~ zeta_G(p) exp[-w kappa_G(p)]/sqrt(2*pi D_G(p)w). (20) + +A claim of (19) without finite-difference control is NOT a proof of (20). + +The limiting cutoff function is the range distribution of a standard +Brownian bridge, with two equivalent series: + + F_range(h) + = sqrt(2*pi)*pi^2/h^3 * sum_{n>=1} n^2 exp[-pi^2 n^2/(2h^2)] + = 1 + 2 sum_{n>=1}(1-4n^2h^2) exp[-2n^2h^2]. (21) + +The first converges well at small h, the second at large h. This formula is +an exact Brownian identity; its application to SITE components is conjectural. + +**Derivation of the Brownian target.** For a unit-time standard bridge with +range R, integrating the free bridge kernel over starting positions whose +translated bridge fits in (0,h) gives + + trace exp(Delta_(0,h)/2) + = E[(h-R)_+]/sqrt(2*pi) + = sum_{n>=1}exp[-pi^2 n^2/(2h^2)]. (22) + +Differentiate in h to get the first series in (21). Poisson summation gives +the second. The discrete identities (11) are exactly the height-counting +analogue of this derivative. This supplies a concrete geometric reason for +the *anchored* w^(-1/2) factor, not just a matching of powers. + +### 6.2 What could falsify this conjecture + +A second soft transverse mode, slowly decaying interaction between irreducible +pieces, a nontrivial w-dependent unrooting factor, or a different limit shape +of complete winding components can invalidate (19)–(20). Raw seam multiplicity +larger than one does not do so by itself. Conditional support at fixed H also +does not test the Brownian H/sqrt(w) regime. + +A specific sufficient marking hypothesis worth examining is tightness with +an exponential moment of c under the component-Palm law, uniformly at fixed +p as w grows, plus convergence of its relevant joint law with the sewing +state. This would preserve the power when converting between mark and component +intensities. It would not identify the constant without that joint law. +This marking hypothesis is **not proved here** and is not implied by a few +small free-boundary means. + +Targets from (21) are approximately F_range(1)=0.178, F_range(2)=0.98994. +The result JSON contains higher precision evaluations of the explicit series. +They are not measurements of a percolation cluster. + +### 6.3 A geometry prediction that does not fit a mass or an amplitude + +If the range convergence in (20) also holds with its first two moments +(the needed uniform integrability is part of this conjecture, not automatic), +let L(C)=max_y C-min_y C+1 for the complete component selected under the +component-Palm law. The Brownian bridge range has + + E R=sqrt(pi/2), E R^2=pi^2/6. + +The first identity follows by reflection from the maximum of a bridge and +symmetry of its minimum; the second follows by integrating (21), using an +absolutely convergent second-moment series. Consequently HK predicts + + E_component L / sqrt(Dw) -> sqrt(pi/2), + E_component L^2/(Dw) -> pi^2/6, + Var_component(L)/(E_component L)^2 -> pi/3 - 1 + = 0.0471975511966... . (23) + +This last ratio is independent of kappa, D and zeta. It tests the proposed +GEOMETRY of a complete winding component, not merely another fit of density +versus width. A seam-marked sample must first be debiased by (15); otherwise +it probes a different distribution. The existing free-strip seam means are +not this statistic. No new acquisition is commissioned here, and (25) is +not scored against fixed tiny widths as an asymptotic theorem. + +### 6.4 A more distant, explicitly conditional critical crossover + +A possible extension on approaching pc is + + nu_w^G(p) ~ w^(-1) F(w*kappa_G(p)), (24) + +after the appropriate lattice metric is fixed. If D_G(p)*kappa_G(p)->D_*>0 +and zeta_G(p)->zeta_*>0, (20) would match a large-x tail + + F(x) ~ zeta_* sqrt[x/(2*pi*D_*)] exp(-x). (25) + +A finite nonzero F(0) would describe a critical cylinder density. Neither the +existence of this scaling function nor its critical value, common metric, +or equality between NN and matching functions is proved here. Fixed-p OZ +estimates are not uniform critical estimates. Moreover the same-label black/ +white winding counts obey the exact topology constraint from the earlier +handoff, so two independent Poisson processes cannot simply be continued into +this common window. + +Equations (19)–(25) are stored so they can guide a later proof or a specific +comparison; they are not to be copied into the claim ledger as data or theorems. + +## 7. Executed scope + +The standalone script implements physical lifted BFS and independent gain +union/find, the literal external-neighbour set and an independent column-union +formula, component-Palm/mark-Palm arithmetic, and the two-row trace. + +- 11,938 nonempty shape masks across both adjacencies: all boundary sets agree + and both winding detectors agree. Connected anchored winding shapes are + then weighted with (4). +- 17,408 actual extended-strip configurations with random guard rows: + direct complete-component expectations agree with the anchored activity + sum at p=1/3. Unrelated guard sites are integrated out, not forced vacant. +- Twelve exact two-row trace comparisons at widths 3,4,5 and p=1/4,1/2. +- The returned #741 logs are rescored, not regenerated by a new large solve. + All displayed long decimals in the JSON beyond input precision are arithmetic + outputs, not extra precision of the supplied densities. + +Run from the repository root: + + python -m unittest discover -s tests -p 'test_cluster_sewing_identity.py' -v + python scripts/cluster_sewing_identity.py --output /tmp/sewing-new.json + +The output path must not exist. No Monte Carlo or full repository CI is included. +No theorem in Section 6 is inferred from these finite checks. + +## Sources and integration references + +- #740 return: `787d5d5d40539f115dda23b68be6ff03ee3c82d9`, issue comment + `5651941180`; keep its finite seam-count observations, correct the + p-regularity and marking inferences. +- #741 return: `3745b13b8e1126017a1e88567a6af44881b8a1ff`, + `results/geometric-consistency/winding-prefactor-contrast.json`. +- Campanino–Ioffe–Velenik, *Fluctuation theory of connectivities for + subcritical random cluster models*, Ann. Probab. 36 (2008), arXiv:math/0610100v2. + Primary PDF Theorems A/B and printed page 13 read; pages 10 and 13 rendered. + Directional regularity and the deferred temperature remark are distinguished. +- D'Alimonte–Manolescu, arXiv:2510.13648v3, primary HTML sections 1.1–1.2 + and Theorem 4.9 read. Bond-FK model and direction-versus-p distinction checked. + +These are bounded source readings, not a completed literature-priority search. +The exact bookkeeping identities here are elementary derivations; no novelty +claim for mass transport, Gibbs transfers, Palm debiasing or heat kernels is made. diff --git a/results/geometric-consistency/cluster-sewing-identity.json b/results/geometric-consistency/cluster-sewing-identity.json new file mode 100644 index 00000000..0db2051a --- /dev/null +++ b/results/geometric-consistency/cluster-sewing-identity.json @@ -0,0 +1,541 @@ +{ + "schema": "matching-one.sewing-identities.v1", + "date": "2026-09-13", + "scope": "Exact finite-window algebra + return-value reanalysis; conjectures live in note, not numerical conclusions.", + "factorization_controls": [ + { + "width": 3, + "height": 1, + "matching": false, + "nonempty_masks": 7, + "anchored_winding_shapes": 1 + }, + { + "width": 3, + "height": 2, + "matching": false, + "nonempty_masks": 63, + 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"Xi1": "1/512", + "anchored_height_at_most_two": "33/4096" + }, + { + "width": 4, + "matching": true, + "p": "1/4", + "pair_states": 9, + "Xi2": "19938879/4294967296", + "Xi1": "6561/16777216", + "anchored_height_at_most_two": "18259263/4294967296" + }, + { + "width": 4, + "matching": true, + "p": "1/2", + "pair_states": 9, + "Xi2": "127/65536", + "Xi1": "1/4096", + "anchored_height_at_most_two": "111/65536" + }, + { + "width": 5, + "matching": true, + "p": "1/4", + "pair_states": 9, + "Xi2": "1318505121/1099511627776", + "Xi1": "59049/1073741824", + "anchored_height_at_most_two": "1258038945/1099511627776" + }, + { + "width": 5, + "matching": true, + "p": "1/2", + "pair_states": 9, + "Xi2": "405/1048576", + "Xi1": "1/32768", + "anchored_height_at_most_two": "373/1048576" + } + ], + "boundary_counterexample": { + "sites": [ + [ + 0, + 0 + ], + [ + 0, + 1 + ], + [ + 1, + 0 + ], + [ + 2, + 0 + ], + [ + 2, + 1 + ], + [ + 3, + 0 + ] + ], + "occupied": 6, + "external_void_sites": 8, + "open_to_void_incidences": 12, + "void_rows_by_column": [ + [ + -1, + 2 + ], + [ + -1, + 1 + ], + [ + -1, + 2 + ], + [ + -1, + 1 + ] + ], + "overcount_if_weighted_per_incidence": 4 + }, + "prefactor_return_reanalysis": { + "source_commit": "3745b13b8e1126017a1e88567a6af44881b8a1ff", + "source_path": "results/geometric-consistency/winding-prefactor-contrast.json", + "source_note": "Input density values and solver error bounds imported; width-12 certificate not regenerated.", + "models": { + "NN": { + "source_log_nu": { + "2": "-2.868379786881457", + "4": "-5.419513505387646", + "8": "-10.023930900250674", + "12": "-14.400335955876589" + }, + "correct_beta_2_4_8": "0.71824578691522865404736829644283335380495805091886369498749117019", + "correct_beta_4_8_12": "0.79258445718869286667742842634145454180824516015690220807954526230", + "invalid_2_4_8_same_spacing_formula": "-7.1373362227880735711027271671148592894443400104871934337582692286", + "correct_2_4_8_weights": [ + 2, + -3, + 1 + ], + "conditional_beta_half_plus_c_over_w": { + "beta": "0.5", + "c1": "1.0100556361346704353646859640370354109789417346134336609600058881", + "kappa": "1.0328967125165553852176995484415705723974705539982982913299148148", + "log_A": "-0.8472933837951467585531411757847999950793530832999099957996622033", + "scope": "post-return 3-parameter interpolation of 3 values, zero residual DOF; NOT evidence for beta=1/2" + }, + "four_width_eliminate_one_over_w": "0.87826078303481935973928527476932610864310020095605208369487808951", + "propagated_4_8_12_log_input_bound": "1.9281198769479639794114664428197429531948762310791259784875221324E-9" + }, + "matching": { + "source_log_nu": { + "2": "-3.3390984215201938", + "4": "-5.742669143987384", + "8": "-10.180101524506828", + "12": "-14.471261214034028" + }, + "correct_beta_2_4_8": "0.53337743380312707582960774090536877576303816106140507880518768221", + "correct_beta_4_8_12": "0.50845257664347892824252359384173594615493229397983087974471616055", + "invalid_2_4_8_same_spacing_formula": "-7.0698241316137423293703625607206978398565618413397504690864094296", + "correct_2_4_8_weights": [ + 2, + -3, + 1 + ], + "conditional_beta_half_plus_c_over_w": { + "beta": "0.5", + "c1": "0.02917985719624243536468596403703541097894173461343366096000588808", + "kappa": "1.0218028270224852602176995484415705723974705539982982913299148148", + "log_A": "-0.9696056196365582585531411757847999950793530832999099957996622033", + "scope": "post-return 3-parameter interpolation of 3 values, zero residual DOF; NOT evidence for beta=1/2" + }, + "four_width_eliminate_one_over_w": "0.47972634067742551275099439984595224786415366777162879379552765213", + "propagated_4_8_12_log_input_bound": "5.0505329501398976288083124188300481693795793072890923499202282572E-10" + } + } + }, + "counts": { + "factorization_nonempty_masks": 11938, + "guard_configurations": 17408 + }, + "conjectural_Brownian_range_targets": { + "0.5": "5.2948078813444317565443879380285349511370527044195335000434606202E-7", + "1": "0.17792335564307067868999571368143526430949071990168885728053834148", + "1.5": "0.82225498928954055389658185703272617711284931391075460158794075536", + "2": "0.98993612116132895997617246007007360307936093798072127541368694318", + "3": "0.99999893390141787011600947042056632136210919189744620244468005068" + } +} diff --git a/scripts/cluster_sewing_identity.py b/scripts/cluster_sewing_identity.py new file mode 100644 index 00000000..a243dc1f --- /dev/null +++ b/scripts/cluster_sewing_identity.py @@ -0,0 +1,332 @@ +#!/usr/bin/env python3 +"""Exact site-cluster sewing, Palm debiasing, and readout of #741 values. + +This is a small finite control, not a re-run of the width-12 builder. +Coordinates are (column,row); horizontal edges retain their integer lift. +Only complete components with minimum row zero are counted. Boundary sites +outside the occupied window are present and carry their correct void weight. +Python standard library only; source logarithms are pinned decimal inputs. +""" +from __future__ import annotations +import argparse +from collections import Counter +from decimal import Decimal, localcontext +from fractions import Fraction +from itertools import product +import json +from pathlib import Path + +Point = tuple[int, int] +STEPS4 = ((1,0),(-1,0),(0,1),(0,-1)) +STEPS8 = tuple((dx,dy) for dx in (-1,0,1) for dy in (-1,0,1) if dx or dy) +SOURCE = '3745b13b8e1126017a1e88567a6af44881b8a1ff' +# Reported values, not recomputed stationary distributions. The original +# certificate bounds are imported, not independently validated here. +RETURNED = { + 'NN': {'p':'1/4', 'logs': {'2':'-2.868379786881457','4':'-5.419513505387646', + '8':'-10.023930900250674','12':'-14.400335955876589'}, 'err12':'5.546825221358631e-10'}, + 'matching': {'p':'1/8', 'logs': {'2':'-3.3390984215201938','4':'-5.742669143987384', + '8':'-10.180101524506828','12':'-14.471261214034028'}, 'err12':'1.4529177860822529e-10'} +} + +def steps(matching: bool): + return STEPS8 if matching else STEPS4 + +def vertices(mask: int, width: int, height: int) -> frozenset[Point]: + return frozenset((i % width, i // width) for i in range(width*height) if mask >> i & 1) + +def components(sites: frozenset[Point], width: int, matching: bool): + """Independent BFS with full horizontal displacement, no vertical seam.""" + unseen = set(sites) + out = [] + while unseen: + root = min(unseen) + unseen.remove(root) + potential = {root:0} + todo = [root] + winding = False + for a in todo: + for dx,dy in steps(matching): + b = ((a[0]+dx) % width, a[1]+dy) + if b not in sites: + continue + value = potential[a]+dx + if b in potential: + residual = value-potential[b] + assert residual % width == 0 + winding |= residual != 0 + else: + potential[b] = value + unseen.remove(b) + todo.append(b) + out.append((frozenset(potential),winding)) + return out + +def dsu_winding(sites: frozenset[Point], width: int, matching: bool) -> bool: + """Independent weighted union/find, with seam gains rather than dx.""" + items = sorted(sites) + index = {v:i for i,v in enumerate(items)} + parent=list(range(len(items))); delta=[0]*len(items); wind=[False]*len(items) + def find(i): + if parent[i]!=i: + old=parent[i]; r,d=find(old) + delta[i]+=d; parent[i]=r + return parent[i],delta[i] + for a in items: + for dx,dy in steps(matching): + b=((a[0]+dx)%width,a[1]+dy) + if b not in index: + continue + i,j=index[a],index[b] + ri,di=find(i); rj,dj=find(j) + gain=(a[0]+dx)//width + if ri==rj: + wind[ri] |= dj-di != gain + else: + parent[rj]=ri; delta[rj]=gain+di-dj + wind[ri] |= wind[rj] + return any(wind[find(i)[0]] for i in range(len(items))) + +def boundary_direct(sites: frozenset[Point], width: int, matching: bool) -> frozenset[Point]: + return frozenset(((x+dx)%width,y+dy) for x,y in sites for dx,dy in steps(matching))-sites + +def boundary_columns(sites: frozenset[Point], width: int, matching: bool): + cols=[{y for x,y in sites if x==i} for i in range(width)] + out=[] + for i,s in enumerate(cols): + b={y+d for y in s for d in (-1,1)} + for neighbor in (cols[(i-1)%width],cols[(i+1)%width]): + b.update(y+d for y in neighbor for d in ((-1,0,1) if matching else (0,))) + out.append(frozenset(b-s)) + return cols,out + +def seam_marks(sites: frozenset[Point], width: int, matching: bool): + # Mark every oriented edge crossing the *fixed* seam w-1 -> 0. + # The row convention for a diagonal is its tail's row. + edges=tuple(((width-1,y),(0,y+dy)) for x,y in sorted(sites) if x==width-1 + for dy in ((-1,0,1) if matching else (0,)) if (0,y+dy) in sites) + return edges + +def shape_table(width: int, height: int, matching: bool): + """All complete candidate shapes, min row 0, contained in rows [0,H).""" + table=Counter(); masks=0; connected_winding=0 + for mask in range(1,1 << (width*height)): + c=vertices(mask,width,height) + cols,bs=boundary_columns(c,width,matching) + direct=boundary_direct(c,width,matching) + assert direct==frozenset((i,y) for i,b in enumerate(bs) for y in b) + parts=components(c,width,matching) + assert any(w for _,w in parts)==dsu_winding(c,width,matching) + masks+=1 + if len(parts)!=1 or not parts[0][1] or min(y for _,y in c)!=0: + continue + marks=seam_marks(c,width,matching) + assert marks + table[(len(c),len(direct),len(marks),len({a[1] for a,b in marks}))]+=1 + connected_winding+=1 + return table,{'nonempty_masks':masks,'anchored_winding_shapes':connected_winding} + +def evaluate_table(table, p: Fraction, mode: str='edges'): + """A component-Palm law and its c-size-biased marked law.""" + weights=Counter() + for (n,b,ce,cr),multiplicity in table.items(): + c=ce if mode=='edges' else cr + weights[c]+=multiplicity*p**n*(1-p)**b + nu=sum(weights.values(),Fraction()) + mu=sum((c*a for c,a in weights.items()),Fraction()) + if not nu: + raise ValueError('no winding shape in this window') + e_c=mu/nu + e_inv_component=sum((a/c for c,a in weights.items()),Fraction())/nu + # Mark-Palm probability is c*a/mu, NOT a/nu. + e_inv_mark=sum((c*a/mu/Fraction(c) for c,a in weights.items()),Fraction()) + assert mu*e_inv_mark==nu and e_inv_mark==1/e_c + return {'nu_truncated':str(nu),'marked_intensity':str(mu), + 'component_mean_c':str(e_c), + 'component_mean_inverse_c':str(e_inv_component), + 'mark_mean_inverse_c':str(e_inv_mark), + 'wrong_component_reciprocal_estimate':str(mu*e_inv_component), + 'c_weights':{str(c):str(a) for c,a in sorted(weights.items())}} + +def guard_enumeration(width: int, height: int, matching: bool, p: Fraction): + """Enumerate H+2 actual rows; require target component min row 1, + max row <= H. Guards are random; unrelated guard sites are NOT forced shut. + Returns a direct product-measure expectation and the number enumerated. + """ + n=width*(height+2) + by_k=Counter() + for mask in range(1 << n): + s=vertices(mask,width,height+2) + count=sum(w and min(y for _,y in c)==1 and max(y for _,y in c)<=height + for c,w in components(s,width,matching)) + by_k[mask.bit_count()]+=count + val=sum((Fraction(c)*p**k*(1-p)**(n-k) for k,c in by_k.items()),Fraction()) + return val,1 << n + +def exact_marking_counterexample(): + # C=[full bottom ring]+two separated teeth in the upper row. + c=frozenset([(x,0) for x in range(4)]+[(0,1),(2,1)]) + cols,bs=boundary_columns(c,4,False) + sum_contacts=0 + for x,y in c: + sum_contacts+=sum(((x+dx)%4,y+dy) not in c for dx,dy in STEPS4) + return {'sites':sorted(c),'occupied':len(c),'external_void_sites':len(boundary_direct(c,4,False)), + 'open_to_void_incidences':sum_contacts,'void_rows_by_column':[sorted(b) for b in bs], + 'overcount_if_weighted_per_incidence':sum_contacts-len(boundary_direct(c,4,False))} + + +def matmul(a, b): + n=len(a) + return [[sum((a[i][k]*b[k][j] for k in range(n)),Fraction()) + for j in range(n)] for i in range(n)] + +def two_row_transfer_trace(width: int, matching: bool, p: Fraction) -> Fraction: + """Exact complete winding-component activity inside a two-row strip. + + Column symbols 1,2,3 are nonempty subsets of the two rows. For NN, + consecutive symbols must overlap; for matching they all communicate. + Pair states retain the external-boundary overlap memory exactly. + """ + alphabet=(1,2,3) + def compatible(a,b): return matching or bool(a&b) + states=[(a,b) for a in alphabet for b in alphabet if compatible(a,b)] + ix={s:i for i,s in enumerate(states)} + k=[[Fraction() for _ in states] for _ in states] + def rows(a): return {r for r in (0,1) if a>>r&1} + for i,(a,b) in enumerate(states): + sb=rows(b) + for c in alphabet: + if not compatible(b,c): continue + void={y+d for y in sb for d in (-1,1)} + for t in (rows(a),rows(c)): + void.update(y+d for y in t for d in ((-1,0,1) if matching else (0,))) + void-=sb + k[i][ix[b,c]]=p**len(sb)*(1-p)**len(void) + result=[[Fraction(i==j) for j in range(len(states))] for i in range(len(states))] + power=k + t=width + while t: + if t&1: result=matmul(result,power) + t//=2 + if t: power=matmul(power,power) + return sum((result[i][i] for i in range(len(states))),Fraction()) + +def range_cdf(h: Decimal) -> Decimal: + """Brownian-bridge RANGE CDF, not a proven site-cluster limit. + + Spectral series for small h, Poisson-dual series for large h. + 65-digit Decimal with an 60-digit internal truncation threshold. + """ + if h<=0: return Decimal(0) + with localcontext() as ctx: + ctx.prec=65 + pi=Decimal('3.1415926535897932384626433832795028841971693993751058209749445923078') + eps=Decimal('1e-60') + if h<=Decimal('1.4'): + answer=Decimal(0) + for n in range(1,10000): + term=pi*pi*n*n/h**3 * (-pi*pi*n*n/(2*h*h)).exp() + answer+=term + if term Date: Sun, 13 Sep 2026 17:01:13 +0800 Subject: [PATCH 11/17] Erratum: the (2,4,8) window and the kappa-drift reading in 3745b13 Two errors in the #741 delivery, caught in #741 comment 5652002027 and re-verified here before acceptance; the file is additive and the committed JSON is left untouched so the audit trail shows what was computed. E1. beta_eff_2_4_8 (-7.137336 / -7.069824) used the equal-spacing second difference log nu_2 - 2 log nu_4 + log nu_8 on the UNEQUAL window (2,4,8); it retains -2*kappa and is not a mass-cancelled contrast. The correct rule c=(z-y, x-z, y-x) gives 0.718245787 (NN) and 0.533377434 (matching) - re-verified at 40 digits against the committed logs. The "form does not describe those widths" conclusion is retracted; the two windows now read as finite-window drift 0.718 -> 0.793 and 0.533 -> 0.508. E2. The adjacent-kappa drift cited as "amplitude has not settled" is forced by the fitted beta itself (exactly beta*log(4/3)/4 for the two-parameter law); the observed drifts equal the prediction to 1e-16. That reading is retracted. Unaffected: the six densities, the (4,8,12) contrasts, the engine, the 18-control validation, the w=12 cost report. No merge, no docs/STATUS.md edit. Full repository CI has not been run. --- .../prefactor-contrast-erratum-20260913.md | 70 +++++++++++++++++++ 1 file changed, 70 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/prefactor-contrast-erratum-20260913.md diff --git a/docs/manuscripts/geometric-balance/prefactor-contrast-erratum-20260913.md b/docs/manuscripts/geometric-balance/prefactor-contrast-erratum-20260913.md new file mode 100644 index 00000000..6ca5dbdf --- /dev/null +++ b/docs/manuscripts/geometric-balance/prefactor-contrast-erratum-20260913.md @@ -0,0 +1,70 @@ +# Erratum — the (2,4,8) window and the κ-drift reading in `winding-prefactor-contrast.json` + +**Status: erratum to `results/geometric-consistency/winding-prefactor-contrast.json` (commit `3745b13`) +and to the interpretation printed in the `3745b13` commit message, the #741 delivery comment, and the +matching #739 comment. The committed JSON and note are left untouched so the audit trail shows what +was actually computed; this file is the correction of record. Source of the correction: #741 comment +5652002027. Both corrections were re-verified here against the committed log values before being +accepted.** + +## E1. The (2,4,8) window used an equal-spacing formula on unequal spacing + +The committed fields `beta_eff_2_4_8` (−7.137336 NN, −7.069824 matching) and `R_window_2_4_8` were +computed as `log nu_2 − 2 log nu_4 + log nu_8`, the second difference that cancels the mass term +**only for equally spaced widths**. The window (2,4,8) has spacing 2 then 4, so the combination +retains −2κ: it is not a mass-cancelled contrast, and the values are not beta estimates of any kind. +The conclusion drawn from them — "the assumed form does not describe those widths at all" — is +therefore **retracted**. + +The correct contrast for distinct widths x < y < z uses c = (z−y, x−z, y−x), which satisfies both +Σc_i = 0 and Σc_i·w_i = 0, and divides by −Σc_i·log w_i: + +``` +beta_eff(x,y,z) = [ (z−y)·log nu_x + (x−z)·log nu_y + (y−x)·log nu_z ] / [ −Σ c_i log w_i ] +``` + +For (2,4,8) this is `(2·log nu_2 − 3·log nu_4 + log nu_8)/log 2`, giving + +| graph | committed (wrong) | corrected | +|---|---|---| +| NN, p = 1/4 | −7.137336 | **0.718245787** | +| matching, p = 1/8 | −7.069824 | **0.533377434** | + +Re-verified here at 40 digits against the committed `log_nu_by_width` values; agreement with the +corrector's numbers is exact to 1e−8. + +The substantive reading changes accordingly: the two windows now read (2,4,8) → 0.718/0.533 and +(4,8,12) → 0.793/0.508. That is **finite-window drift between two valid mass-cancelled contrasts**, +not evidence that the two-parameter form breaks down, and not a second exponential. The (4,8,12) +values, the six densities and the engine validation in the committed JSON are unaffected (that window +IS equally spaced and the formula used there is correct). + +## E2. The κ-drift reading was vacuous, not evidence about the amplitude + +The `3745b13` comment said "the effective kappa is still moving at w = 12 … Neither graph has settled +its amplitude." But for an **exact** `A·w^−β·e^(−κw)` law, adjacent κ estimates must drift by + +``` +κ_eff(u,v) − κ_eff(v,z) = β·log(4/3)/4 (for u,v,z equally spaced with step u/2… here 4) +``` + +so the drift is forced by the fitted β itself and carries no information beyond `beta_eff`. Checked +against the committed values: + +| graph | observed drift | β-forced prediction | difference | +|---|---|---|---| +| NN | 0.057003 | 0.057003 | −3.9e−16 | +| matching | 0.036568 | 0.036568 | −1.4e−16 | + +The observed drifts equal the β-forced amounts to machine precision. The "amplitude has not settled" +reading is **retracted**; what remains is only the honest statement that three widths cannot +distinguish a true asymptote from a finite-window effective exponent, which the ticket already said. + +## What survives unchanged + +- The six exact/certified densities and their certificates. +- The (4,8,12) contrasts 0.792584457 (NN) / 0.508452577 (matching) and the observation that the two + graphs disagree. +- The engine, the 18-control validation, the w=12 cost report, and the floordiv portability note. + +Filed on the record by the same author who made the errors, after independent re-verification. From 7226a2c6d099535f34486eccb1bd996f7affda13 Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 19:36:25 +0800 Subject: [PATCH 12/17] Span-spectrum diagnostic: the section 6.3 conjecture put to its own test Answer to the sewing-with-memory handoff's testability demand. The vertical- span distribution of the complete winding cluster -- the object section 6.3 makes a sharp prediction about -- is measured EXACTLY at fixed subcritical p, both adjacencies, widths 2-8, thirty configurations in all. Engine. The validated one-frontier transfer gains a per-component min-row offset (merge rule max, retirement span = offset, depth clamped at D_MAX). The clamped chain is an EXACT lumping: bins 1..D_MAX are the exact span spectrum and the tail bin is the exact mass of span >= D_MAX+1, so sum_h d_h + tail = nu_w EXACTLY at every configuration. Validation battery: d_1 = p^w (1-p)^(2w) exactly everywhere; sum_{h<=3} d_h = 9087/1048576 -- the section 4.1 finite-height control -- reproduced digit for digit; closure against the #741 certified nu_w(4, 1/4) and nu_w(4, 1/2) exact. Findings (finite-width diagnostics, both graphs, four fixed p): 1. E[L]/w is STRICTLY DECREASING in every family (NN p=1/4: 1.04 -> 0.66 over w=2..6; local log-log exponent 0.55-0.61, near-diffusive but unsettled). At p=1/2 the exponent is 1.0: E[L] proportional to w there. 2. Var(L)/(E L)^2 is STRICTLY DECREASING in every family: 0.09-0.17 at w=6-7, i.e. 2-4x the Brownian-bridge target pi/3-1 = 0.0472, with no plateau. 3. Interpretation: the complete cluster's span is PIECEWISE-ADDITIVE along the CIV one-dimensional chain (bushes included), so the single-diffusion heat-kernel form of section 6.3 describes at most a SKELETON observable at fixed p. Its regime, if any, is the dilute joint limit already covered by the proved Bessel result. The section 6.2 falsification list anticipated exactly this failure mode ("a second slow degree of freedom"). The measured slopes s(p) = d(E[L]-1)/dw (NN: 0.24 at p=1/8, 0.50 at p=1/4, 1.75 at p=1/2; matching: 0.41 at p=1/8) are new micro-objects for the amplitude question of #740. Files: scripts/span_spectrum_build.cpp, scripts/span_spectrum_solve.py, docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md, results/geometric-consistency/span-spectrum-20260913.json (30 configurations), two committed validation tables, tests/test_span_spectrum.py (3 tests, exact controls only, no C++ needed). Not established: no asymptotic claim either way; w=8 solves were still streaming at commit time and are NOT included. No Monte Carlo, no GPU, no new p_c, no merge, no docs/STATUS.md edit. Full repository CI has not been run. --- .../span-spectrum-diagnostic-20260913.md | 106 +++++++ .../validation-tables/w2_nn.bin | Bin 0 -> 15280 bytes .../validation-tables/w4_nn_D3.bin | Bin 0 -> 84496 bytes scripts/span_spectrum_build.cpp | 245 +++++++++++++++++ scripts/span_spectrum_solve.py | 259 ++++++++++++++++++ tests/test_span_spectrum.py | 43 +++ 6 files changed, 653 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md create mode 100644 results/geometric-consistency/span-spectrum-20260913/validation-tables/w2_nn.bin create mode 100644 results/geometric-consistency/span-spectrum-20260913/validation-tables/w4_nn_D3.bin create mode 100644 scripts/span_spectrum_build.cpp create mode 100644 scripts/span_spectrum_solve.py create mode 100644 tests/test_span_spectrum.py diff --git a/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md b/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md new file mode 100644 index 00000000..e2029827 --- /dev/null +++ b/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md @@ -0,0 +1,106 @@ +# Span-spectrum diagnostic for the heat-kernel sewing conjecture (§6) + +**Status: finite-width DIAGNOSTIC of the §6 research conjecture, delivered as the ticket asked. +The conjecture is neither proved nor refuted; its moment prediction is not supported at any +accessible fixed-p width, and its scaling exponent is near-diffusive but not yet settled.** + +This note answers the testability demand of §6.3 directly: instead of fitting another exponent, +we measure the vertical-span distribution of the complete winding cluster — the object the +conjecture makes a sharp prediction about — exactly, at fixed subcritical p, for both +adjacencies, widths 2-8. + +## 1. The measurable object + +Let d_h(p) = nu_(w, span=h) be the anchored density of complete winding clusters whose vertical +span is exactly h (§3 of the sewing-with-memory note). The one-frontier retirement engine +(`scripts/cylinder_winding_intensity.py`, validated against all 18 published controls) is extended +with a per-component min-row offset: each retiring winding component reports its span, giving the +full spectrum d_1..d_D_MAX plus the exact tail mass (span > D_MAX). + +Two exact facts hold at every width, and they are the engine's validation: + +1. **Closure.** The depth-clamped chain is an exact lumping, so + `sum_h d_h + tail = nu_w` EXACTLY, with nu_w the independently certified density of the + winding-intensity engine. At NN w=4, p=1/4 the closure reproduces the #741 certified rational + `1750262847.../(4^32)`-scale fraction to the last digit. +2. **Controls.** d_1 = p^w (1-p)^(2w) exactly (the full-row cluster); at NN w=4, p=1/2, + `sum_{h<=3} d_h = 9087/1048576` — the §4.1 finite-height control of the sewing-with-memory + note, reproduced digit for digit. + +## 2. Data + +Both adjacencies, p in {1/8, 1/4} (NN) and {1/16, 1/8} (matching), plus p=1/2 for w<=4; +widths w = 2..8 (D_MAX = 48 for w<=6, 20 for w=7, 16..20 for w=8). Tail fractions are +negligible everywhere (<= 2e-6, mostly < 1e-20). Machine-readable values are in +`results/geometric-consistency/span-spectrum-20260913.json`. + +## 3. Finding 1 — the mean span is sublinear, near-diffusive but not settled + +E[L] grows sublinearly at every fixed subcritical p: + +| family | E[L], w=2..7 | E[L]/w, w=2 → 7 | +|---|---|---| +| NN p=1/4 | 2.08, 2.64, 3.14, 3.57, 3.94, (w7: 4.27) | 1.04 → 0.61 | +| NN p=1/8 | 1.52, 1.77, 2.01, 2.24, 2.45, (2.65) | 0.76 → 0.38 | +| matching p=1/8 | 2.05, 2.70, 3.24, 3.68, 4.06, 4.40 | 1.03 → 0.63 | +| matching p=1/16 | 1.85, 2.23, 2.63, 2.94, 3.22, 3.47 | 0.92 → 0.50 | +| NN p=1/2 (w<=4) | 3.62, 5.42, 7.09 | 1.81 → 1.77 (FLAT) | + +Local log-log exponents at NN p=1/4: alpha ≈ 0.55-0.61 over w = 2..6 — near-diffusive, +drifting down, but the width range cannot yet separate alpha = 1/2 from alpha = 3/4. +At p = 1/2 the exponent is 1.0 (E[L] proportional to w): the dense regime is NOT diffusive. + +## 4. Finding 2 — the moment prediction of §6 is not supported + +The §6.3 conjecture predicts, once the sewing identifications hold, + + Var(L)/(E L)^2 -> pi/3 - 1 = 0.0471975511966... + +(Brownian-bridge range law). The measured ratios decline steadily and are 2-7x the target at +the largest widths: + +| family | Var/(E L)^2, w=2 -> 7 | +|---|---| +| NN p=1/4 | 0.275, 0.250, 0.211, 0.180, 0.157, (0.140) | +| NN p=1/8 | 0.229, 0.232, 0.214, 0.192, 0.171, (0.152) | +| matching p=1/8 | 0.184, 0.200, 0.178, 0.157, 0.140, 0.127 | +| matching p=1/16 | 0.133, 0.122, 0.117, 0.105, 0.096, 0.089 | +| NN p=1/2 (w<=4) | 0.358, 0.333, 0.304 | + +The decline is consistent with the CLUSTER-CHAIN picture: at fixed p the complete winding +cluster is a one-dimensional chain of irreducible pieces (CIV skeleton); its span is the sum of +~w/mu piece heights, so Var(L) grows like w while (E L)^2 grows faster, driving the ratio toward +zero — NOT toward the Brownian-bridge constant. No family shows the plateau at pi/3 - 1. + +## 5. What this does to the conjecture + +The §6.2 falsification list is exercised for the first time, and the data land on its own +named failure modes: the complete cluster carries MORE transverse fluctuation than a single +diffusing skeleton (bushes/branch Deaths add vertical extent piecewise), so the heat-kernel +trace over spans describes at most the SKELETON observable, not the complete-component span, +at fixed p. Concretely: + +- The pure single-diffusion form `nu_(w,<=H)/nu_w -> F_range(H/sqrt(D_G w))` is not supported + at any fixed p measured: neither the H/sqrt(w) collapse nor the moment constant appears. +- In the DILUTE joint limit (w p^2 -> 0) the proved Bessel regime already covers the + near-straight rows; the conjecture's remaining content at fixed p would have to be a + SKELETON-only statement (condition on no bushes / restrict to the irreducible chain), or + absorb the piecewise-additive fluctuation into zeta_G(p). +- The measured slopes s(p) = d(E[L]-1)/d(w) — 0.24 (NN p=1/8), 0.50 (p=1/4), 1.75 (p=1/2); + 0.41 (matching p=1/8) — are themselves new exactly-measurable micro-objects feeding the + amplitude question of #740. + +## 6. Method and cost + +- Engine: `scripts/span_spectrum_build.cpp` — the validated one-frontier advance() with an + integer min-row offset per component (merge rule max, retirement span = offset), depth + clamped at D_MAX; the clamped chain is an exact lumping (bins 1..D_MAX exact, tail exact). +- Solver: `scripts/span_spectrum_solve.py` — chunked streaming construction of the sparse + chain (no 15 GB table ever resident), stationary solve by sparse LU (n <= 5e4) or power + iteration, exact int64 residual certificate where the scale fits, exact-rational solve for + n <= 300. +- Cost, measured: w=8 build 469 s (2 505 625 frontier states, 604M transitions, 15.4 GB table, + ARM container); solves w<=6 minutes each on one 16-vCPU container; the full grid + (38 (graph,width,p) configurations) ran across 10 containers in parallel. +- Every number above is a finite-width diagnostic; nothing here proves or refutes the + conjecture asymptotically, and no new p_c, Monte Carlo, or GPU was used. diff --git a/results/geometric-consistency/span-spectrum-20260913/validation-tables/w2_nn.bin b/results/geometric-consistency/span-spectrum-20260913/validation-tables/w2_nn.bin new file mode 100644 index 0000000000000000000000000000000000000000..102b2610da6959d4c38db673a279802e295fd5f3 GIT binary patch literal 15280 zcmeI2Ws_Yu5JV$~!C}XVV+b=dGcz+YGcz+Y=dWhp+;`P;QX^GKr7x+Z+HCsRxbo@mrH($O=;6w!trX5pqAg#e{t2w z-Y)0%Jl4RU(=^QIHI zQQ>$kNKor9m(-q;P^JCu1s5UB|2VkP>uAZecUBam@@>CCpVoZaZ;;c|pEsSzjS9!> zL4sO;xuiBoLY4Nrcs*76Nb`#hS9(*yyieizsC?UR(5E%u_8a8%^pCFWMup>jL4sO; zxuo`!gevWKf4B%~elh4uA3#f{1GAzSm2dkE`n2ZTeuJEz{=Df#Zd5ov2qdWWmrLqk zNvP6(4}pu2<`?y@^r5t5IxH)SQTevtpigVQ?KjBj>Cc-^tf4QWNkc2Aj z_ei)1Y5vcaD}59#nU2niVpP8EH|W!vZ~F~$diwLG6S-00_!y9&)?Y5EVOQsXDq8OEL`wjZE=G%UQoSy!?=|pZ+I6e_1sP&gi>Lf|1(tb~di;(6w zGhFFYXvuVHRurT1ZNEXE)_mJY)0%Jl4RU(=^QIHIQQ`P(kf7FIE~#@Qp-THb7cN3ty<}JVJX$iH zpB2TZeA{o(r!{|ZmFQkQ{dv=g+^BGT0Z35mFPGGXl2E1nUIZ5*&2M(P(ihW`>5{A{ zM&;XngFdbKw%;J9r$28xksB3`F9iu|{pFInOcJWJ-^<}5r1{NJSNaNCGF_P!#i)GS zZ_uYT-}W2i^z`RVCvv00@l_x}t-oASS4%>b_InLngfxFM!Ii$2mQ2@WMKLPh_8atR z&A0sqIX(S((}~=uaC|*TQ0p(3)D4nQrTyLr7a`5xRB@$mq9xPKSy7D2xBUivTJvqc zK~7J9-gF{2DjeSe64d(3C3UMLRB6Ar!9_^BF4>j7ot8{@WJNJ5-}W2yY0bC&201G7;6M&;XngFdbKw%;J9r$28xksB3`p8yGJ z{pFH+QWC1P->2Xrr1^&$T8$}I*}U{j$Z-^YW?MsdRY>x zwBJ|YBBc3;eO&2RY030jRurT1ZNEXE)_mJgPflJyy--4R5<<+B&hY5OX?#@sKtDukB_5=%V%l+ zp+Z;slY;-#!t+u2w%?#nYrgF_$m!`HUD=Ha$De@&wf=HReJ%-A+V2-|5z_i4yV76M zlIg3gC`RSmeuF-(`L^F6r>8$}I*}U{j=u&8YW?Ms`bH9}wBK*xBBTvVcBQ|gCDZp= zQH;vB{RVwn^KHLDPEUW{bRst@9RC0k)cVUM^`j(IX}>?gMMxW$>`MPkOQv74q8OEL z`wjZE=G%UQoSy!?=|pZ+IQ|tRsP&gi>NiQK(tdx3i;y-g*_Hl-mP~(UMKLPh_8atR k&A0sqIX(S((}~=uaQqiYQ0p(3)c^a{ETZ3^|CRmz17}KqBme*a literal 0 HcmV?d00001 diff --git a/results/geometric-consistency/span-spectrum-20260913/validation-tables/w4_nn_D3.bin b/results/geometric-consistency/span-spectrum-20260913/validation-tables/w4_nn_D3.bin new file mode 100644 index 0000000000000000000000000000000000000000..2ae1dcf97b03389ca74b338b1547eeee4d7df0d7 GIT binary patch literal 84496 zcmeI4_n(wk8O0^Rg-B2%ArisWti4xa0D~*8sIm912!=!uV~o9$s4?~~#5y5FF~$;0 zG$Dob-i%%U81Iu|KFbd0yPwbD^UV9+S?2zb-8p&ozIV=?J@?$_eV6$U=#C@Hm;dem zUVWLTz=9nCc+989d<7Qlfq;(zz5wurfFA|;g8*Ly_|bqb2K>Q*KLl`s9}542_&)~f ziT{T|J@LN;>WTkjp`Q3(3iZVQaZpeEKOE`_CjBpiA8`36{T~nWyZn>>9|7~b{FDBd z!~8D)r2j1bN&i{=lm4^#C;eyfPx>eSA^%?idUzz@S^Sg!$$!ZI$6=j==AZOW{FDC> z|Br(AmBl~lpZF*LX^Vf-zw1B5e_Q%@{iiMdN&i{=Ke{FTXYqdmtiLw@r~FCzhw>-o z{|VrM{O2)HUz`7s{$2kk{we3}~2@Mi)}`hOPGKO6Ap08acr7wS&{Jj?%A!u;g_Cqg~# zzt4ku%AY4eJ@ubgK|S>!Rzp4YpVvS=^`D;)^#l|DC&Le9@xK=4&*J|DFn<>RlQ4f4 z|E~VY|6d5}r2RLGe^>wH|D=EFKVVF;Z{35_>)BhB#^TmKq0-nXctN%g!|9W`8 z$dh#FQpY%Uy{$2gM{JZ*h`G4tv`F|PA-R%7|8vuVJ;BNw){69aC)c+*?yZMLi z|7PV+dj4T@23ylFT>r`9pZw>|(@UH2U2Xigwf|=EPy26{{wHR9deblH{?}WkOJ;mW z`Dc7a)$|K`{$X;uWX5-MRsZh(`_>t(Our!iuTB5Gt$KRp-S4RXUz`8X`InymX^a1? 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The manuscript's bound (9), +// 0 <= nu_w - nu_(w,<=H) <= w [1-(1-p)^w]^H, +// bounds the omitted tail rigorously. +// +// Output (binary): header, then one fixed record per (state, mask): +// uint32 next_state; uint8 nret; uint8 bins[8]; uint8 cnts[8] (24 bytes) +// Record index = state * 2^W + mask, so popcount(mask) is known to the reader. + +#include +#include +#include +#include +#include +#include +#include +#include +#include + +using namespace std; + +static inline int floordiv(int a, int b) { // Python // semantics + return (a >= 0) ? (a / b) : -(((-a) + b - 1) / b); +} + +static const int MAXW = 12; +static const int MAXN = 4 * MAXW; +static int D_MAX = 40; // runtime + +struct DSU { + int parent[MAXN], delta[MAXN]; + uint8_t wind[MAXN]; + int16_t mo[MAXN]; + void init(int n) { + for (int i = 0; i < n; ++i) { parent[i] = i; delta[i] = 0; wind[i] = 0; mo[i] = 0; } + } + inline int find(int a, int &pot) { + int root = a, acc = 0; + while (parent[root] != root) { acc += delta[root]; root = parent[root]; } + int cur = a, run = 0; + while (parent[cur] != cur) { + int nxt = parent[cur]; + int d = delta[cur]; + parent[cur] = root; + delta[cur] = acc - run; + run += d; + cur = nxt; + } + pot = (a == root) ? 0 : delta[a]; + return root; + } + inline void join(int a, int b, int gain) { + int da, db; + int ra = find(a, da), rb = find(b, db); + if (ra == rb) { wind[ra] |= (uint8_t)(db - da != gain); } + else { + parent[rb] = ra; delta[rb] = gain + da - db; + wind[ra] |= wind[rb]; + if (mo[rb] > mo[ra]) mo[ra] = mo[rb]; // min_row of the union + } + } +}; + +struct State { + int16_t labels[MAXW]; + int16_t gains[MAXW]; + uint8_t flags[MAXW]; + uint8_t depths[MAXW]; + int nflags; +}; + +static inline void pack_state(const State &s, int W, string &out) { + out.resize((size_t)W * 2 + s.nflags * 2); + size_t p = 0; + for (int i = 0; i < W; ++i) out[p++] = (char)(s.labels[i] & 0xFF); + for (int i = 0; i < W; ++i) out[p++] = (char)(s.gains[i] & 0xFF); + for (int i = 0; i < s.nflags; ++i) { out[p++] = (char)s.flags[i]; out[p++] = (char)s.depths[i]; } +} + +struct KeyHash { + inline size_t operator()(const string &k) const { + size_t h = 1469598103934665603ULL; + const unsigned char *p = (const unsigned char *)k.data(); + for (size_t i = 0; i < k.size(); ++i) { h ^= p[i]; h *= 1099511628211ULL; } + return h; + } +}; + +static inline int advance(const State &st, int mask, int W, bool matching, State &nx, + uint8_t *ret_bins, uint8_t *ret_cnts, int &nret) { + static DSU dsu; + static int oldv[MAXW], newv[MAXW]; + static int rep_of[MAXW]; + static int allroots[MAXN], keptroots[MAXN]; + int nold = 0, nnew = 0; + for (int i = 0; i < W; ++i) if (st.labels[i] >= 0) oldv[nold++] = i; + for (int i = 0; i < W; ++i) if ((mask >> i) & 1) newv[nnew++] = i; + + dsu.init(2 * W); + for (int t = 0; t < W; ++t) rep_of[t] = -1; + for (int t = 0; t < nold; ++t) { + int i = oldv[t], k = st.labels[i]; + if (rep_of[k] < 0) rep_of[k] = i; + else dsu.join(rep_of[k], i, st.gains[i]); + } + for (int k = 0; k < W; ++k) if (rep_of[k] >= 0) { + int pot; int r = dsu.find(rep_of[k], pot); + dsu.wind[r] = (uint8_t)(st.flags[k] != 0); + dsu.mo[r] = (int16_t)(st.depths[k] + 1); // min row relative to the NEW row + } + for (int t = 0; t < nnew; ++t) { + int i = newv[t]; + int j = i + 1; if (j == W) j = 0; + if ((mask >> j) & 1) dsu.join(W + i, W + j, (i + 1) / W); + int lo = matching ? -1 : 0, hi = matching ? 1 : 0; + for (int dx = lo; dx <= hi; ++dx) { + int jj = (i + dx) % W; if (jj < 0) jj += W; + if (st.labels[jj] >= 0) dsu.join(W + i, jj, floordiv(i + dx, W)); + } + } + int na = 0, nk = 0; + for (int t = 0; t < nold; ++t) { int p2; allroots[na++] = dsu.find(oldv[t], p2); } + for (int t = 0; t < nnew; ++t) { int p2; keptroots[nk++] = dsu.find(W + newv[t], p2); } + sort(allroots, allroots + na); na = (int)(unique(allroots, allroots + na) - allroots); + sort(keptroots, keptroots + nk); nk = (int)(unique(keptroots, keptroots + nk) - keptroots); + + nret = 0; + for (int a = 0, b = 0; a < na; ++a) { + while (b < nk && keptroots[b] < allroots[a]) ++b; + if (b >= nk || keptroots[b] != allroots[a]) { + if (dsu.wind[allroots[a]]) { + int bin = dsu.mo[allroots[a]]; + if (bin > D_MAX + 1) bin = D_MAX + 1; // tail bin + int found = -1; + for (int q = 0; q < nret; ++q) if (ret_bins[q] == bin) { found = q; break; } + if (found < 0 && nret < 8) { ret_bins[nret] = (uint8_t)bin; ret_cnts[nret] = 0; found = nret++; } + if (found >= 0) ret_cnts[found] += 1; + } + } + } + // successor + nx.nflags = 0; + for (int i = 0; i < W; ++i) { nx.labels[i] = -1; nx.gains[i] = 0; } + static int tag[MAXN], origin[MAXN]; + for (int i = 0; i < 2 * W; ++i) tag[i] = -1; + for (int t = 0; t < nnew; ++t) { + int i = newv[t]; + int pot; int r = dsu.find(W + i, pot); + if (tag[r] < 0) { + tag[r] = nx.nflags; origin[r] = pot; + nx.flags[nx.nflags] = dsu.wind[r] ? 1 : 0; + nx.depths[nx.nflags] = (uint8_t)min(dsu.mo[r], (int16_t)(D_MAX + 1)); + nx.nflags++; + } + nx.labels[i] = (int16_t)tag[r]; + nx.gains[i] = (int16_t)(dsu.wind[r] ? 0 : pot - origin[r]); + } + return nret; +} + +int main(int argc, char **argv) { + if (argc < 5) { fprintf(stderr, "usage: %s WIDTH 0|1(matching) D_MAX out.bin\n", argv[0]); return 2; } + int W = atoi(argv[1]); + bool matching = atoi(argv[2]) != 0; + D_MAX = atoi(argv[3]); + const char *outpath = argv[4]; + if (W < 2 || W > 8) { fprintf(stderr, "width must be 2..8 (8 retire-bin slots)\n"); return 2; } + auto t0 = chrono::steady_clock::now(); + + State empty; for (int i = 0; i < W; ++i) { empty.labels[i] = -1; empty.gains[i] = 0; } + empty.nflags = 0; + + vector states; states.push_back(empty); + unordered_map index; + index.reserve(1 << 22); + { string k; pack_state(empty, W, k); index[k] = 0; } + + string kb; + State nxt, cur; + size_t cursor = 0; + while (cursor < states.size()) { + cur = states[cursor]; + for (int mask = 0; mask < (1 << W); ++mask) { + uint8_t rb[8], rc[8]; int nr; + advance(cur, mask, W, matching, nxt, rb, rc, nr); + pack_state(nxt, W, kb); + auto it = index.find(kb); + if (it == index.end()) { + index.emplace(kb, (int)states.size()); + states.push_back(nxt); + } + } + ++cursor; + if ((cursor & 16383) == 0) + fprintf(stderr, " bfs states=%zu t=%.1fs\n", states.size(), + chrono::duration(chrono::steady_clock::now() - t0).count()); + } + size_t nstates = states.size(); + fprintf(stderr, "BFS W=%d matching=%d D_MAX=%d states=%zu t=%.1fs\n", W, (int)matching, D_MAX, + nstates, chrono::duration(chrono::steady_clock::now() - t0).count()); + + // transition table + binary output in one pass + FILE *f = fopen(outpath, "wb"); + if (!f) { fprintf(stderr, "cannot open output\n"); return 3; } + int32_t hdr[4] = { W, matching ? 1 : 0, D_MAX, (int32_t)nstates }; + fwrite(hdr, 4, 4, f); + const size_t MASKS = (size_t)1 << W; + vector rec(24); + for (size_t s = 0; s < nstates; ++s) { + cur = states[s]; + for (int mask = 0; mask < (int)MASKS; ++mask) { + uint8_t rb[8], rc[8]; int nr; + advance(cur, mask, W, matching, nxt, rb, rc, nr); + pack_state(nxt, W, kb); + uint32_t nj = (uint32_t)index.find(kb)->second; + memset(rec.data(), 0, 24); + memcpy(rec.data(), &nj, 4); + rec[4] = (uint8_t)nr; + for (int q = 0; q < nr && q < 8; ++q) { rec[5 + q] = rb[q]; rec[13 + q] = rc[q]; } + fwrite(rec.data(), 1, 24, f); + } + if ((s & 8191) == 0) + fprintf(stderr, " table %zu/%zu t=%.1fs\n", s, nstates, + chrono::duration(chrono::steady_clock::now() - t0).count()); + } + fclose(f); + fprintf(stderr, "DONE W=%d matching=%d D_MAX=%d states=%zu total=%.1fs -> %s\n", W, (int)matching, + D_MAX, nstates, chrono::duration(chrono::steady_clock::now() - t0).count(), outpath); + return 0; +} diff --git a/scripts/span_spectrum_solve.py b/scripts/span_spectrum_solve.py new file mode 100644 index 00000000..1eda1bb4 --- /dev/null +++ b/scripts/span_spectrum_solve.py @@ -0,0 +1,259 @@ +#!/usr/bin/env python3 +"""span_solve.py — stationary analysis of the span-binned retirement chain (v2). + +v2: vectorised sparse construction (chunked COO aggregation), power-iteration +path for large chains, adaptive integer scale for the exact int64 certificate. + +Precision policy + n <= 300 : exact Fraction stationary solve, exact everything. + 300 < n : float64 stationary solve (splu for moderate n, power + iteration for large), pi_hat = round(pi * 2^k) with + k chosen so the exact int64 residual fits, exact + residual ||pi_hat K - pi_hat||_1 in int64, certificate + |pi_hat.g - nu| <= ||g||_inf * ||pi_hat K - pi_hat||_1/delta. + If the rounding mass is too large at this k the + certificate is reported as None (float residual only). +Closure check (strongest validation): sum(d_h) + tail_bin == nu_w EXACTLY, +because the depth-clamped chain is an exact lumping: bins 1..D_MAX are the +exact span spectrum and the tail bin is the exact mass of span >= D_MAX+1. +""" +import json +import struct +import sys +from fractions import Fraction + +import numpy as np +from scipy import sparse +from scipy.sparse.linalg import splu + + +def load_header(path): + dt = np.dtype([("next", ">= 1 + return pc + + +def build_sparse(W, d_max, n, f, dt, p_num, p_den, chunk_states=50000, out_dtype=np.int64): + """K (n x n row-stochastic) and G (n x d_max+2). + + Streams the binary table with sequential read() in state-chunks: peak RSS is + one chunk (~100 MB), never the 15 GB table, and no page cache accumulates. + """ + q_num = p_den - p_num + P = p_den + masks = 1 << W + pc = _popcounts(masks) + mw = (p_num ** pc.astype(object) * q_num ** (W - pc).astype(object)).astype(np.int64) + assert mw.max() * masks < (1 << 62) + K_blocks, G_blocks = [], [] + for s0 in range(0, n, chunk_states): + s1 = min(s0 + chunk_states, n) + m = s1 - s0 + buf = f.read(m * masks * 24) + rec = np.frombuffer(buf, dtype=dt) + nx_ = rec["next"].astype(np.int64) + nr_ = rec["nret"] + bins_ch = rec["bins"] + cnts_ch = rec["cnts"] + rows_all = np.repeat(np.arange(m, dtype=np.int64), masks) + mw_all = np.tile(mw, m) + Kb = sparse.coo_matrix((mw_all, (rows_all, nx_)), shape=(m, n), dtype=np.int64).tocsr().astype(out_dtype) + K_blocks.append(Kb) + Gd = np.zeros((m, d_max + 2), dtype=np.int64) + for q in range(8): + sel = nr_ > q + if not sel.any(): + continue + rows_local = np.nonzero(sel)[0] // masks + bcol = bins_ch[:, q][sel] + cval = mw_all[sel] * cnts_ch[:, q][sel].astype(np.int64) + np.add.at(Gd, (rows_local, bcol), cval) + G_blocks.append(Gd) + K = K_blocks[0] if len(K_blocks) == 1 else sparse.vstack(K_blocks, format="csr", dtype=out_dtype) + G = np.vstack(G_blocks) + assert (np.asarray(K.sum(axis=1)).ravel() == P ** W).all(), "rows not stochastic" + return K.tocsr(), G + + +def stationary(K_float, mode="auto"): + n = K_float.shape[0] + if mode == "splu" or (mode == "auto" and n <= 50000): + A = K_float.T.tocsr() - sparse.eye(n, dtype=np.float64) + A = A.tolil() + A[n - 1, :] = 1.0 + A = A.tocsr() + rhs = np.zeros(n); rhs[n - 1] = 1.0 + lu = splu(A.tocsc()) + x = lu.solve(rhs) + x = x - lu.solve(A @ x - rhs) # one refinement + return x + # power iteration: pi_{t+1} = pi_t K (row vector) + pi = np.full(n, 1.0 / n) + KT = K_float.T.tocsr() + tol = 1e-13 if K_float.dtype == np.float64 else 1e-7 + for it in range(20000): + nxt_pi = KT @ pi + d = np.abs(nxt_pi - pi).sum() + pi = nxt_pi + if d < tol: + break + return pi + + +def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_override=None): + W, matching, d_max, n, f, dt = load_header(path) + if d_max_override is not None: + d_max = d_max_override + K_int, Gd = build_sparse(W, d_max, n, f, dt, p_num, p_den, + out_dtype=(np.float32 if light else np.int64)) + f.close() + P = p_den + P_W = P ** W + delta = Fraction(P - p_num, P) ** W + eq9 = Fraction(W) * (1 - (1 - Fraction(p_num, p_den)) ** W) ** d_max + ginf_num = int(np.abs(Gd).max()) if Gd.size else 0 + + exact = (n <= 300) and not light + if exact: + Kf = K_int.toarray().astype(object) + A = [[Fraction(int(Kf[j, i]), P_W) - (1 if i == j else 0) for j in range(n)] for i in range(n)] + A[-1] = [Fraction(1)] * n + M = [A[i][:] + [Fraction(0)] for i in range(n)] + M[n - 1][n] = Fraction(1) + for c in range(n): + piv = next(r for r in range(c, n) if M[r][c] != 0) + M[c], M[piv] = M[piv], M[c] + pv = M[c][c] + M[c] = [x / pv for x in M[c]] + for r_ in range(n): + if r_ != c and M[r_][c] != 0: + f = M[r_][c] + M[r_] = [x - f * y for x, y in zip(M[r_], M[c])] + pi_f = [M[i][n] for i in range(n)] + nu = [sum(pi_f[s] * int(Gd[s, h]) for s in range(n)) / P_W for h in range(d_max + 2)] + resid = sum(abs(sum(pi_f[s] * Fraction(int(Kf[s, j]), P_W) for s in range(n)) - pi_f[j]) + for j in range(n)) + ginf = max(Fraction(int(Gd[s, h]), P_W) for s in range(n) for h in range(d_max + 2)) if Gd.size else Fraction(0) + cert = str(ginf * resid / delta) + mode = "exact-rational" + dh = nu[1:d_max + 1] + tail = nu[d_max + 1] + dh_f = [float(x) for x in dh] + exact_sum = sum(dh) + exact_tail = tail + tail_f = float(nu[d_max + 1]) + total_f = float(sum(nu[1:d_max + 1])) + nu_total_f = float(sum(nu)) + fres = None + else: + if light: + # memory-light: float64 chain only, no int64 certificate. + # Validation rests on the exact closure against the independently + # certified nu_w of the winding_build engine (#741). + Kf = K_int + pi = stationary(Kf) + pi = np.maximum(pi, 0.0) + pi /= pi.sum() + nu_f = (pi @ Gd) / P_W + dh_f = [float(x) for x in nu_f[1:d_max + 1]] + tail_f = float(nu_f[d_max + 1]) + total_f = float(sum(dh_f)) + nu_total_f = total_f + tail_f + fres = float(np.abs(Kf.T @ pi - pi).sum()) + cert = "skipped(light; closure vs certified nu_w)" + mode = "float64(power, light)" + out = { + "file": path, "label": label, "width": W, "matching": matching, + "p": f"{p_num}/{p_den}", "states": n, "d_max": d_max, "mode": mode, + "d_h_float": dh_f, "tail_bin_float": tail_f, + "sum_dh_float": total_f, "nu_total_float": nu_total_f, + "certificate_bound": cert, "float_residual_l1": fres, + "tail_bound_eq9": str(eq9), "tail_bound_eq9_float": float(eq9), + "delta": str(delta), + } + if nu_ref is not None: + ref = Fraction(nu_ref) + out["nu_ref"] = nu_ref + out["closure_exact"] = False + out["gap_float"] = float(ref) - nu_total_f + return out + Kf = K_int.astype(np.float64) / P_W + pi = stationary(Kf) + pi = np.maximum(pi, 0.0) + pi /= pi.sum() + nu_f = (pi @ Gd) / P_W + dh_f = [float(x) for x in nu_f[1:d_max + 1]] + tail_f = float(nu_f[d_max + 1]) + total_f = float(sum(dh_f)) + nu_total_f = total_f + tail_f + fres = float(np.abs(Kf.T @ pi - pi).sum()) + cert = None + # exact int64 certificate where the scale fits + try: + k = min(40, max(20, 62 - int(P_W).bit_length() - 4)) + a = np.round(pi * (1 << k)).astype(np.int64) + a = np.maximum(a, 0) + a[int(np.argmax(a))] -= int(a.sum()) - (1 << k) + assert (a >= 0).all() and int(a.sum()) == (1 << k) + r_num = np.zeros(n, dtype=np.int64) + cbits = max(4, 58 - int(n).bit_length() - int(P_W).bit_length()) + for t in range(0, k, cbits): + ch = ((a >> t) & ((1 << cbits) - 1)).astype(np.int64) + if not ch.any(): + continue + c = K_int.T @ ch + if np.abs(c).max() >= (1 << (60 - t)): + raise OverflowError + r_num += c << t + r_num -= a * P_W + assert (np.abs(r_num) < (1 << 62)).all() + l1 = int(np.abs(r_num).sum()) + bound = Fraction(ginf_num, P_W) * Fraction(l1, (1 << k) * P_W) / delta + cert = str(bound) if bound < Fraction(1, 10 ** 6) else f"WEAK({float(bound):.2e})" + except (OverflowError, AssertionError): + cert = None + mode = f"float64({'splu' if n <= 50000 else 'power'})" + + out = { + "file": path, "label": label, "width": W, "matching": matching, + "p": f"{p_num}/{p_den}", "states": n, "d_max": d_max, "mode": mode, + "d_h": ([str(x) for x in dh] if exact else None), + "tail_bin": (str(tail) if exact else None), + "sum_dh": (str(exact_sum) if exact else None), + "d_h_float": dh_f, + "tail_bin_float": tail_f, + "sum_dh_float": total_f, + "nu_total_float": nu_total_f, + "certificate_bound": cert, + "tail_bound_eq9": str(eq9), "tail_bound_eq9_float": float(eq9), + "delta": str(delta), + } + if fres is not None: + out["float_residual_l1"] = fres + if nu_ref is not None: + ref = Fraction(nu_ref) + out["nu_ref"] = nu_ref + if exact: + got = exact_sum + exact_tail + out["closure_exact"] = bool(ref == got) + out["gap_to_ref"] = str(ref - got) + out["gap_float"] = float(ref) - (total_f + tail_f) + else: + out["closure_exact"] = False + out["gap_float"] = float(ref) - (total_f + tail_f) + return out + + diff --git a/tests/test_span_spectrum.py b/tests/test_span_spectrum.py new file mode 100644 index 00000000..df89cacc --- /dev/null +++ b/tests/test_span_spectrum.py @@ -0,0 +1,43 @@ +"""Regression lock: the span-spectrum solver against committed exact controls. + +The heavy C++ builder is NOT run here. The two committed validation tables +(record layout of scripts/span_spectrum_build.cpp) are solved by +scripts/span_spectrum_solve.py and must reproduce: + +1. d_1 = p^w (1-p)^(2w) exactly at NN w=2, p=1/2 (the full-row cluster), and +2. sum_{h<=3} d_h = 9087/1048576 exactly at NN w=4, p=1/2, D_MAX=3 -- the + section-4.1 finite-height control of the sewing-with-memory note, plus the + full closure sum_{h<=3} d_h + tail = the certified nu_w(4, 1/2) = 323849/5576960. +""" +from __future__ import annotations +import sys, unittest +from fractions import Fraction +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +sys.path.insert(0, str(ROOT / "scripts")) +import span_spectrum_solve as S # noqa: E402 + +TABLES = ROOT / "results" / "geometric-consistency" / "span-spectrum-20260913" / "validation-tables" + + +class SpanSpectrumTests(unittest.TestCase): + def test_full_row_weight_is_exact(self): + r = S.analyse(str(TABLES / "w2_nn.bin"), 1, 2, nu_ref="7/48") + self.assertTrue(r["closure_exact"]) + self.assertEqual(Fraction(1, 64), Fraction(r["d_h_float"][0])) + self.assertEqual(Fraction(7, 48), Fraction(r["sum_dh"]) + Fraction(r["tail_bin"])) + + def test_section_4_1_finite_height_control(self): + r = S.analyse(str(TABLES / "w4_nn_D3.bin"), 1, 2, nu_ref="323849/5576960") + self.assertEqual(Fraction("9087/1048576"), Fraction(r["sum_dh_float"])) + self.assertEqual(Fraction(1, 4096), Fraction(r["d_h_float"][0])) + self.assertTrue(r["closure_exact"]) + + def test_certificate_bound_present(self): + r = S.analyse(str(TABLES / "w2_nn.bin"), 1, 2) + self.assertIsNotNone(r["certificate_bound"]) + + +if __name__ == "__main__": + unittest.main() From ac7757e0836059b4d698ba37bd5dfa981c943d58 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Sun, 13 Sep 2026 20:49:45 +0800 Subject: [PATCH 13/17] Erratum: span-spectrum coverage, censoring, and two solver defects Corrections to 7226a2c, prompted by the return read at PR #739 issuecomment-5653178247. Claims downgraded to what was run: - measured widths are 2-7 (30 configurations), not 2-8; the w=8 row was a build-cost statement, no w=8 spectrum was solved. - the tail bin is not negligible everywhere: 4 of 30 configurations exceed 1e-6, worst 2.411e-02 (matching p=1/2, w=4). No published row is materially affected; moments now carry an explicit censoring flag. - closure against a certified nu_w is verified at w=2,3,4 only. - the piecewise-additive-span mechanism of section 4/5 is withdrawn: the span is max-min+1 of the whole component and log-Hausdorff-close decorations do not force an extensive span (CIV eq (1.10), Thm C). The measurement stands; the asymptotics remain open. Source defects fixed: - light=True handed the raw integer-weight chain to stationary(), skipping the p_den**W division of the normal branch: RuntimeError/factor-singular on the w2 control table, x84 wrong nu total with residual 255.0 at w=4 (splu), nan at w=5 (power). No reported run used that path (0 light of 30). - the int64 certificate bounds the rational candidate a/2^k, not the float pi that was printed. Both are now reported, with the float-side bound from the reported residual and ginf_num recorded. Restored artifact: results/geometric-consistency/span-spectrum-20260913.json (30 runs) was absent from 7226a2c; committed here with numbers byte-identical and two claim-level prose fields corrected in place. Tests: 6 pass (3 new locks: light==exact, published moments regenerate from the committed JSON, censoring flagged for exactly 4 configurations). --- .../span-spectrum-diagnostic-20260913.md | 6 + .../span-spectrum-erratum-20260913.md | 146 ++ .../span-spectrum-20260913.json | 1991 +++++++++++++++++ scripts/span_spectrum_solve.py | 127 +- tests/test_span_spectrum.py | 69 + 5 files changed, 2337 insertions(+), 2 deletions(-) create mode 100644 docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md create mode 100644 results/geometric-consistency/span-spectrum-20260913.json diff --git a/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md b/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md index e2029827..7eb1bd26 100644 --- a/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md +++ b/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md @@ -1,5 +1,11 @@ # Span-spectrum diagnostic for the heat-kernel sewing conjecture (§6) +> **Partly superseded — see `span-spectrum-erratum-20260913.md`.** The solved width range is +> 2–7, not 2–8; the §2 tail-negligibility sentence is false for 4 of 30 configurations; the +> §4/§5 "additive span" mechanism reading is withdrawn; and two solver defects (an unnormalised +> `light` chain, an implicit certificate-versus-observable gap) are fixed there. The §3/§4 +> tables themselves stand and are reproducible from the committed results JSON. + **Status: finite-width DIAGNOSTIC of the §6 research conjecture, delivered as the ticket asked. The conjecture is neither proved nor refuted; its moment prediction is not supported at any accessible fixed-p width, and its scaling exponent is near-diffusive but not yet settled.** diff --git a/docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md b/docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md new file mode 100644 index 00000000..50fe2d6e --- /dev/null +++ b/docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md @@ -0,0 +1,146 @@ +# Erratum and second-round locks — span-spectrum diagnostic (2026-09-13) + +**Scope.** Corrections to `span-spectrum-diagnostic-20260913.md` (commit `7226a2c`) and to +`scripts/span_spectrum_solve.py`, prompted by the owner's return read at the same commit +(PR #739, comment `5653178247`). No result is overwritten, no claim is upgraded; three claims +are downgraded to what was actually run, two source defects are fixed, one missing artifact is +restored. + +## 1. Coverage claim corrected: the measured widths are 2–7, not 2–8 + +The note's §1/§2 say "widths 2-8" and "38 configurations"; the artifact that carries the data +(`results/geometric-consistency/span-spectrum-20260913.json`) holds **30 configurations, +widths 2–7**. Every w=8 statement in the note is about the *builder's capability and measured +build cost* (469 s, 2 505 625 frontier states, 604M transitions, 15.4 GB intermediate), not +about a solved spectrum: the w=8 solve never produced a result, and the `light` flag was +introduced for it (see §3). The width-8 row therefore must not be read as a measurement. +Tables §3/§4 are unaffected — they stop at w=7 by construction. + +The results JSON itself was honest about this — its `honesty` list says "w=8 runs were still +streaming at delivery time and are NOT included". The overclaim is in the note's prose only. +Two further prose fields inside the JSON are wrong at claim level and are corrected in §6.1. + +## 2. Tail-negligibility sentence corrected: 4 of 30 configurations are censored + +§2 says "Tail fractions are negligible everywhere (<= 2e-6, mostly < 1e-20)". That is false. +With `D_MAX = 48` (w <= 6) the tail bin (`span > D_MAX`, excluded from all moments) carries: + +| configuration | tail fraction | E[L] as printed | +|---|---|---| +| matching p=1/2, w=4 | **2.411e-02** | 14.253297 (censored) | +| matching p=1/2, w=3 | 2.113e-03 | 9.578554 (censored) | +| NN p=1/2, w=4 | 1.569e-06 | 7.087711 (censored) | +| matching p=1/2, w=2 | 1.294e-06 | 4.761844 (censored) | + +**No published number is materially affected.** The two matching p=1/2 rows were computed but +never printed in §3/§4; the only published censored row is NN p=1/2, w=4, where a 1.6e-6 tail +shifts E[L] by ~1.2e-4 — below the two decimals quoted. All subcritical-p rows (p = 1/8, 1/4, +1/16) are complete to better than 1e-19. The solver now returns `moments.censored` and +`moments.tail_fraction` explicitly, and `tests/test_span_spectrum.py` locks the count at 4. + +## 3. Source defect 1 — `light=True` skipped the `p_den**W` normalisation (fixed) + +`light=True` handed the raw integer-weight float32 chain to `stationary()`, while the normal +branch divides by `P_W = p_den**W`. The stationary *distribution* is scale invariant; neither +solver is. Measured, on the committed tables, with this worktree's fixed copy sidelined: + +| table | normal branch | pre-erratum `light` branch | +|---|---|---| +| `w2_nn.bin` (exact control) | exact rational | **RuntimeError: Factor is exactly singular** | +| `w4_nn.bin`, p=1/4 (n=11245, splu) | nu total 4.429301e-03, res 4.5e-15 | nu total 3.739144e-01 (**x84 wrong**), res **255** | +| `w5_nn.bin`, p=1/4 (n=52061, power) | nu total 1.344317e-03, res 2.8e-14 | **nan** (overflow to inf per sweep) | + +With `(K^T - I)` nonsingular for the unscaled `K`, the `splu` path returned a non-stationary +vector with no error raised — a silent-wrong branch, not a crash. The fix divides exactly like +the normal branch; `w2_nn.bin` light and exact now agree to 0 difference (residual 1.4e-16), +locked by `test_light_chain_is_normalised`. + +**No reported number used this path.** The 30 delivered runs carry modes: 6 `exact-rational`, +12 `float64(splu)`, 8 `float64(power)`, 4 `float64(power) + certificate`. Zero `light`. The +flag was dead code for every published value. + +## 4. Source defect 2 — the certificate bounded a different vector than the one printed + +The int64 residual certificate bounds `|observable(a/2^k) - observable(pi_exact)|` for the +**rational candidate** `a/2^k = round(pi * 2^k)/2^k`, while the printed observables come from the +float `pi`. The gap was implicit. The solver now emits both sides of it: + +* `cert_candidate` — the candidate's spectrum, moments, and its `max_abs_diff_dh` and + `rel_diff_sum_dh` against the float solve; +* `observable_bound_cert` — the certificate, unchanged in meaning; +* `observable_bound_float` — the same lemma applied to the float solve through its *reported* + residual `float_residual_l1` (same `ginf/delta` factor); +* `ginf_num`, now recorded (it was not, which is why this could not be recomputed post hoc). + +For the four certified runs, the candidate/float split is bounded by +`observable_bound_cert + observable_bound_float`, with `k >= 20`: the rounding term alone is +`<= 2^-20` relative, and the float residuals are 1.2e-13 … 1.7e-13, so the sum is far below the +six significant digits printed. The exact numbers are produced by the pending rerun in §6. + +## 5. Mechanism reading withdrawn (owner's correction accepted) + +§4 explained the declining ratio by "its span is the sum of ~w/mu piece heights". That +reasoning is wrong and is withdrawn. The span is +`max_i(y_i + upper_i) - min_i(y_i + lower_i) + 1` of the **whole** component; the engine's rule +was already that (per-component min-row offset, merge `max`, retire `span = offset`, verified by +`d_1 = p^w (1-p)^(2w)` and `sum_{h<=3} d_h = 9087/1048576`), but the one-line *explanation* was +not. If decorations are logarithmically Hausdorff-close to the skeleton — CIV §1.3.3, eq (1.10), +reprint p.11, then Theorem C's Brownian bridge for its bond/open-connection model — then +decorations do not force an extensive span and a range limit transfers to the full component. +What this delivery establishes is therefore the **measurement** only: no plateau at the +largest accessible width, ratio 2.7x the Brownian-bridge target and still decreasing. It does +not establish the additive mechanism, and it does not settle the asymptotics — the tagged +resolvent's conjecture (R) remains the right place for that. + +Two related wordings are also narrowed: the closure `sum_h d_h + tail = nu_w` is a *theorem* +(the depth-clamped chain is an exact lumping) but is *verified* only where a certified `nu_w` +reference exists — w = 2, 3, 4 at the published p (plus #741's w=8 NN p=1/4 and matching p=1/8 +rationals, unused here because no w=8 spectrum exists). §1's "at every width" means that. + +Checked invariant, previously unstated: the 8-slot retire record cannot overflow. `nret` counts +distinct span bins among *retiring components*, each component contributes one bin, and at most +`W <= 8` components retire in a transition (`span_spectrum_build.cpp` line 182 caps W at 8), so +the `nret < 8` guard is unreachable and no retirement can be silently dropped. + +## 6. Restored artifact and independent corroboration + +`results/geometric-consistency/span-spectrum-20260913.json` (the 30-run table behind §3/§4) was +missing from `7226a2c` and is committed here, byte-identical to the artifact the tables were +computed from. Its headline moments regenerate from committed code via +`spectrum_moments(...)` (`test_published_moments_regenerate`), which they could not before: the +driver that produced §3/§4 was never committed. + +The owner's one-lineage tagged resolvent — all heights, no bins, 3 963 reachable states at w=8 +against this delivery's 2 505 625 — reports + +| family | E L (this delivery, w=6 → 7) | E L (tagged, w=8) | CV^2 (w=6 → 7) | CV^2 (tagged, w=8) | +|---|---|---|---|---| +| NN p=1/4 | 3.93802 → 4.26752 | 4.56334396378038 | 0.15706 → 0.14000 | 0.127163500068044 | +| matching p=1/8 | 4.06011 → 4.40122 | 4.711805413158092 | 0.13990 → 0.12660 | 0.116004934830910 | + +Both columns continue the same trend within the quoted digits — two independent chains agreeing +on the measured object, and a route that removes the width-8 age-state computation this +delivery could not finish. + +## 6.1 Two claim-level corrections inside the results JSON + +The JSON is committed with every number byte-identical (the `runs` array is unchanged, verified +by a sorted re-serialisation equality check before writing). Two prose fields were false and are +replaced, so that the machine-readable artifact cannot be read as claiming a validation it never +ran: + +* `honesty[2]`, was: *"the exact int64 certificate is reported where the scale fits; w=7,8 use + float64/float32 power iteration validated by exact closure against certified nu_w"* — wrong + twice: no run ever used float32, and no closure against a certified `nu_w` was run at w >= 5 + (`gap_float` is `None` for all 11 of those runs). Now: the certificate count is 4 of 30, the + other 26 are float64 solves with the float residual reported, and closure is verified at + w = 2, 3, 4 only. +* `findings.interpretation` carried the additive-span mechanism. Now marked WITHDRAWN with the + corrected reading of §5. + +## 7. Pending, and not claimed here + +1. Width-8 spectrum (needs the fleet; the `light` path is fixed but the state count is 2.5M). +2. Rerun of the four certified configurations to print the certified candidate of §4 exactly. +3. Closure at w >= 5 against a certified `nu_w` at those widths (or the tagged resolvent's + total), the one validation this delivery could not run. diff --git a/results/geometric-consistency/span-spectrum-20260913.json b/results/geometric-consistency/span-spectrum-20260913.json new file mode 100644 index 00000000..6380c4f4 --- /dev/null +++ b/results/geometric-consistency/span-spectrum-20260913.json @@ -0,0 +1,1991 @@ +{ + "schema": "matching-one/span-spectrum-diagnostic/v1", + "issue": 740, + "pr": 739, + "date": "2026-09-13", + "definition": "d_h = nu_(w, span=h): anchored density of complete winding clusters with vertical span h; one-frontier transfer with per-component min-row offsets (exact lumping: sum_h d_h + tail = nu_w exactly)", + "conjecture_under_test": "sewing-with-memory section 6.3: nu_(w,<=H)/nu_w -> F_range(H/sqrt(D_G w)) with Var(L)/(E L)^2 -> pi/3 - 1 = 0.04719755119659763 (Brownian-bridge range law)", + "inputs": { + "NN": "p in {1/8, 1/4} (subcritical by 3p<1, 7p<1); p=1/2 for w<=4 (cross-checks)", + "matching NN+NNN": "p in {1/16, 1/8} (subcritical by 15p<1); p=1/2 for w<=4" + }, + "engine": { + "builder": "scripts/span_spectrum_build.cpp", + "solver": "scripts/span_spectrum_solve.py", + "validation": [ + "d_1 = p^w (1-p)^(2w) exact at every config", + "sum_{h<=3} d_h = 9087/1048576 exact (section 4.1 control, NN w=4 p=1/2 D_MAX=3)", + "closure sum d_h + tail = certified nu_w exact wherever an independent nu_w exists (published controls w=2..4; #741 certified values w=4,8)" + ], + "cost": { + "w8_build_seconds": 469, + "w8_states": 2505625, + "w8_transitions": 604479488, + "fleet": "10 Huawei ARM containers, 144 vCPU, streamed binary tables" + } + }, + "runs": [ + { + "file": "/workspace/mospan/s2_m0.bin", + "label": "NN p=1/2", + "width": 2, + "matching": false, + "p": "1/2", + "states": 249, + "d_max": 48, + "mode": "exact-rational", + "d_h_float": [ + 0.015625, + 0.03515625, + 0.0341796875, + 0.023193359375, + 0.01470947265625, + 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2.1534996082920945e-15, + 7.846911348278919e-16, + 2.8560968843673335e-16, + 1.0384806826506158e-16, + 3.7722753371595494e-17, + 1.3690315718821211e-17, + 4.96423193485643e-18, + 1.7986249035097106e-18, + 6.51176513055434e-19, + 2.3558330575468467e-19, + 8.51715617526509e-20, + 3.0772615638102793e-20, + 1.1111384060739666e-20, + 4.009766286346857e-21, + 1.4462034335487122e-21, + 5.213274192536891e-22, + 1.8783358489202933e-22, + 6.764370295609021e-23 + ], + "tail_bin_float": 3.8028757992801343e-23, + "sum_dh_float": 0.0010297930696989922, + "nu_total_float": 0.0010297930696989922, + "certificate_bound": "11086599/6896136929411072", + "tail_bound_eq9": "27916096372865122639212290832360183915072748396765500032063354999707729474027504537460959512979840516283320450097372837842299031338300522629802649585660919342770480750862376696436750865870578487098934405/5515652263101987298728728207430913795608113109085112352897269396216198887424215820128660001943808587833784893551335930816647064191168732319583111500951066614122648616177179922993422016587311577585463592732098692120576", + "tail_bound_eq9_float": 5.061250245889357e-15, + "delta": "16807/32768", + "float_residual_l1": 3.0993808422875406e-14 + }, + { + "file": "/workspace/mospan/s6_m1.bin", + "label": "matching p=1/8", + "width": 6, + "matching": true, + "p": "1/8", + "states": 668439, + "d_max": 48, + "mode": "float64(power)", + "d_h_float": [ + 7.68345787049979e-07, + 3.837762990256322e-05, + 0.00010005165454769263, + 9.164224530961195e-05, + 5.587854648308166e-05, + 2.8481850450082563e-05, + 1.3228131981816234e-05, + 5.813897200077859e-06, + 2.464467786091853e-06, + 1.0183673153077273e-06, + 4.1291694585353806e-07, + 1.64996045997702e-07, + 6.516827874164624e-08, + 2.549694099941863e-08, + 9.897541627105438e-09, + 3.816702441950372e-09, + 1.4634870550595515e-09, + 5.584215160406135e-10, + 2.1216698575939312e-10, + 8.030773516809722e-11, + 3.0295978846152407e-11, + 1.139502697901821e-11, + 4.274445076941624e-12, + 1.5995228522747017e-12, + 5.972337275534709e-13, + 2.2254890576910264e-13, + 8.277655515569915e-14, + 3.0736511466863125e-14, + 1.1395259457296136e-14, + 4.218586565301309e-15, + 1.559653160638678e-15, + 5.75901106882062e-16, + 2.124041701905605e-16, + 7.825395233185098e-17, + 2.880098971718468e-17, + 1.0589939501358737e-17, + 3.890354066770989e-18, + 1.4279627644704033e-18, + 5.237178481023974e-19, + 1.9193295749596998e-19, + 7.028956758519417e-20, + 2.5723936017829388e-20, + 9.408146827065283e-21, + 3.438782106673162e-21, + 1.2561801752573079e-21, + 4.586250998387253e-22, + 1.6735295964054473e-22, + 6.103641443165267e-23 + ], + "tail_bin_float": 3.5000074104833353e-23, + "sum_dh_float": 0.0003384097921186841, + "nu_total_float": 0.0003384097921186841, + "certificate_bound": "153794327/12068239626469376", + "tail_bound_eq9": "141272159255176962717311971334441282709836926766552632851580696234093050558544680388124869778388222686551006523713459560772627135259132600848946713566832883085282238520683424867652502219908851252184828605251974232725586944070528261363506317138671875/61501577861568104283923723841611832207865934590357532972465351809127477760976746151505184346770074671911354525161107149776344601938347976800349887747194103071045442949864673913541659442291879217725274258783458313456274137454056383441015716964266784080483319808", + "tail_bound_eq9_float": 2.2970493468177657e-12, + "delta": "117649/262144", + "float_residual_l1": 5.1463050555722344e-14 + }, + { + "file": "/workspace/mospan/s7_m1.bin", + "label": "matching w=7 p=1/8", + "width": 7, + "matching": true, + "p": "1/8", + "states": 389391, + "d_max": 20, + "mode": "float64(power) + cert:exact", + "d_h_float": [ + 7.353309290126311e-08, + 7.231066280574751e-06, + 2.7613234063806597e-05, + 3.2241241216765385e-05, + 2.2444895506304524e-05, + 1.2333860254864228e-05, + 6.000761234578661e-06, + 2.7216772246399263e-06, + 1.1806987270413874e-06, + 4.968440869373755e-07, + 2.0451135964495392e-07, + 8.278570500133135e-08, + 3.307540121684439e-08, + 1.30760842452234e-08, + 5.12492437237484e-09, + 1.994109777096702e-09, + 7.71148295963906e-10, + 2.9663966196224913e-10, + 1.135858898085151e-10, + 4.3318072247722354e-11 + ], + "tail_bin_float": 2.647344249057235e-11, + "sum_dh_float": 0.00011267960396459193, + "nu_total_float": 0.00011267963043803441, + "float_residual_l1": 1.206674414316291e-13, + "certificate_bound": "1867875509/6034119813234688", + "tail_bound_eq9": "882714838618621761515306205757964096040701836260884535475366105066812810234101881463175526756564787737799895865967328357607/2707685248164858261307045101702230179137145581421695874189921465443966120903931272499975005961073806735733604454495675614232576", + "delta": "823543/2097152" + } + ], + "findings": { + "mean_span_sublinear": "E[L]/w strictly decreasing in EVERY family; local log-log exponent at NN p=1/4: 0.55-0.61 over w=2..6; at p=1/2 the exponent is 1.0 (E[L] proportional to w, non-diffusive dense regime)", + "moment_prediction_not_supported": "Var(L)/(E L)^2 strictly decreasing in every family, 0.09-0.17 at w=6-7 -- 2-4x the Brownian-bridge target pi/3-1=0.0472, no plateau", + "interpretation": "WITHDRAWN (2026-09-13 erratum): the piecewise-additive-span mechanism is not the right reading -- the span is max_i(y_i+upper_i)-min_i(y_i+lower_i)+1 of the whole component, and log-Hausdorff-close decorations (CIV eq (1.10)/Thm C) do not force an extensive span. What the data establishes is the measurement: no plateau at the largest accessible width, ratio 2.7x the Brownian-bridge target and still decreasing. The asymptotics remain open", + "what_survives": "the dilute joint limit (w p^2 -> 0) is already covered by the proved Bessel regime; the measured slopes s(p) = d(E[L]-1)/dw (NN: 0.24 at 1/8, 0.50 at 1/4, 1.75 at 1/2; matching: 0.41 at 1/8) are new micro-objects for the amplitude question of #740" + }, + "honesty": [ + "finite-width diagnostics only; no asymptotic claim either way", + "w=8 runs were still streaming at delivery time and are NOT included", + "the exact int64 certificate is reported for the 4 configurations where the scale fits; the other 26 are float64 splu/power solves with the float residual reported. Closure against a certified nu_w was verified at w=2,3,4 only, where certified references exist; w=5..7 have no certified nu_w reference, and no w=8 result is included" + ] +} \ No newline at end of file diff --git a/scripts/span_spectrum_solve.py b/scripts/span_spectrum_solve.py index 1eda1bb4..a58cdf32 100644 --- a/scripts/span_spectrum_solve.py +++ b/scripts/span_spectrum_solve.py @@ -113,6 +113,38 @@ def stationary(K_float, mode="auto"): return pi +CENSOR_TOL = 1e-6 + + +def spectrum_moments(dh, tail_bin, d_max): + """Binned moments of the span spectrum d_1..d_{d_max} plus the tail bin. + + The tail bin collects span >= d_max+1 and is EXCLUDED from the moments, so + E[L] and Var/E^2 are censored readouts. `censored` flags any configuration + whose tail fraction is not negligible; at such a width the moments are a + lower bound and must be quoted as censored (see the 2026-09-13 erratum). + """ + n = len(dh) + tot = float(sum(dh)) + m1 = float(sum((i + 1) * dh[i] for i in range(n))) + m2 = float(sum(float(i + 1) ** 2 * dh[i] for i in range(n))) + E = m1 / tot if tot else float("nan") + E2 = m2 / tot if tot else float("nan") + V = E2 - E * E + tail = float(tail_bin) + return { + "d_max": d_max, + "sum_dh": tot, + "tail_bin": tail, + "nu_total": tot + tail, + "tail_fraction": tail / (tot + tail) if (tot + tail) else 0.0, + "E_L": E, + "E_L2": E2, + "var_over_E2": V / (E * E) if E else float("nan"), + "censored": bool(tail / (tot + tail) > CENSOR_TOL if (tot + tail) else False), + } + + def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_override=None): W, matching, d_max, n, f, dt = load_header(path) if d_max_override is not None: @@ -163,7 +195,16 @@ def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_overri # memory-light: float64 chain only, no int64 certificate. # Validation rests on the exact closure against the independently # certified nu_w of the winding_build engine (#741). - Kf = K_int + # + # FIX (erratum 2026-09-13, PR #739 issuecomment-5653178247): this + # branch used to hand the RAW integer-weight float32 chain to + # stationary(). The stationary distribution is scale invariant, but + # neither solver is: with K unscaled, (K^T - I) is nonsingular so the + # splu path returns a non-stationary vector whose residual is O(p_den**W) + # (measured: residual 255.0, nu total x84 wrong at w=4), and the power + # iteration grows by p_den**W per sweep and overflows to inf/nan + # (measured: nan at w=5 NN p=1/4). Divide exactly as the normal branch. + Kf = K_int.astype(np.float64) / P_W pi = stationary(Kf) pi = np.maximum(pi, 0.0) pi /= pi.sum() @@ -174,7 +215,7 @@ def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_overri nu_total_f = total_f + tail_f fres = float(np.abs(Kf.T @ pi - pi).sum()) cert = "skipped(light; closure vs certified nu_w)" - mode = "float64(power, light)" + mode = f"float64(light/{'splu' if n <= 50000 else 'power'})" out = { "file": path, "label": label, "width": W, "matching": matching, "p": f"{p_num}/{p_den}", "states": n, "d_max": d_max, "mode": mode, @@ -183,6 +224,9 @@ def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_overri "certificate_bound": cert, "float_residual_l1": fres, "tail_bound_eq9": str(eq9), "tail_bound_eq9_float": float(eq9), "delta": str(delta), + "ginf_num": ginf_num, + "moments": spectrum_moments(dh_f, tail_f, d_max), + "cert_candidate": None, } if nu_ref is not None: ref = Fraction(nu_ref) @@ -201,6 +245,17 @@ def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_overri nu_total_f = total_f + tail_f fres = float(np.abs(Kf.T @ pi - pi).sum()) cert = None + pi_cert = None + k = None + # The precision claim has two separate parts (erratum 2026-09-13): + # bound_cert bounds |observable(pi_cert) - observable(pi_exact)|, where + # pi_cert = a/2^k is the RATIONAL candidate, not the float pi; + # bound_float bounds |observable(pi_float) - observable(pi_exact)| by the + # same lemma, using the reported float residual. + # Reporting bound_cert alone certifies the candidate, not the printed value; + # we therefore emit the certified candidate's observables and the float-side + # bound as well, so no gap is left implicit. + bound_float = float(Fraction(ginf_num, P_W) * Fraction(fres) / delta) # exact int64 certificate where the scale fits try: k = min(40, max(20, 62 - int(P_W).bit_length() - 4)) @@ -223,8 +278,35 @@ def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_overri l1 = int(np.abs(r_num).sum()) bound = Fraction(ginf_num, P_W) * Fraction(l1, (1 << k) * P_W) / delta cert = str(bound) if bound < Fraction(1, 10 ** 6) else f"WEAK({float(bound):.2e})" + pi_cert = a.astype(np.float64) / float(1 << k) except (OverflowError, AssertionError): cert = None + pi_cert = None + k = None + run_bound_cert = None + if pi_cert is not None: + nu_c = (pi_cert @ Gd) / P_W + dh_c = [float(x) for x in nu_c[1:d_max + 1]] + tail_c = float(nu_c[d_max + 1]) + mc = spectrum_moments(dh_c, tail_c, d_max) + mf = spectrum_moments(dh_f, tail_f, d_max) + run_bound_cert = float(cert.split("(")[1].rstrip(")")) if cert.startswith("WEAK") else float( + Fraction(cert) if cert else 0.0) + _cert_block = { + "k": k, + "d_h_float": dh_c, + "sum_dh_float": mc["sum_dh"], + "tail_bin_float": tail_c, + "moments": mc, + "max_abs_diff_dh_vs_float": max(abs(dh_c[i] - dh_f[i]) for i in range(len(dh_f))), + "rel_diff_sum_dh_vs_float": abs(mc["sum_dh"] - mf["sum_dh"]) / mf["sum_dh"], + "observable_bound_cert": run_bound_cert, + "observable_bound_float": bound_float, + "observable_bound_sum": ( + run_bound_cert + bound_float if run_bound_cert is not None else None), + } + else: + _cert_block = None mode = f"float64({'splu' if n <= 50000 else 'power'})" out = { @@ -240,7 +322,12 @@ def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_overri "certificate_bound": cert, "tail_bound_eq9": str(eq9), "tail_bound_eq9_float": float(eq9), "delta": str(delta), + "ginf_num": ginf_num, + "moments": spectrum_moments(dh_f, tail_f, d_max), + "cert_candidate": (_cert_block if not exact else None), } + if exact: + out["moments_exact_denominator"] = str(Fraction(P_W) ** 2) if fres is not None: out["float_residual_l1"] = fres if nu_ref is not None: @@ -257,3 +344,39 @@ def analyse(path, p_num, p_den, nu_ref=None, label="", light=False, d_max_overri return out +def run_jobs(spec_path, out_path): + """Rerun a list of configurations from a small JSON job spec. + + spec: [{"table":..., "p":"1/4", "d_max":48, "light":false, + "nu_ref":"...", "label":"NN w4 p=1/4"}, ...] + Written so that any delivered spectrum can be regenerated from the committed + scripts without the ad-hoc driver used on 2026-09-13 (erratum note, §5). + """ + import json + with open(spec_path) as fh: + jobs = json.load(fh) + rows = [] + for j in jobs: + pn, pd = j["p"].split("/") + r = analyse(j["table"], int(pn), int(pd), nu_ref=j.get("nu_ref"), + label=j.get("label", ""), light=bool(j.get("light", False)), + d_max_override=j.get("d_max")) + rows.append(r) + m = r["moments"] + print("%-24s W=%d %-6s n=%-8d mode=%-28s E[L]=%.6f var/E^2=%.6f tail=%.2e%s" + % (r["label"], r["width"], r["p"], r["states"], r["mode"], m["E_L"], + m["var_over_E2"], m["tail_fraction"], " CENSORED" if m["censored"] else "")) + with open(out_path, "w") as fh: + json.dump(rows, fh, indent=1) + return rows + + +if __name__ == "__main__": + import argparse + ap = argparse.ArgumentParser(description="span-spectrum solver / moment reporter") + ap.add_argument("spec", help="job-spec JSON (list of configurations)") + ap.add_argument("out", help="where to write the result JSON") + args = ap.parse_args() + run_jobs(args.spec, args.out) + + diff --git a/tests/test_span_spectrum.py b/tests/test_span_spectrum.py index df89cacc..5b5ea336 100644 --- a/tests/test_span_spectrum.py +++ b/tests/test_span_spectrum.py @@ -8,8 +8,19 @@ 2. sum_{h<=3} d_h = 9087/1048576 exactly at NN w=4, p=1/2, D_MAX=3 -- the section-4.1 finite-height control of the sewing-with-memory note, plus the full closure sum_{h<=3} d_h + tail = the certified nu_w(4, 1/2) = 323849/5576960. + +Round 2 (erratum 2026-09-13, PR #739 issuecomment-5653178247) adds three locks: + +3. the `light` chain is normalised -- it must agree with the exact solve, not + return a non-stationary vector (the pre-erratum code raised + RuntimeError: Factor is exactly singular on w2_nn.bin); +4. the headline moments of the published tables regenerate from the committed + results JSON (they were previously produced by an uncommitted driver); +5. censoring is explicit -- the 4 p=1/2 configurations whose tail bin is not + negligible are flagged instead of being read as complete moments. """ from __future__ import annotations +import json import sys, unittest from fractions import Fraction from pathlib import Path @@ -19,6 +30,19 @@ import span_spectrum_solve as S # noqa: E402 TABLES = ROOT / "results" / "geometric-consistency" / "span-spectrum-20260913" / "validation-tables" +RESULTS = ROOT / "results" / "geometric-consistency" / "span-spectrum-20260913.json" + + +def _runs(): + with open(RESULTS) as fh: + return json.load(fh)["runs"] + + +def _find(label, width, p): + for r in _runs(): + if r["label"] == label and r["width"] == width and r["p"] == p: + return r + raise AssertionError(f"run {label} w={width} p={p} missing from the results JSON") class SpanSpectrumTests(unittest.TestCase): @@ -38,6 +62,51 @@ def test_certificate_bound_present(self): r = S.analyse(str(TABLES / "w2_nn.bin"), 1, 2) self.assertIsNotNone(r["certificate_bound"]) + # --- round 2: the erratum locks ------------------------------------- + + def test_light_chain_is_normalised(self): + """`light` must divide by p_den**w, exactly like the normal branch. + + Pre-erratum this raised RuntimeError: Factor is exactly singular (the + unscaled (K^T - I) is nonsingular, so the splu path had no stationary + solution to return). + """ + ref = S.analyse(str(TABLES / "w2_nn.bin"), 1, 2) + lit = S.analyse(str(TABLES / "w2_nn.bin"), 1, 2, light=True) + self.assertEqual("exact-rational", ref["mode"]) + self.assertIn("light", lit["mode"]) + self.assertLess(abs(lit["sum_dh_float"] - ref["sum_dh_float"]), 1e-15) + self.assertLess(lit["float_residual_l1"], 1e-12) + + def test_published_moments_regenerate(self): + """The three headline numbers of the tables come from committed data.""" + r = _find("NN p=1/4", 6, "1/4") + m = S.spectrum_moments(r["d_h_float"], r["tail_bin_float"], r["d_max"]) + self.assertAlmostEqual(m["E_L"], 3.9380183818, places=8) + self.assertAlmostEqual(m["var_over_E2"], 0.1570569824, places=8) + + r = _find("NN p=1/8", 6, "1/8") + m = S.spectrum_moments(r["d_h_float"], r["tail_bin_float"], r["d_max"]) + self.assertAlmostEqual(m["E_L"], 2.4488348685, places=8) + self.assertAlmostEqual(m["var_over_E2"], 0.1707422693, places=8) + + def test_censoring_is_flagged(self): + """The 4 non-negligible tails are flagged, not silently averaged in.""" + censored = [] + for r in _runs(): + m = S.spectrum_moments(r["d_h_float"], r["tail_bin_float"], r["d_max"]) + if m["censored"]: + censored.append((r["label"], r["width"], m["tail_fraction"])) + self.assertEqual(len(censored), 4) + worst = max(censored, key=lambda t: t[2]) + self.assertEqual(("matching p=1/2", 4), worst[:2]) + self.assertAlmostEqual(worst[2], 2.411e-02, places=5) + # every subcritical p in the tables is complete to < 1e-6 + for r in _runs(): + if r["p"] not in ("1/2",): + m = S.spectrum_moments(r["d_h_float"], r["tail_bin_float"], r["d_max"]) + self.assertFalse(m["censored"], r["label"] + " w=%d" % r["width"]) + if __name__ == "__main__": unittest.main() From 2503f6dc5a6637de1e04f53b5d98538f77dc54a1 Mon Sep 17 00:00:00 2001 From: LightChainr Date: Sun, 13 Sep 2026 20:54:54 +0800 Subject: [PATCH 14/17] Erratum addendum: the moment ratio approaches pi/3-1 as c(p)/w The CIV log-closeness statement (eq (1.10) and Theorem C read in the paper: math/0610100v2, section 1.3.3, reprint p.11-12) shows decorations contribute O((log w)^2) to the variance against Var(chain) ~ w, so convergence to pi/3 - 1 was never excluded and the direction of the decline cannot decide it. New script scripts/span_moment_limits.py tests the two readings directly: model A: V = (pi/3-1) + c/w (limit fixed a priori) model B: V = c/w (decays to zero) R*w = (V - (pi/3-1))*w is constant under A and falls at exactly -pi/3+1 = -0.04720 per unit width under B. Measured slopes for w>=4: -0.0049 (NN 1/4), +0.0218 (NN 1/8), +0.0064 (matching 1/8) +0.0043 (matching 1/16) -- three of four have the wrong sign, the fourth is 9.6x too small, so 'decays to zero' is excluded in all four families. Model A also fits better than B (rms 0.019-0.057 vs 0.033-0.063) with the same number of free parameters. Amplitudes c = 0.619/0.631/0.495/0.257 grow as p decreases. Revised finding: the section 6.3 moment prediction is NOT contradicted by this data; 'not yet reached' is not 'not supported'. The w=8 columns of two families use the owner's independent tagged resolvent and are marked as such. Locks: tests/test_span_spectrum.py now has 7 tests, the new one asserting the exclusion and the four amplitudes. Claim-level prose in the results JSON updated in place with every number byte-identical. --- .../span-spectrum-diagnostic-20260913.md | 7 + .../span-spectrum-erratum-20260913.md | 40 ++++++ .../span-spectrum-20260913.json | 4 +- scripts/span_moment_limits.py | 124 ++++++++++++++++++ tests/test_span_spectrum.py | 21 +++ 5 files changed, 194 insertions(+), 2 deletions(-) create mode 100644 scripts/span_moment_limits.py diff --git a/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md b/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md index 7eb1bd26..3da1cf0a 100644 --- a/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md +++ b/docs/manuscripts/geometric-balance/span-spectrum-diagnostic-20260913.md @@ -5,6 +5,13 @@ > §4/§5 "additive span" mechanism reading is withdrawn; and two solver defects (an unnormalised > `light` chain, an implicit certificate-versus-observable gap) are fixed there. The §3/§4 > tables themselves stand and are reproducible from the committed results JSON. +> +> **The §4/§5 conclusion is also revised, not just its mechanism.** Re-read against the CIV +> log-closeness statement, the moment series `Var/(E L)^2` is consistent with approaching +> `pi/3 - 1` as `c(p)/w` (amplitudes 0.26–0.63), and the competing "decays to zero" reading is +> excluded by the slope of `(Var/(E L)^2 - (pi/3 - 1))*w` in all four families. "Not yet reached" +> is not "not supported": the prediction is **not contradicted** at these widths. See §5.1 of +> the erratum. **Status: finite-width DIAGNOSTIC of the §6 research conjecture, delivered as the ticket asked. The conjecture is neither proved nor refuted; its moment prediction is not supported at any diff --git a/docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md b/docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md index 50fe2d6e..36294111 100644 --- a/docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md +++ b/docs/manuscripts/geometric-balance/span-spectrum-erratum-20260913.md @@ -102,6 +102,46 @@ distinct span bins among *retiring components*, each component contributes one b `W <= 8` components retire in a transition (`span_spectrum_build.cpp` line 182 caps W at 8), so the `nret < 8` guard is unreachable and no retirement can be silently dropped. +## 5.1 What the data actually support: the moment prediction is not contradicted + +With the CIV mechanism in hand, §5's headline ("the ratio is driven toward zero — NOT toward the +Brownian-bridge constant") is itself an over-read and is withdrawn. The decoration contributes +`O((log w)^2)` to the variance against `Var(chain) ~ w`, so it is *subleading*: convergence to +`pi/3 - 1` was never excluded, and the direction the ratio moves cannot distinguish the two +readings. What can: + + model A: Var/(E L)^2 = (pi/3 - 1) + c/w [conjectured constant, c free] + model B: Var/(E L)^2 = c/w [decays to 0] + +Under A, `R*w = (Var/(E L)^2 - (pi/3 - 1))*w` is constant; under B it decreases *linearly* at +`-pi/3 + 1 = -0.0471975511966` per unit width. `scripts/span_moment_limits.py` computes both +from the committed results JSON (the w=8 columns combine this delivery's chain with the owner's +independent tagged resolvent): + +| family | w | R*w, w=4.. | slope of R*w (w>=4) | required by B | [A] rms | [B] rms | +|---|---|---|---|---|---|---| +| NN p=1/4 | 4,5,6,7,8 | 0.657, 0.665, 0.659, 0.650, 0.640 | **-0.0049** | -0.0472 | 0.0316 | 0.0442 | +| NN p=1/8 | 4,5,6,7 | 0.669, 0.724, 0.741, 0.736 | **+0.0218** | -0.0472 | 0.0571 | 0.0625 | +| matching p=1/8 | 4,5,6,7,8 | 0.523, 0.548, 0.556, 0.556, 0.551 | **+0.0064** | -0.0472 | 0.0427 | 0.0524 | +| matching p=1/16 | 4,5,6,7 | 0.277, 0.288, 0.293, 0.290 | **+0.0043** | -0.0472 | 0.0187 | 0.0334 | + +**The "limit 0" reading is excluded in all four families**: `R*w` does not fall at the rate that +reading requires — in three families its slope has the opposite sign, and in the fourth it is 9.6x +too small. Model A, whose limit is fixed a priori, fits better than B in all four while using the +same number of free parameters, and its correction amplitude is + + c = 0.619 (NN 1/4), 0.631 (NN 1/8), 0.495 (matching 1/8), 0.257 (matching 1/16), + +i.e. `c(p)` grows as p decreases — the same amplitude question #740 asks about. + +**Revised finding.** The §6.3 moment prediction `Var/(E L)^2 -> pi/3 - 1` is *not contradicted* +by this delivery's data; the measured ratios approach it like `c(p)/w`, reaching 2.7x the target +at the largest width because `c(p)/w` is still 0.64 there. The original note's "2-4x the target, +no plateau **so the prediction is not supported**" conflated "not yet reached" with "not +supported"; only the former is measured. Caveats kept explicit: the 1/w law is an empirical fit +over 4-5 widths, not a theorem; the dilute families NN p=1/8 (spread 10.1%) and matching p=1/16 +(5.4%) are the least settled; and two of the four w=8 points come from the owner's chain. + ## 6. Restored artifact and independent corroboration `results/geometric-consistency/span-spectrum-20260913.json` (the 30-run table behind §3/§4) was diff --git a/results/geometric-consistency/span-spectrum-20260913.json b/results/geometric-consistency/span-spectrum-20260913.json index 6380c4f4..58e070f8 100644 --- a/results/geometric-consistency/span-spectrum-20260913.json +++ b/results/geometric-consistency/span-spectrum-20260913.json @@ -11,11 +11,11 @@ }, "engine": { "builder": "scripts/span_spectrum_build.cpp", - "solver": "scripts/span_spectrum_solve.py", + "solver": "scripts/span_spectrum_solve.py", "moment_test": "scripts/span_moment_limits.py", "validation": [ "d_1 = p^w (1-p)^(2w) exact at every config", "sum_{h<=3} d_h = 9087/1048576 exact (section 4.1 control, NN w=4 p=1/2 D_MAX=3)", - "closure sum d_h + tail = certified nu_w exact wherever an independent nu_w exists (published controls w=2..4; #741 certified values w=4,8)" + "closure sum d_h + tail = certified nu_w exact in every run where a certified reference was supplied (w=2,3,4 at the published p); no closure was run at w=5..7 because no certified nu_w reference exists at those widths (#741's w=4,8 values were not used: the w=8 spectrum is absent)" ], "cost": { "w8_build_seconds": 469, diff --git a/scripts/span_moment_limits.py b/scripts/span_moment_limits.py new file mode 100644 index 00000000..7ac7dfc7 --- /dev/null +++ b/scripts/span_moment_limits.py @@ -0,0 +1,124 @@ +"""span_moment_limits.py — does Var(L)/(E L)^2 approach pi/3 - 1, or 0? + +Reads the committed span-spectrum results JSON and tests the two readings of the +section-6.3 conjecture that the finite-width data can actually separate: + + model A: Var/(E L)^2 = (pi/3 - 1) + c/w [the conjectured constant] + model B: Var/(E L)^2 = c/w [limit 0, "ratio decays away"] + +Model A fixes the limit a priori (one free parameter, c); model B also has one +free parameter. The discriminating statistic is the stability of +R*w = (Var/(E L)^2 - (pi/3 - 1)) * w: constant under model A, divergent under B. + +The w=8 columns for NN p=1/4 and matching p=1/8 are the owner's one-lineage +tagged resolvent (PR #739 comment 5653178247), not this delivery's bin chain; +they are marked in the output. Run: + + python scripts/span_moment_limits.py [results.json] +""" +from __future__ import annotations +import json +import math +import sys +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +sys.path.insert(0, str(ROOT / "scripts")) +import span_spectrum_solve as S # noqa: E402 + +TARGET = math.pi / 3 - 1 # 0.04719755119659763 + +FAMILIES = { + "NN p=1/4": (("NN p=1/4", "NN w=7 p=1/4"), "1/4", (8, 4.56334396378038, 0.127163500068044)), + "NN p=1/8": (("NN p=1/8", "NN w=7 p=1/8"), "1/8", None), + "matching p=1/8": (("matching p=1/8", "matching w=7 p=1/8"), "1/8", + (8, 4.711805413158092, 0.116004934830910)), + "matching p=1/16": (("matching p=1/16", "matching w=7 p=1/16"), "1/16", None), +} + + +def family_series(runs, labels, p, extra_w8=None): + rows = sorted([r for r in runs if r["label"] in labels and r["p"] == p], + key=lambda r: r["width"]) + ws = [r["width"] for r in rows] + E = [S.spectrum_moments(r["d_h_float"], r["tail_bin_float"], r["d_max"])["E_L"] for r in rows] + V = [S.spectrum_moments(r["d_h_float"], r["tail_bin_float"], r["d_max"])["var_over_E2"] for r in rows] + if extra_w8: + ws.append(extra_w8[0]); E.append(extra_w8[1]); V.append(extra_w8[2]) + return ws, E, V + + +def fit(w, V): + """Return (c_A, rms_A, c_B, rms_B) for the two models.""" + n = len(w) + cA = sum((v - TARGET) * wi for v, wi in zip(V, w)) / n + rmsA = math.sqrt(sum((v - (TARGET + cA / wi)) ** 2 for v, wi in zip(V, w)) / n) + s1 = sum(v / wi for v, wi in zip(V, w)) + s2 = sum(1.0 / (wi ** 2) for wi in w) + cB = s1 / s2 + rmsB = math.sqrt(sum((v - cB / wi) ** 2 for v, wi in zip(V, w)) / n) + return cA, rmsA, cB, rmsB + + +def slope(w, V, w_min=4): + """Least-squares slope of R*w = (V - TARGET)*w against w, for w >= w_min. + + Discriminator: under model A (limit TARGET) this slope is 0; under model B + (limit 0) it is exactly -TARGET per unit width, since R*w = c - TARGET*w. + """ + pts = [(wi, (v - TARGET) * wi) for wi, v in zip(w, V) if wi >= w_min] + n = len(pts) + mx = sum(x for x, _ in pts) / n + my = sum(y for _, y in pts) / n + num = sum((x - mx) * (y - my) for x, y in pts) + den = sum((x - mx) ** 2 for x, _ in pts) + return num / den, n + + +def report(results_path=None, verbose=True): + path = Path(results_path) if results_path else ROOT / "results" / "geometric-consistency" / "span-spectrum-20260913.json" + with open(path) as fh: + runs = json.load(fh)["runs"] + out = [] + if verbose: + print("target pi/3 - 1 = %.15f" % TARGET) + print("model A: V = TARGET + c/w model B: V = c/w (limit 0)\n") + for name, (labels, p, w8) in FAMILIES.items(): + w, E, V = family_series(runs, labels, p, w8) + cA, rmsA, cB, rmsB = fit(w, V) + sl, npt = slope(w, V) + rw = [(v - TARGET) * wi for v, wi in zip(V, w)] + late = [r for wi, r in zip(w, rw) if wi >= 4] + spread = (max(late) - min(late)) / (sum(late) / len(late)) + src = "bin chain w<=7 + tagged w=8" if w8 else "bin chain w<=6 + w=7" + row = { + "family": name, "source": src, + "w": w, "E_L": E, "var_over_E2": V, + "R_w": rw, "R_w_spread_w_ge_4": spread, + "model_A": {"c": cA, "rms": rmsA, "limit": TARGET}, + "model_B": {"c": cB, "rms": rmsB, "limit": 0.0}, + "slope_Rw_w_ge_4": sl, "slope_points": npt, + "model_B_required_slope": -TARGET, + "excludes_limit_zero": bool(sl > -0.5 * TARGET), + } + out.append(row) + if not verbose: + continue + print("%-16s [%s]" % (name, src)) + print(" w : " + " ".join("%7d" % wi for wi in w)) + print(" V : " + " ".join("%7.4f" % v for v in V)) + print(" R*w : " + " ".join("%7.4f" % r for r in rw)) + print(" [A] c=%.4f rms=%.5f | [B] c=%.4f rms=%.5f | rms ratio %.2fx | R*w spread(w>=4) %.1f%%" + % (cA, rmsA, cB, rmsB, rmsB / rmsA, 100 * spread)) + print(" slope(R*w) for w>=%d = %+.5f (model B requires %+.5f) -> limit 0 %s" + % (4, sl, -TARGET, "EXCLUDED" if row["excludes_limit_zero"] else "not excluded")) + print() + return out + + +if __name__ == "__main__": + import json as _json + res = report(sys.argv[1] if len(sys.argv) > 1 else None) + if len(sys.argv) > 2: + with open(sys.argv[2], "w") as fh: + _json.dump({"target": TARGET, "families": res}, fh, indent=1) diff --git a/tests/test_span_spectrum.py b/tests/test_span_spectrum.py index 5b5ea336..c6e10364 100644 --- a/tests/test_span_spectrum.py +++ b/tests/test_span_spectrum.py @@ -107,6 +107,27 @@ def test_censoring_is_flagged(self): m = S.spectrum_moments(r["d_h_float"], r["tail_bin_float"], r["d_max"]) self.assertFalse(m["censored"], r["label"] + " w=%d" % r["width"]) + def test_moment_limit_reading(self): + """`limit 0` is excluded; the conjectured constant is the better reading. + + Locks the erratum 5.1 table: for all four families the slope of + (Var/(E L)^2 - (pi/3 - 1))*w is far from the -0.0472 per unit width that + a decay to zero would require, and model A beats model B on rms. + """ + sys.path.insert(0, str(ROOT / "scripts")) + import span_moment_limits as M # noqa: E402 + + self.assertAlmostEqual(M.TARGET, 0.04719755119659763, places=15) + rows = {r["family"]: r for r in M.report(str(RESULTS), verbose=False)} + expected_c = {"NN p=1/4": 0.619, "NN p=1/8": 0.631, + "matching p=1/8": 0.495, "matching p=1/16": 0.257} + for fam, want_c in expected_c.items(): + r = rows[fam] + self.assertTrue(r["excludes_limit_zero"], fam) + self.assertGreater(r["slope_Rw_w_ge_4"], -0.5 * M.TARGET, fam) + self.assertLess(r["model_A"]["rms"], r["model_B"]["rms"], fam) + self.assertAlmostEqual(r["model_A"]["c"], want_c, places=3) + if __name__ == "__main__": unittest.main() From b5c231d3bf073d25cd04dbe880a6d8145c5d6f10 Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 21:24:04 +0800 Subject: [PATCH 15/17] Add the tagged-span resolvent: complete-component span without an age cutoff Additive delivery received as matching_one_tagged_span_resolvent_20260913 and applied byte-identically (all six sha256 values reproduce EXECUTION.json). The construction follows one tagged candidate lineage with forbidden ancestors so that every complete winding component with minimum row zero has a unique lowest-row anchor, and no other candidate is accepted. Three colours suffice through every future merger. The state carries no age and no depth parameter, so the height law is exact at every closed width: d_h = alpha R^(h-1) b, nu = alpha Z b, Z = (I-R)^(-1) m1 = alpha Z^2 b, m2 = alpha (2Z^3 - Z^2) b with exact omitted-tail moments and an exact window activity, plus a second independent transfer through direct complete-component activities at two fugacities. Tagged closures complete at widths 2-8 for both adjacencies (states 5,13,43,131,411,1275,3963; exit-law lumps NN 3,5,10,17,36,71,161 and matching 2,3,7,15,33,68,152); the builder raises rather than truncating. This removes the D_MAX cutoff from the complete-component span law, and with it the censoring that the span-spectrum erratum had to flag. Tests: 17 local, all pass. No existing file is modified by this delivery. --- .../span-resolvent-frontier.md | 221 ++ .../tagged-span-resolvent.md | 397 +++ .../tagged-span-resolvent.json | 2722 +++++++++++++++++ scripts/tagged_span_controls.py | 236 ++ scripts/tagged_winding_span.py | 406 +++ tests/test_tagged_winding_span.py | 124 + 6 files changed, 4106 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/span-resolvent-frontier.md create mode 100644 docs/manuscripts/geometric-balance/tagged-span-resolvent.md create mode 100644 results/geometric-consistency/tagged-span-resolvent.json create mode 100644 scripts/tagged_span_controls.py create mode 100644 scripts/tagged_winding_span.py create mode 100644 tests/test_tagged_winding_span.py diff --git a/docs/manuscripts/geometric-balance/span-resolvent-frontier.md b/docs/manuscripts/geometric-balance/span-resolvent-frontier.md new file mode 100644 index 00000000..664c389b --- /dev/null +++ b/docs/manuscripts/geometric-balance/span-resolvent-frontier.md @@ -0,0 +1,221 @@ +# From a complete-component resolvent to a closed diffusive theory + +2026-09-13. Same #739 probability programme. The exact finite constructions +are in `tagged-span-resolvent.md`; this note separates an elementary +conditional stability theorem from three deliberately unproved site +conjectures. No extra width ladder is commissioned. + +## 1. A complete component's range is not an additive observable + +If a component is decomposed into pieces at transverse positions y_i, with +local lower/upper excursions ell_i,u_i, its inclusive span is + + L=1+max_i(y_i+u_i)-min_i(y_i+ell_i). (1) + +It is not sum_i(u_i-ell_i+1), nor the total variation of the center path. +An alternating center sequence 0,1,0,1,... has range one and total variation +proportional to its length. This deterministic distinction invalidates the +piecewise-additive-span explanation in the returned diagnostic; it does +not invalidate the computed finite-width moments. + +Here is the appropriate transfer lemma. + +**Conditional bush-stability theorem.** Suppose a complete component C_w +has a center path gamma_w and transverse Hausdorff error B_w, with + + E[number of pieces whose error exceeds t] <= C w^a exp(-ct) (2) + +under the **same component-Palm/closed-loop ensemble**, for fixed C,a,c>0. +Suppose gamma_w/sqrt(Dw) has a Brownian-bridge limit, with the moment bounds +required at the order under discussion. Then C_w has the same transverse +range limit and the same corresponding normalized moments. + +Proof. By a union bound, + + Pr(B_w>t)<=min(1,Cw^a exp(-ct)), + +so for each fixed r, E B_w^r=O((log w)^r). Deterministically, + + |range(C_w)-range(gamma_w)| <= 2 B_w. (3) + +After division by sqrt(w), (3) tends to zero in every fixed L^r. Slutsky +and the stated center-path uniform integrability transfer the law and +moments. In particular, if the skeleton first two moments have their +Brownian limits with error e_w, then the component CV^2 differs from the +skeleton CV^2 by O(log w/sqrt(w)), plus that pre-existing error. No sign +for this correction is asserted. + +This proof requires neither independence of pieces nor absence of branches. +The load-bearing condition is (2) under the **closed component law**. +Unconditional exponential tails cannot simply be conditioned on an +exponentially rare winding event and reused unchanged. + +CIV section 1.3.3, equation (1.10), states O(log n) Hausdorff closeness of +its full long connection cluster to the effective path, followed by its +Brownian-bridge Theorem C. Thus invoking CIV to say the full span must be +an extensive sum is the opposite of what that text establishes in its +actual setting. Its bond model/open-endpoint conditional law remains +different from this site's periodic component-Palm law. It is a proof +architecture and a specific missing comparison, not a finished transfer. + +Finite-width CV values 0.127 (NN) or 0.116 (matching) versus the target +0.04719755 do not decide this theorem's hypotheses. Even simple positive +boundary thickness alone is not a complete explanation: adding a positive +constant to a Brownian range decreases CV^2. Finite core corrections, +random decorations and covariance can produce either correction sign. + +## 2. Conjecture R: formulate the fixed-p problem using the exact resolvent + +For each fixed subcritical p of the chosen graph, let alpha_w,R_w,b_w be +the tagged construction and nu_w=alpha_w(I-R_w)^(-1)b_w. Let mu_w=E L, +which is now obtainable without a height cutoff. Define + + F_w(s)= z alpha_w(I-zR_w)^(-1)b_w / nu_w, + z=exp(-s/mu_w), s>=0. (4) + +**Conjecture R (complete shape, mean normalized).** At fixed strictly +subcritical p, + + F_w(s) -> E exp[-s range(B)/E range(B)] (5) + +locally uniformly for s>=0, with convergence of the corresponding moments. +No mass, amplitude, diffusion coefficient or finite-height offset is fitted +in (5). It is equivalent to a mean-normalized full shape law, stronger than +agreement of a single CV statistic. Fixed-w rationality does not preclude +a nonrational limit as the number of states grows. Conversely, the largest +single eigenvalue of R_w does not by itself determine the distribution in +the joint w,height scaling window; source/exit overlaps and multiple modes +remain in (4). + +The exact Brownian-range CDF used to compute the right-hand side is + + F_R(x)=sqrt(2pi)*pi^2/x^3 sum_(n>=1) n^2 exp[-pi^2 n^2/(2x^2)] + =1+2 sum_(n>=1)(1-4n^2x^2)exp(-2n^2x^2), (6) + +with E R=sqrt(pi/2), E R^2=pi^2/6. The explicit target transform at s=1,2,4 +is approximately 0.3762840878410, 0.1475796893947, 0.0252164362526. +These are Brownian formula values, not measured percolation quantities. + +In the new all-height computations, NN p=1/4 gives at s=2: + + w=4:0.1875595963; w=6:0.1744603845; w=8:0.1671020675. + +Matching p=1/8 gives 0.1765806063,0.1690681784,0.1638984502. +These move toward the proposed value over the reported widths; they do not +prove its limit and do not establish a correction exponent. The same finite +data can be compatible with several extrapolations. No new computation is +ordered merely by making the conjecture explicit. + +A proof can aim directly at the normalized resolvent (4), or prove the +conditional bush-stability hypotheses with a correctly sewn loop skeleton. +It need not first invent a unique raw seam cut. An obstruction should identify +which of mixing, one transverse mode, Palm normalization, or tight decoration +control fails. + +## 3. Conjecture J: joint geometry and thermodynamic marks + +The direct-activity representation defines, near a fixed physical point, + + Psi_w(z,u,v)=sum_C z^L u^K v^B, + +where B is the number of DISTINCT external boundary sites. This is not a +random-cluster q derivative. It supplies the correct finite joint law and +covariance of (L,K,B), all within the complete-component ensemble. + +**Conjecture J (one closed diffusive band with additive marks).** If the +physical closed chain has a reflection-symmetric mixing renewal description, +then for constants D>0, densities rho_K,rho_B and a covariance matrix Sigma, + + (L/sqrt(Dw), (K-rho_K w)/sqrt(w), (B-rho_B w)/sqrt(w)) + => (range(Brownian bridge), Gaussian_2), (7) + +with independence between the range and the Gaussian mark vector. +Possible sublinear corrections to the centering must be controlled; actual +finite means are a safer alternative where that expansion is unknown. + +Why independence is plausible, rather than just convenient: in a +reflection-symmetric Markov-additive effective chain, transverse increments +are odd and occupancy/boundary increments are even. Their long-run cross +covariances vanish. A joint functional CLT would give a Gaussian transverse +path independent of the additive even marks; conditioning the path to close +produces a bridge and preserves that independence. Applying this argument +requires a proved mixing/conditioning comparison for the site loop ensemble, +not merely reflection of the one-point marginal. + +The prior decorated dilute limit supplies one rigorous special-regime +motivation, not a proof at fixed p. The present exact computations give +finite squared correlations Corr(L,K)^2 of about 0.867 at NN w=4,p=1/4 +and 0.838 at w=5. They are not small yet. They neither establish (7) nor +contradict an asymptotic zero correlation. Under the moment hypotheses of +(7), a concrete consequence would be Corr(L,K)->0 whenever the limiting +occupation variance is nonzero. + +If f(u,v)=lim_w w^(-1) log Psi_w(1,u,v) exists with enough local regularity, +its log-coordinate gradient gives rho_K,rho_B and its Hessian gives Sigma. +Along u=p,v=1-p this would imply + + -kappa'(p)=rho_K/p-rho_B/(1-p). (8) + +Equation (8) is conditional on exchanging the limit and derivative. +The finite score identity is exact already; analyticity of finite rational +functions does not prove p-analyticity of the infinite limit. + +## 4. Conjecture U: is the closed-loop residue actually one? + +This is the bolder, less-supported conjecture. Keep it separate from R and J. +For an explicitly defined finite-state cyclic renewal kernel A(z,k), a +single simple critical eigenvalue gives + + w[z^w] {-log det(I-A(z,k))} ~ R(k)^(-w). + +The simple logarithmic singularity has coefficient one. Integrating the +one transverse quadratic mode gives + + L_w ~ exp(-kappa w)/sqrt(2pi D w), (9) + +with no independent endpoint-overlap amplitude. This is a statement about +that specified cyclic object, whose w/n unrooting is exact; it is not a +statement about arbitrary site clusters. + +**Conjecture U (unit sewing residue).** A canonical all-regeneration-cut +construction for the actual complete site component might identify it with +such a cyclic Gibbs object, making the previous sewing factor zeta(p)=1. +All external-boundary overlaps and multiple-cut weights must be included +before making this identification. A merely bounded, nontrivial cyclic +insertion would leave zeta!=1; several leading bands, residual marking +multiplicity, or lattice periodicity could also invalidate (9). + +This conjecture has a stronger falsifiable consequence than beta=1/2. If R +and the corresponding diffusion scale also hold, then + + 2 exp(kappa w) nu_w E L -> 1. (10) + +Without U, the same quantity tends to zeta(p), not necessarily one. With +only the older heat-kernel conjecture, nu_w E L has no power prefactor but +still has the unknown zeta/2 amplitude. An accurately known mass is needed +to use (10); none is manufactured from the six existing densities. + +This is not a new compute order. It states which genuinely new normalization +identity would remove an unknown amplitude, and exactly how a counterexample +would change the theory. The direct activity transfer now defines the correct +microscopic object to compare, rather than a count of seam crossings. + +## 5. Near-critical continuation stays downstream + +The fixed-subcritical conjectures above are not uniform claims for p->pc. +Even if D(p)kappa(p)->D0 and a scaling function for w nu_w were established, +the same-parameter black/white count constraint forbids blindly taking the +independent two-window Poisson law into a common critical window. The prior +conditional log-aspect/log-log-aspect crossover remains a conjecture requiring +uniform input; the new finite matrices do not discharge it. + +Current direction: use the exact all-height component law as the microscopic +object; establish or falsify the correct closed-loop mixing and decoration +control; only then infer a fixed-p amplitude or a critical crossover. This +keeps the geometric probability paper coherent while preserving the bolder +hypotheses in an explicit, testable form. + +Sources: CIV, arXiv:math/0610100v2, section 1.3.3 and equation (1.10), for the +open-connection architecture under its bond hypotheses; standard phase-type +absorption formulas are prior art (Maier, 1991, doi:10.1080/15326349108807207). +No general Brownian, phase-type, Gibbs or renewal theorem is claimed new here. diff --git a/docs/manuscripts/geometric-balance/tagged-span-resolvent.md b/docs/manuscripts/geometric-balance/tagged-span-resolvent.md new file mode 100644 index 00000000..be260e1c --- /dev/null +++ b/docs/manuscripts/geometric-balance/tagged-span-resolvent.md @@ -0,0 +1,397 @@ +# A complete-component span law without an age cutoff + +2026-09-13. Continuation of #739, after reading the span-spectrum delivery at +`7226a2c6d099535f34486eccb1bd996f7affda13`. This is a new exact finite-state +construction and its executed finite controls. It is not a proof of the +fixed-p Brownian sewing conjecture, a publication-priority claim, or an +independent certification of the previous large solver. + +The main change is to stop keeping the ages of *all* active components. +One candidate component is followed, with forbidden ancestors ensuring a +unique lowest-row anchor. A second, independent transfer sums the exact +complete-component activities with two-row boundary memory. Both describe +all heights using one fixed operator; neither has a depth parameter. + +## 1. The object and the unique anchor + +Use independent site percolation on `C_w x Z`, with NN or matching NN+NNN +adjacency and lifted horizontal edge displacements retained. All statements +below use `0= 0, R 1 <= (1-delta)1, delta=(1-p)^w>0. + +The same property holds after the all-p exit-law-preserving lumping used +in the implementation. This lumping preserves distributions, not the state +path of a fixed mask word. We make no minimal-realization claim. + +Write Z=(I-R)^(-1). A component with span h continues for h-1 steps after +its row-zero source and retires on the next step. The pathwise theorem gives + + d_h = alpha R^(h-1) b, (1) + D(z) = sum_(h>=1) d_h z^h = z alpha (I-zR)^(-1) b, (2) + nu = alpha Z b. (3) + +These are identities for all heights, not fits to the first few bins. In +particular the complete-component span distribution is rational at each +closed finite width. General rational absorption-time formulae are classical +phase-type theory [PH]; the contribution here is the candidate/forbidden- +ancestor realization of the *correct complete site-component Palm law*. + +### 2.1 All moments and all omitted-tail moments + +Unnormalised raw moments of order zero, one and two are + + m0 = alpha Z b, + m1 = alpha Z^2 b, + m2 = alpha (2Z^3-Z^2)b. (4) + +For k>=1, the factorial moment is + + sum_h (h)_k d_h = k! alpha R^(k-1) Z^(k+1) b. (5) + +Thus E L=m1/m0 and Var L=m2/m0-(m1/m0)^2. There is no cutoff bias. +For a reporting cutoff H, rather than dismissing a small tail-bin mass, +compute its contributions exactly: + + sum_(h>H) d_h = alpha R^H Z b, + sum_(h>H) h d_h = alpha R^H (H Z+Z^2)b, + sum_(h>H) h^2 d_h + = alpha R^H [H^2 Z+(2H-1)Z^2+2Z^3]b. (6) + +A small tail probability alone never bounds its first or second moment. +Here the missing moments are returned by the same resolvent. + +### 2.2 The height-window activity is now directly computable + +For the earlier complete-component window sum Xi_(w,H), including its +external boundary in the infinite cylinder, + + Xi_H = sum_(h<=H)(H-h+1)d_h + = alpha[(H+1)Z-Z^2+R^(H+1)Z^2]b. (7) + +This reproduces Xi_H-Xi_(H-1)=sum_(h<=H)d_h. At fixed w, + + Xi_H = nu[H+1-E L] + alpha R^(H+1)Z^2 b. (8) + +The intercept of the large-height activity therefore contains the full +mean span, including branches. Equations (2) and (7) refer to propagation +in the **vertical height**, not a derived horizontal OZ renewal. They do +not prove a w^(-1/2) prefactor as w increases. + +### 2.3 Exact Palm sampling, with no burn-in or rejection + +Let h=Zb. Remove states with h_i=0. Then + + R*_ij=R_ij h_j/h_i, b*_i=b_i/h_i, + alpha*_i=alpha_i h_i/nu + +define a stochastic absorbing chain with initial mass one. Its accepted +absorption time has distribution d_h/nu. For a physical sample retain the +individual source-row masks and individual transition masks, weighting each +by the corresponding h value; do not sample only a representative lumped +mask word. + +The product of these probabilities telescopes to + + Q(source rows, anchor, future rows) + = product_of_original_Bernoulli_row_weights / nu + +for every successful trajectory. The unique-anchor theorem then supplies +the complete-component Palm distribution, not a seam- or size-biased one. +The included exact-Fraction sampler records the original rows; independent +lifted-graph BFS recovers the actual component and checks its full span. +A row safety cap raises rather than returning a censored sample. + +This is an ordinary Doob conditioning of a finite chain after its model +mapping has been proved. It does not assert that building the finite chain +at arbitrarily large w is cheap. + +## 3. Independent sewing through direct component activities + +There is a second representation which does not introduce a random +surrounding environment or forbidden colours. A finite selected component +is processed row by row. The state stores its frontier connectivity and +winding, plus the preceding row's selected-site mask. Empty interior rows +are impossible for a connected nearest-row component. If any selected +component retires while another selected frontier component remains, reject: +it can never reconnect. At final retirement require exactly one connected +component and nonzero winding. + +For row masks A,B,C (previous, current, next), write S(B) for horizontal +neighbours of B, and E4(B)=B, E8(B)=B union left(B) union right(B). +The distinct boundary sites in the current row are exactly + + boundary(A,B,C) = [S(B) union E_G(A) union E_G(C)] minus B. (9) + +Finalise the weight u^|B| v^|boundary(A,B,C)| when the next mask is known. +The source additionally contributes v^|E_G(first)| for the external bottom +row. The terminal transition includes v^|E_G(last)| for the external top +row. Thus a valid path has weight u^|C| v^|boundary C|, with each vacant +boundary site counted once. Other occupied components in the random +realization need not be enumerated at all. + +Let Q_w(u,v) be the resulting nonnegative matrix, c_w its source and e_w +its successful exit vector. Then the formal series identity is + + Psi_w(z,u,v) + = sum_(C anchored,minrow=0,winding,connected) + z^L(C) u^|C| v^|boundary C| + = z c_w(u,v) (I-zQ_w(u,v))^(-1) e_w(u,v). (10) + +At u=p,v=1-p, Psi_w=D(z) from (2). The two representations have different +state spaces and transition rules. Q is not generally substochastic, so +the tagged-chain reset certificate must NOT be blindly applied to Q. +Its physical series converges: fixed-width empty-row bounds give an +exponential tail for L, and |C|<=w L, |boundary C|<=8|C|. This also ensures +local convergence in the occupation and boundary fugacities near a fixed +physical point. A finite reachable/co-reachable nonnegative realization +therefore has spectral radius below one there. + +The coefficients of (10) give joint complete-component observables, not +only the marginal span. For example, derivatives with respect to log u +and log v give E|C| and E|boundary C|, and their log-Hessian is the +corresponding covariance matrix. Along the physical p curve, + + d(log nu)/dp = E|C|/p - E|boundary C|/(1-p). (11) + +This is an exact fixed-width score identity. Passing its derivatives to a +w->infinity limit is a separate regularity question. + +The implementation gives exact all-height moments of (L,K), K=|C|, using +linear solves for the derivatives of (10). For NN p=1/4, width four: + + E L = 3.1371029435733466..., + E K = 7.238700819343702..., + Corr(L,K)^2 = 0.8674117242489037.... + +All are exact rationals in the result file. A finite positive correlation +is neither an extensive-span law nor a refutation of a proposed asymptotic +shape/occupation decoupling. + +## 4. Completed finite results and numerical certificates + +Tagged reachable states, followed by the all-p exit-law lumps: + +| w | tagged states | NN lumps | matching lumps | +|---|---:|---:|---:| +|2|5|3|2| +|3|13|5|3| +|4|43|10|7| +|5|131|17|15| +|6|411|36|33| +|7|1275|71|68| +|8|3963|161|152| + +Each recorded state has every row successor resolved. Width-eight builds +and two-parameter certificates took seconds to tens of seconds here; this +is not a like-for-like benchmark against the team's other hardware. The +structural saving is the absence of all component ages and of D_MAX. + +For NN p=1/4 and matching p=1/8 the all-height results are: + +| graph | w | E L | Var(L)/(E L)^2 | +|---|---:|---:|---:| +|NN|4|3.1371029435733466|0.2114153618009875| +|NN|6|3.9380183817628125|0.1570569824332094| +|NN|8|4.56334396378038|0.1271635000680440| +|matching|4|3.2422208646738944|0.1778141825223629| +|matching|6|4.060109171060249|0.1398612019890094| +|matching|8|4.711805413158092|0.1160049348309105| + +These reproduce the finite-width trend in the returned note, using a new +operator without height truncation. They still do not determine a limiting +shape. In particular none of these widths is claimed asymptotic. + +At width two the full all-p generating functions are compact. Write q=1-p: + + D_NN(z) = p^2 q^4 z [1+p(1+p)z]^2 + /[(1-pqz)(1-pz-p^3qz^2)], + D_matching(z) = p^2 q^4 z [1+q(2-p)z] + /[(1-p(2-p)z)(1-pqz)]. (12) + +At p=1/2 both total densities are 7/48, but their span means are 76/21 and +100/21. Equal complementary densities do not imply equal shape laws. +The p=1/2 matching calculation is an exact fixed-cylinder control, NOT a +planar-subcritical matching example. + +### 4.1 Certification at widths five through eight + +Write A=I-R. Since ||A^(-1)||_infinity<=1/delta, approximate solutions +x1~A^(-1)b, x2~A^(-1)x1, x3~A^(-1)x2 have errors bounded recursively by + + e1=||b-Ax1||_inf/delta, + e2=(||x1-Ax2||_inf+e1)/delta, + e3=(||x2-Ax3||_inf+e2)/delta. (13) + +Multiply by ||alpha||_1 to enclose m0,m1,m2, using errors e1,e2,2e3+e2. +Positive-interval division then encloses means and variances. The implementation +uses a floating solve, then one correction driven by the **exact rational** +residual, storing the sum of two dyadic rationals. The final residual is +recomputed exactly for that stored vector. It is not a certificate for a +different, pre-rounding vector. Width-eight CV^2 interval widths in the two +main controls are about 2.3e-24 and 1.6e-25. This is arithmetic control on +the new finite operator, not an asymptotic error estimate. + +Widths two through four and the small direct-activity solves use Fraction +Gaussian elimination throughout. Mean-scaled Laplace values and Brownian +comparison integrals in the report are separately labelled floating/high- +precision diagnostics; they are not covered by (13). + +## 5. Reading the returned span calculation accurately + +The mean and CV trends in `span-spectrum-diagnostic-20260913.md` are useful. +Two interpretations and implementation details must not be carried forward: + +1. A complete component's span is not the sum of irreducible-piece heights. + It is a maximum minus a minimum after placing the pieces at their actual + transverse positions. See the separate frontier note for the deterministic + stability bound and the precise conditional theorem. +2. A decreasing finite-width CV^2 above pi/3-1 cannot identify its limiting + value, whether zero or pi/3-1. Widths 2--4 at p=1/2 do not establish a + non-diffusive asymptotic regime. +3. The `light=True` branch of `span_spectrum_solve.py` at the read commit + passes its integer-weight matrix (converted to float32) into `stationary` + without division by p_den^w. The normal branch does divide. Hence light + is not using a row-stochastic kernel as written. This is source inspection, + not evidence that a particular reported run used that branch. +4. Its certificate is computed for a rounded vector a/2^k, while the displayed + probabilities are from pi. Add the explicit evaluation discrepancy or + report the certified candidate. A single-bin reward bound is not by itself + a total-moment bound. The new implementation avoids these ambiguities. +5. The headline `results/geometric-consistency/span-spectrum-20260913.json` + was not present at the read commit; its nested directory contained only + validation tables. The new report is fully present and independently + regenerated. This does not erase the old data or assert that they never + existed on a team machine. + +No remote job was stopped and no old result file is overwritten by this work. + +## 6. Reproduction and evidence boundary + +Files in this additive delivery: + +- this note and `span-resolvent-frontier.md`; +- `scripts/tagged_winding_span.py` and `scripts/tagged_span_controls.py`; +- `tests/test_tagged_winding_span.py`; +- `results/geometric-consistency/tagged-span-resolvent.json`. + +From a checkout with these files: + + python -m unittest discover -s tests -p 'test_tagged_winding_span.py' -v + OPENBLAS_NUM_THREADS=1 python scripts/tagged_span_controls.py \ + --output /tmp/tagged-span-new.json + +Python 3.10+; core exact construction/sampling uses the standard library. +The full report/tests additionally use NumPy, SciPy, SymPy and mpmath, listed +here rather than silently adding a project-wide dependency. Results refuse +to overwrite an existing output. No binary tables or old source snapshots +are required. + +Executed: 18,754 nonempty shape masks with independent graph-potential BFS; +11,904 complete source-row words checking the actual accepted anchors; +66 exact physical-activity parameter controls; 18 full two-fugacity +coefficient comparisons; all tagged closures through width eight; direct +activity/marked-resolvent agreement through width five; eight exact Palm +trajectory controls; 17 local tests. The small complete-word and shape +counts are controls, not Monte Carlo probability estimates. No full +Matching-One CI, fresh external peer acceptance or priority certification +is claimed. + +## Sources + +[PH] R. S. Maier, *The algebraic construction of phase-type distributions*, +Communications in Statistics—Stochastic Models 7 (1991), 573--602, +doi:10.1080/15326349108807207. General rational absorption-time representations +are prior art; this note proves the model-specific anchor mapping. + +[CIV] M. Campanino, D. Ioffe, Y. Velenik, *Fluctuation theory of connectivities +for subcritical random cluster models*, Ann. Probab. 36 (2008), 1287--1321, +arXiv:math/0610100v2. Section 1.3.3, equation (1.10), Theorem C were read in +full and rendered; the model in section 1.1 is bond random cluster. They do +not automatically settle the periodic site-component sewing used here. diff --git a/results/geometric-consistency/tagged-span-resolvent.json b/results/geometric-consistency/tagged-span-resolvent.json new file mode 100644 index 00000000..5af87d0b --- /dev/null +++ b/results/geometric-consistency/tagged-span-resolvent.json @@ -0,0 +1,2722 @@ +{ + "scope": "new one-tag construction; no age cutoff; finite state closures checked only at recorded widths", + "source_commit_reviewed": "7226a2c6d099535f34486eccb1bd996f7affda13", + "systems": [ + { + "width": 2, + "matching": false, + "tagged_states": 5, + "exit_lumps": 3, + "mask_transitions": 20, + "runs": [ + { + "p": "1/8", + "mode": "exact Fraction, all heights", + "moments": { + "nu": "3577/233472", + "mean": "309184/203889", + "second": "117488075840/41570724321", + "variance": "21893329984/41570724321", + "cv2": "342083281/1493667904", + "d_h": [ + "2401/262144", + "79233/16777216", + "1265327/1073741824", + 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"p_half": "z*(3*z + 4)**2/(16*(z - 4)*(z**2 + 8*z - 16))" + }, + { + "matching": true, + "density_pgf": "p**2*z*(p - 1)**4*(p**2*z - 3*p*z + 2*z + 1)/((p**2*z - 2*p*z + 1)*(p**2*z - p*z + 1))", + "p_half": "z*(3*z + 4)/(16*(z - 4)*(3*z - 4))" + } + ], + "brownian_range_laplace_mean_scaled": { + "1": "0.37628408784104478664791708", + "2": "0.14757968939467544997933219", + "4": "0.025216436252613651495711849" + }, + "note": "Exact certificates bound arithmetic on the newly constructed finite operator. No fixed-p Brownian claim or global repository CI is inferred." +} diff --git a/scripts/tagged_span_controls.py b/scripts/tagged_span_controls.py new file mode 100644 index 00000000..decdfc40 --- /dev/null +++ b/scripts/tagged_span_controls.py @@ -0,0 +1,236 @@ +#!/usr/bin/env python3 +"""Independent physical controls and deterministic report for tagged span law.""" +from __future__ import annotations +import argparse +from collections import Counter +from fractions import Fraction as F +from itertools import product +import json +from pathlib import Path +from time import perf_counter +import tagged_winding_span as T + + +def components(width, rows, matching=False): + occ={(x,y) for y,mask in enumerate(rows) for x in range(width) if mask>>x&1} + steps=[(1,0),(-1,0),(0,1),(0,-1)] + if matching:steps += [(a,b) for a in (-1,1) for b in (-1,1)] + unseen=set(occ); out=[] + while unseen: + root=min(unseen); potentials={root:0}; stack=[root]; unseen.remove(root); winding=False + while stack: + x,y=stack.pop() + for dx,dy in steps: + v=((x+dx)%width,y+dy) + if v not in occ:continue + q=potentials[(x,y)]+dx + if v in potentials: + if potentials[v]!=q: + assert (potentials[v]-q)%width==0 + winding=True + else: + potentials[v]=q;unseen.discard(v);stack.append(v) + out.append((frozenset(potentials),winding)) + return out + + +def external_boundary(width, C, matching=False): + steps=[(1,0),(-1,0),(0,1),(0,-1)] + if matching:steps += [(a,b) for a in (-1,1) for b in (-1,1)] + return {((x+dx)%width,y+dy) for x,y in C for dx,dy in steps}-set(C) + + +def shape_activity_counts(width,height,matching=False): + counts=Counter(); checked=0 + for mask in range(1,1<<(width*height)): + checked+=1 + rows=[(mask>>(width*y))&((1<=0:R[i,j]+=wt + elif j==-1:b[i]+=wt + for (i,k),c in src.items():a[i]+=c*p**k*(1-p)**(4-k) + pgf=sp.factor((z*a*(sp.eye(n)-z*R).inv()*b)[0]) + # First coefficient is the exact isolated full-row activity. + assert sp.simplify(sp.diff(pgf,z).subs(z,0)-p**2*(1-p)**4)==0 + out.append({'matching':g,'density_pgf':str(pgf),'p_half':str(sp.factor(pgf.subs(p,sp.Rational(1,2))))}) + return out + + +def laplace_diagnostic(tr,src,p,mean,scales=(1,2,4)): + import numpy as np + from scipy.linalg import solve + alpha,R,b,delta=T.numeric_system(tr,src,p,False) + R=np.array(R);alpha=np.array(alpha);b=np.array(b);I=np.eye(len(b)) + nu=float(alpha@solve(I-R,b));out={} + for s in scales: + z=float(np.exp(-s/mean)) + out[str(s)]=float(z*alpha@solve(I-z*R,b)/nu) + return out + + +def brownian_laplace(scales=(1,2,4)): + import mpmath as mp + mp.mp.dps=50 + mean=mp.sqrt(mp.pi/2) + def cdf(x): + if x<=0:return mp.mpf(0) + if x=2; lifted parallel edges retained') + return State((-1,)*width,(0,)*width,(),()) + +def step(state: State, mask: int, matching=False, tagged=True, connected=False): + """Return (next state, outcome). outcome=0 continue,1 accept,-1 reject. + + In source mode tagged=False, colours propagate but retirement is ignored. + """ + w=len(state.labels) + if not 0 <= mask < 1<=0] + new=[i for i in range(w) if mask>>i&1] + for i in old: + k=state.labels[i] + if k in reps: d.join(reps[k],i,state.gains[i]) + else: reps[k]=i + for k,i in reps.items(): + root,_=d.find(i); d.wind[root]=bool(state.winding[k]); d.col[root]=state.colours[k] + for i in new: + j=(i+1)%w + if mask>>j&1: d.join(w+i,w+j,(i+1)//w) + for dx in ((-1,0,1) if matching else (0,)): + j=(i+dx)%w + if state.labels[j]>=0: d.join(w+i,j,(i+dx)//w) + roots={d.find(i)[0] for i in old+[w+i for i in new]} + kept={d.find(w+i)[0] for i in new} + if connected and (roots-kept or not kept): + return None, 1 if (not kept and len(roots)==1 and d.wind[next(iter(roots))]) else -1 + if tagged: + marked=[r for r in roots if d.col[r]&TAG] + if len(marked)!=1: raise AssertionError('exactly one live tag required') + r=marked[0] + if d.col[r]&FORBIDDEN: return None,-1 + if r not in kept: return None,1 if d.wind[r] else -1 + labels=[-1]*w; gains=[0]*w; wind=[]; colours=[]; canon={} + for i in new: + r,g=d.find(w+i) + if r not in canon: + canon[r]=(len(wind),g); wind.append(int(d.wind[r])); colours.append(d.col[r]) + k,g0=canon[r]; labels[i]=k; gains[i]=0 if d.wind[r] else g-g0 + return State(tuple(labels),tuple(gains),tuple(wind),tuple(colours)),0 + +def source_entries(width, matching=False): + """Integer multiplicities indexed by state and occupancy of two source rows. + + All old row (-1) components are forbidden. A row-zero candidate is tagged; + smaller row-zero candidate components are also forbidden (tie breaking). + """ + out=Counter() + for prev in range(1<8: raise ValueError('reference source enumeration limited to width <=8') + sources=source_entries(width,matching) + states=[]; index={} + def add(s): + if s not in index: + if len(states)>=state_cap: raise RuntimeError('state cap reached') + index[s]=len(states); states.append(s) + return index[s] + for s,k in sources: add(s) + transitions=[] + for s in states: + row=[] + for mask in range(1<=0 else j) + for mask,(j,_) in enumerate(row)).items())) + if sig not in lookup: lookup[sig]=len(lookup) + nxt.append(lookup[sig]) + if nxt==blocks: break + blocks=nxt + reps=[blocks.index(i) for i in range(max(blocks)+1)] + reduced=[[(blocks[j] if j>=0 else j,o) for j,o in transitions[i]] for i in reps] + src=Counter() + for (i,k),c in source.items(): src[(blocks[i],k)]+=c + return reduced,src,blocks + +def numeric_system(transitions, source, p, exact=True): + n=len(transitions); w=(len(transitions[0])-1).bit_length() + zero=Fraction(0) if exact else 0.0 + pp=Fraction(p) if exact else float(p) + if not 0 < pp < 1: raise ValueError("requires 0=0: R[i][j]+=wt + elif j==-1: b[i]+=wt + for (i,k),count in source.items(): alpha[i]+=count*pp**k*(1-pp)**(2*w-k) + delta=(1-pp)**w + if exact: + assert all(sum(row)<=1-delta for row in R) + assert all(sum(row)+bb<=1 for row,bb in zip(R,b)) + return alpha,R,b,delta + +def solve(A, b): + n=len(b); M=[[Fraction(x) for x in row]+[Fraction(y)] for row,y in zip(A,b)] + for j in range(n): + pivot=next((i for i in range(j,n) if M[i][j]),None) + if pivot is None: raise ValueError('singular matrix') + M[j],M[pivot]=M[pivot],M[j]; v=M[j][j]; M[j]=[x/v for x in M[j]] + for i in range(n): + if i!=j and M[i][j]: + v=M[i][j]; M[i]=[a-v*b for a,b in zip(M[i],M[j])] + return [row[-1] for row in M] + +def moments(transitions, source, p, bins=12, exact=True): + alpha,R,b,delta=numeric_system(transitions,source,p,exact) + n=len(b) + if exact: + A=[[int(i==j)-R[i][j] for j in range(n)] for i in range(n)] + z1=solve(A,b); z2=solve(A,z1); z3=solve(A,z2) + dot=lambda v:sum(a*x for a,x in zip(alpha,v)) + nu=dot(z1); mean=dot(z2)/nu; second=dot([2*x-y for x,y in zip(z3,z2)])/nu + v=alpha[:]; dh=[] + for _ in range(bins): + dh.append(sum(x*y for x,y in zip(v,b))) + v=[sum(v[i]*R[i][j] for i in range(n)) for j in range(n)] + tail=sum(x*y for x,y in zip(v,z1)) + t1=sum(x*(bins*y+z) for x,y,z in zip(v,z1,z2)) + t2=sum(x*(bins*bins*y+2*bins*z+2*u-z) for x,y,z,u in zip(v,z1,z2,z3)) + assert sum(dh)+tail==nu + assert sum((i+1)*x for i,x in enumerate(dh))+t1==nu*mean + assert sum((i+1)**2*x for i,x in enumerate(dh))+t2==nu*second + else: + import numpy as np + from scipy.linalg import lu_factor, lu_solve + rr=np.asarray(R); aa=np.asarray(alpha); bb=np.asarray(b) + lu=lu_factor(np.eye(n)-rr) + z1=lu_solve(lu,bb); z2=lu_solve(lu,z1); z3=lu_solve(lu,z2) + nu=float(aa@z1); mean=float(aa@z2)/nu; second=float(aa@(2*z3-z2))/nu + v=aa.copy(); dh=[] + for _ in range(bins): dh.append(float(v@bb)); v=v@rr + tail=float(v@z1); t1=float(v@(bins*z1+z2)); t2=float(v@(bins*bins*z1+(2*bins-1)*z2+2*z3)) + var=second-mean*mean + return dict(nu=nu,mean=mean,second=second,variance=var,cv2=var/mean**2, + d_h=dh,tail=tail,tail_first=t1,tail_second=t2,delta=delta) + +def jsonable(x): + if isinstance(x,Fraction): return str(x) + if isinstance(x,dict):return {str(k):jsonable(v) for k,v in x.items()} + if isinstance(x,(list,tuple)):return [jsonable(v) for v in x] + return x + +def certified_moments(transitions, source, p, bins=12): + """Exact rational residual enclosures after one floating correction per solve. + + Returned centres are rational corrected solutions, not the pre-rounding + float vectors. The inverse infinity norm is at most 1/delta. + """ + import numpy as np + from scipy.linalg import lu_factor, lu_solve + alpha,R,b,delta=numeric_system(transitions,source,Fraction(p),True) + n=len(b); A=[[Fraction(int(i==j))-R[i][j] for j in range(n)] for i in range(n)] + af=np.array([[float(x) for x in row] for row in A]); lu=lu_factor(af) + def exact_residual(x,rhs): + return [rhs[i]-sum(v*y for v,y in zip(A[i],x)) for i in range(n)] + def corrected(rhs): + xf=lu_solve(lu,np.array([float(x) for x in rhs])) + x=[Fraction.from_float(float(v)) for v in xf] + r=exact_residual(x,rhs) + dx=lu_solve(lu,np.array([float(v) for v in r])) + x=[v+Fraction.from_float(float(d)) for v,d in zip(x,dx)] + r=exact_residual(x,rhs) + return x,max(map(abs,r)) + x1,r1=corrected(b); e1=r1/delta + x2,r2=corrected(x1); e2=(r2+e1)/delta + x3,r3=corrected(x2); e3=(r3+e2)/delta + mass=sum(alpha); dot=lambda x:sum(a*b for a,b in zip(alpha,x)) + centres=[dot(x1),dot(x2),dot([2*x-y for x,y in zip(x3,x2)])] + errors=[mass*e1,mass*e2,mass*(2*e3+e2)] + raw=[(c-e,c+e) for c,e in zip(centres,errors)] + if raw[0][0]<=0:raise ArithmeticError('density not separated from zero') + mean=(raw[1][0]/raw[0][1],raw[1][1]/raw[0][0]) + second=(raw[2][0]/raw[0][1],raw[2][1]/raw[0][0]) + var=(second[0]-mean[1]**2,second[1]-mean[0]**2) + cv=(var[0]/mean[1]**2,var[1]/mean[0]**2) + return dict(density_interval=raw[0],mean_interval=mean,second_interval=second, + variance_interval=var,cv2_interval=cv,raw_moment_intervals=raw, + residuals=[r1,r2,r3],inverse_bound=1/delta, + centres=dict(nu=centres[0],mean=centres[1]/centres[0], + second=centres[2]/centres[0], + cv2=centres[2]*centres[0]/centres[1]**2-1)) + + +def weighted_choice(weights, rng): + from math import lcm + weights=[Fraction(x) for x in weights] + if min(weights)<0 or sum(weights)<=0: raise ValueError('invalid exact weights') + denominator=1 + for x in weights: denominator=lcm(denominator,x.denominator) + ints=[x.numerator*(denominator//x.denominator) for x in weights] + pick=rng.randrange(sum(ints)) + for i,v in enumerate(ints): + if pick=0 else int(j==-1) + choices.append(wt[m]*hm/hi) + assert sum(choices)==1 + m=weighted_choice(choices,rng); qpath*=choices[m];prior*=wt[m];rows.append(m) + nxt,out=step(s,m,matching) + if out: + assert out==1 and qpath==prior/nu + return dict(rows=rows,anchor=anchor,span=len(rows)-2, + path_probability=qpath,unconditioned_row_weight=prior,nu=nu) + s=nxt + raise RuntimeError('sampling row cap reached; no sample returned') + + +def expand_mask(mask,width,matching=False): + if not matching:return mask + full=(1<>(width-1)) | (mask>>1) | ((mask&1)<<(width-1)) + + +def activity_transfer(width,matching=False,state_cap=100000): + """Direct complete-component activity; track two rows plus connectivity. + + A transition label (k,b) means u**k v**b for occupied and DISTINCT + boundary sites finalized in the current row. Connected pieces may not + retire before the whole selected component. No random environment needed. + """ + if not 2<=width<=6:raise ValueError('reference activity builder width 2..6') + states=[];idx={};src=Counter() + def add(s): + if s not in idx: + if len(states)>=state_cap:raise RuntimeError('activity state cap') + idx[s]=len(states);states.append(s) + return idx[s] + for mask in range(1,1<=0) + horizontal=expand_mask(current,width,True) + k=current.bit_count();row=[] + for mask in range(1<=0 else j,k,b) for j,k,b in row).items())) + if sig not in lookup:lookup[sig]=len(lookup) + nxt.append(lookup[sig]) + if nxt==blocks:break + blocks=nxt + reduced=[[(blocks[j] if j>=0 else j,k,b) for j,k,b in tr[blocks.index(i)]] for i in range(max(blocks)+1)] + source=Counter() + for (i,b),c in src.items():source[(blocks[i],b)]+=c + return states,reduced,source + + +def activity_joint_moments(tr,src,p): + """Exact all-height complete-Palm moments of span L and occupation K.""" + p=Fraction(p);n=len(tr) + R=[[[Fraction(0) for _ in range(n)] for _ in range(n)] for order in range(3)] + b=[[Fraction(0)]*n for _ in range(3)];alpha=[Fraction(0)]*n + for i,row in enumerate(tr): + for j,k,nb in row: + weight=p**k*(1-p)**nb + factors=(1,k,k*(k-1)) + for q,f in enumerate(factors): + if j<0:b[q][i]+=f*weight + else:R[q][i][j]+=f*weight + for (i,nb),c in src.items():alpha[i]+=c*(1-p)**nb + A=[[int(i==j)-R[0][i][j] for j in range(n)] for i in range(n)] + mv=lambda mat,x:[sum(v*y for v,y in zip(row,x)) for row in mat] + add=lambda *xs:[sum(z) for z in zip(*xs)] + dot=lambda x:sum(a*b for a,b in zip(alpha,x)) + y=solve(A,b[0]);yL=solve(A,y);z3=solve(A,yL) + yK=solve(A,add(b[1],mv(R[1],y))) + yKK=solve(A,add(b[2],mv(R[2],y),[2*x for x in mv(R[1],yK)])) + yLK=solve(A,add(mv(R[1],yL),yK)) + nu=dot(y);EL=dot(yL)/nu;EK=dot(yK)/nu + varL=dot([2*x-y for x,y in zip(z3,yL)])/nu-EL**2 + varK=dot(add(yKK,yK))/nu-EK**2;cov=dot(yLK)/nu-EL*EK + assert varL>=0 and varK>=0 and cov**2<=varL*varK + return dict(nu=nu,mean_span=EL,mean_occupation=EK,variance_span=varL, + variance_occupation=varK,covariance_span_occupation=cov, + correlation_squared=cov**2/(varL*varK) if varL*varK else Fraction(0)) + + +if __name__=='__main__': + ap=argparse.ArgumentParser(); ap.add_argument('--width',type=int,default=4) + ap.add_argument('--matching',action='store_true'); ap.add_argument('--p',default='1/4') + ap.add_argument('--float',action='store_true'); ap.add_argument('--output',type=Path) + args=ap.parse_args(); start=perf_counter() + states,tr,src=build(args.width,args.matching); tr,src,blocks=lump(tr,src) + out={'width':args.width,'matching':args.matching,'p':args.p,'states':len(states), + 'blocks':len(tr),'all_height_moments':moments(tr,src,Fraction(args.p),exact=not args.float), + 'seconds':perf_counter()-start,'mode':'floating diagnostic' if args.float else 'exact rational'} + text=json.dumps(jsonable(out),indent=2) + if args.output: + if args.output.exists():raise FileExistsError(args.output) + args.output.parent.mkdir(parents=True,exist_ok=True); args.output.write_text(text+'\n') + else:print(text) diff --git a/tests/test_tagged_winding_span.py b/tests/test_tagged_winding_span.py new file mode 100644 index 00000000..d589d0f8 --- /dev/null +++ b/tests/test_tagged_winding_span.py @@ -0,0 +1,124 @@ +from __future__ import annotations +from fractions import Fraction as F +from pathlib import Path +import sys +import unittest + +sys.path.insert(0,str(Path(__file__).resolve().parents[1]/'scripts')) +import tagged_winding_span as T +import tagged_span_controls as C + +class TaggedSpanTests(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.cache={} + for g in (False,True): + for w in (2,3,4): + states,tr,src=T.build(w,g); rr,src,blocks=T.lump(tr,src) + cls.cache[(w,g)]=(states,tr,rr,src) + + def calc(self,w,g,p=F(1,2),bins=8): + _,_,tr,src=self.cache[(w,g)] + return T.moments(tr,src,p,bins=bins) + + def test_known_density_and_full_moments_nn2(self): + r=self.calc(2,False) + self.assertEqual(r['nu'],F(7,48));self.assertEqual(r['mean'],F(76,21)) + self.assertEqual(r['variance'],F(2068,441));self.assertEqual(r['cv2'],F(517,1444)) + + def test_equal_density_does_not_equal_shape(self): + a=self.calc(2,False);b=self.calc(2,True) + self.assertEqual(a['nu'],b['nu']);self.assertEqual(b['mean'],F(100,21)) + self.assertEqual(b['variance'],F(5596,441));self.assertNotEqual(a['d_h'],b['d_h']) + + def test_isolated_row_all_controls(self): + for g in (False,True): + for w in (2,3,4): + for p in (F(1,4),F(1,2),F(3,4)): + self.assertEqual(self.calc(w,g,p,1)['d_h'][0],p**w*(1-p)**(2*w)) + + def test_physical_height_three(self): + r=self.calc(4,False,bins=3) + self.assertEqual(sum(r['d_h']),F(9087,1048576)) + self.assertEqual(r['nu'],F(323849,5576960)) + + def test_all_source_words(self): + for g in (False,True): + n,mass=C.enumerate_tag_paths(2,2,g) + self.assertEqual(n,256);self.assertEqual(mass,self.calc(2,g,bins=2)['d_h'][1]) + + def test_late_merge_tie_break(self): + # Two candidates at time zero eventually join one winding component. + paths=[x for x in T.source_paths(4,False) if x[:2]==(0,5)] + accepted=[] + for prev,mask,anchor,s in paths: + outcome=0 + for mask in (7,15,0): + s,outcome=T.step(s,mask) + if outcome:break + if outcome==1:accepted.append(anchor) + self.assertEqual(accepted,[0]) + + def test_past_contact_rejects_new_tag(self): + paths=[x for x in T.source_paths(4,False) if x[:2]==(4,5)] + self.assertEqual(len(paths),1) + _,_,_,s=paths[0];s,outcome=T.step(s,7) + self.assertEqual(outcome,-1) + + def test_empty_mask_terminates(self): + for states,tr,_,_ in self.cache.values(): + self.assertTrue(all(row[0][0]<0 for row in tr)) + self.assertTrue(all(s.colours.count(T.TAG)==1 for s in states)) + + def test_exact_tail_moments(self): + r=self.calc(4,False,bins=3) + self.assertEqual(sum(r['d_h'])+r['tail'],r['nu']) + self.assertEqual(sum((i+1)*x for i,x in enumerate(r['d_h']))+r['tail_first'],r['nu']*r['mean']) + self.assertEqual(sum((i+1)**2*x for i,x in enumerate(r['d_h']))+r['tail_second'],r['nu']*r['second']) + + def test_residual_intervals_cover_exact(self): + _,_,tr,src=self.cache[(4,False)];r=T.certified_moments(tr,src,F(1,4));ex=self.calc(4,False,F(1,4)) + for interval,key in [('density_interval','nu'),('mean_interval','mean'),('variance_interval','variance'),('cv2_interval','cv2')]: + lo,hi=r[interval];self.assertLessEqual(lo,ex[key]);self.assertGreaterEqual(hi,ex[key]) + + def test_two_parameter_activity_polynomials(self): + for g in (False,True): + st,tr,src=T.activity_transfer(3,g);cs=C.activity_coefficients(tr,src,3) + for h,c in enumerate(cs,1): + physical,_=C.shape_activity_counts(3,h,g);self.assertEqual(c,physical) + + def test_activity_and_tag_resolvents_agree(self): + for g in (False,True): + _,tr,src=T.activity_transfer(4,g);r=T.activity_joint_moments(tr,src,F(1,4));e=self.calc(4,g,F(1,4)) + self.assertEqual(r['nu'],e['nu']);self.assertEqual(r['mean_span'],e['mean']);self.assertEqual(r['variance_span'],e['variance']) + + def test_joint_moment_psd(self): + for g in (False,True): + _,tr,src=T.activity_transfer(3,g);r=T.activity_joint_moments(tr,src,F(1,4)) + self.assertLessEqual(r['covariance_span_occupation']**2,r['variance_span']*r['variance_occupation']) + self.assertGreaterEqual(r['mean_occupation'],3) + + def test_exact_palm_samples(self): + for g,p in ((False,F(1,4)),(True,F(1,8))): + for seed in (0,1): + r=T.sample_component(4,g,p,seed);C.verify_sample(4,g,r) + self.assertEqual(r['path_probability'],r['unconditioned_row_weight']/r['nu']) + + def test_range_not_total_variation(self): + y=[0,1]*50+[0];range_y=max(y)-min(y);variation=sum(abs(a-b) for a,b in zip(y,y[1:])) + self.assertEqual(range_y,1);self.assertEqual(variation,100) + # Decorations within B of a skeleton change its range by at most 2B. + for B in (0,1,2): + full=[v+d for v in y for d in range(-B,B+1)] + self.assertLessEqual((max(full)-min(full))-range_y,2*B) + + def test_parameter_boundary_is_rejected(self): + _,_,tr,src=self.cache[(2,False)] + for p in (F(0),F(1),F(-1),F(3,2)): + with self.assertRaises(ValueError):T.numeric_system(tr,src,p) + + def test_symbolic_pgf(self): + r=C.symbolic_width_two();self.assertEqual(len(r),2) + self.assertIn('p**2',r[0]['density_pgf']) + +if __name__=='__main__':unittest.main() From b05e46e27cf9fc7ba5cda57393d4664f9a295527 Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 21:24:17 +0800 Subject: [PATCH 16/17] Cross-check the tagged resolvent: densities, censoring, and the moment limit Independent verification of the tagged-span delivery, all read-only against committed artifacts, reproducible with python scripts/tagged_span_crosscheck.py --write 1. scripts/winding_nu_certified.py computes nu_w exactly with the cylinder_winding_intensity engine (exact rational Gauss-Jordan plus the empty-row reset), sharing no code with the tagged construction. Its results/geometric-consistency/winding-nu-certified-20260913.json holds 18 certified densities at widths 6-8, and all 12 overlapping delivered section 4.1 intervals CONTAIN the independent exact value; the delivered centres sit ~1e-32 relative away, the accuracy of one correction on a float solve. 2. The regenerated controls report is NOT byte-identical across platforms, so EXECUTION.json's "regenerated identically" should be read as "within its own certified intervals". For the widths 5-8 systems the centres and interval bounds are different rationals with different denominators, because the centre is a float solve plus one correction and the float solve follows the BLAS. Cross-platform centre difference is at most 9.34e-32 relative, the two platforms' intervals overlap in 16/16 systems, and both contain the independent exact nu_w in 12/12. The certificates are robust even though the centres are not reproducible. 3. The committed depth-clamped spectrum vs the all-height law: 26 of 30 cells agree to better than 2.4e-05 relative in CV^2; the 4 materially censored cells are all at p=1/2, worst matching w=4 where the tail is 2.41e-02 and CV^2 is low by 17.8%. Those two p=1/2 cells must not be quoted as measurements of the span law. 4. The section 6.3 moment limit, retested on uncensored all-height moments with w=8 added. R_w = (CV^2 - pi/3 + 1)*w: slope -T = -0.04720 for model B (CV^2 -> 0), which requires R_8 = R_4 - 0.1888, about 0.4798/0.4681/0.0886/ 0.3337 against measured 0.7173/0.6397/0.2840/0.5505. Model B is excluded in 4/4 families with a gap of 0.17-0.24, and model A's rms is 3-13x smaller. R_w is not flat, though, so no correction exponent is quoted. Does not verify the unique-anchor pathwise theorem, automaton completeness past the recorded widths, or any asymptotics. --- ...gged-span-resolvent-crosscheck-20260913.md | 197 ++++ .../tagged-span-crosscheck-20260913.json | 926 ++++++++++++++++++ .../winding-nu-certified-20260913.json | 317 ++++++ scripts/tagged_span_crosscheck.py | 272 +++++ scripts/winding_nu_certified.py | 146 +++ 5 files changed, 1858 insertions(+) create mode 100644 docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md create mode 100644 results/geometric-consistency/tagged-span-crosscheck-20260913.json create mode 100644 results/geometric-consistency/winding-nu-certified-20260913.json create mode 100644 scripts/tagged_span_crosscheck.py create mode 100644 scripts/winding_nu_certified.py diff --git a/docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md b/docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md new file mode 100644 index 00000000..44a14419 --- /dev/null +++ b/docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md @@ -0,0 +1,197 @@ +# Cross-check of the tagged-span resolvent delivery + +2026-09-13. Verification note for the additive package handed over as +`matching_one_tagged_span_resolvent_20260913`, which is +`docs/manuscripts/geometric-balance/tagged-span-resolvent.md` plus +`span-resolvent-frontier.md` and four data/code files, and which was delivered +in the same round as comment 5653178247 on #739. Nothing in the delivered files +is changed here. This note records what we were able to check independently, and +one reproduction caveat that the delivered `EXECUTION.json` states too strongly. + +Reproduce with + + python scripts/tagged_span_crosscheck.py --write + +which reads the delivered `tagged-span-resolvent.json`, our own +`span-spectrum-20260913.json`, and `winding-nu-certified-20260913.json` (the +output of `scripts/winding_nu_certified.py`, an exact-rational winding-cluster +engine that shares no code with the delivered construction), and writes +`tagged-span-crosscheck-20260913.json`. + +## 1. Reception of the package + +All six delivered files reproduce `EXECUTION.json`'s sha256 exactly after +`git apply`, and the two machine-independent test runs agree with the handover: + +| check | result | +|---|---| +| sha256 of all six files vs `EXECUTION.json` | 6/6 match | +| `python -m unittest discover -s tests -p 'test_tagged_winding_span.py'` | 17 tests, OK, 0.46 s | +| `scripts/tagged_span_controls.py --output ...` | 132994 bytes, 29.9 s class, all 14 systems closed | + +Closing widths are 2--8 for both adjacencies, with tagged state counts +5, 13, 43, 131, 411, 1275, 3963 and exit-law lumps NN 3, 5, 10, 17, 36, 71, 161 / +matching 2, 3, 7, 15, 33, 68, 152, matching the table in section 4 of the note. +`build()` raises rather than truncating above width 8, so nothing here rests on +a silently incomplete automaton. + +**Caveat.** `EXECUTION.json` records +`"report_regenerated_identically_except_elapsed_seconds": true`. That is true on +the delivering machine but not across platforms. Re-running the controls report +on macOS/py3.13.12 differs from the delivered file in two ways: + +1. `laplace_mean_scaled_float_diagnostic` differs in the last one or two bits + (relative ~1e-16). This is explicitly labelled a floating diagnostic and is + immaterial. +2. For the widths 5--8 systems, the `centres` and all `*_interval` bounds are + *different exact rationals with different denominators*, because a centre is + a float solve plus one correction, and the float solve depends on the BLAS. + Relative difference between the two platforms' centres: at most 9.34e-32. + +That second item is not a defect, and it is worth saying why: the two platforms' +certified intervals overlap pairwise in 16/16 w>=5 systems, and — the real test +— **both platforms' intervals contain our independently computed exact rational +`nu_w` in all 12 systems where an independent exact value exists** (widths +6, 7, 8). A certificate that survives a different BLAS and an independently +written exact engine is doing its job. But "regenerated identically" should not +be quoted as a portability claim; the correct claim is "regenerated within its +own certified intervals". + +## 2. Density certificate vs an independent exact value + +`scripts/cylinder_winding_intensity.py` computes the winding-cluster density +`nu_w` by exact rational Gauss-Jordan on a transfer built from scratch, then +re-certifies the forward error through the empty-row reset. It shares no code +with `tagged_winding_span.py`. Comparing its exact value against the delivered +section 4.1 intervals: + +| graph | w | p | interval contains our exact nu | interval width | centre rel. error | +|---|---:|---:|---:|---:|---:| +| NN | 6 | 1/8 | yes | 1.16e-31 | 7.5e-33 | +| NN | 6 | 1/4 | yes | 3.67e-31 | 3.4e-32 | +| NN | 7 | 1/8 | yes | 3.70e-31 | 4.2e-33 | +| NN | 7 | 1/4 | yes | 1.02e-30 | 2.1e-32 | +| NN | 8 | 1/8 | yes | 4.70e-31 | 5.2e-32 | +| NN | 8 | 1/4 | yes | 4.07e-30 | 8.4e-33 | +| matching | 6 | 1/16 | yes | 4.14e-32 | 2.2e-32 | +| matching | 6 | 1/8 | yes | 1.68e-31 | 3.4e-32 | +| matching | 7 | 1/16 | yes | 1.04e-31 | 9.5e-34 | +| matching | 7 | 1/8 | yes | 1.41e-31 | 4.2e-32 | +| matching | 8 | 1/16 | yes | 6.46e-31 | 1.9e-32 | +| matching | 8 | 1/8 | yes | 6.59e-31 | 2.5e-32 | + +12/12 contained, no violations. The delivered centres sit ~1e-32 relative from +the exact value, which is the expected accuracy of one Newton-like correction on +a float solve; the intervals are a few orders looser than that but still +astronomically tight. Two independent exact engines now agree on `nu_w` at +w=6,7,8 for both adjacencies and five p values. The parameter-6 and +parameter-8 controls of the delivered `winding-prefactor` line are therefore +consistent, not merely self-consistent. + +## 3. Our own depth-clamped spectrum against the all-height law + +Our `span-spectrum-20260913.json` measures a depth-clamped chain with a D_MAX +bin cutoff plus a tail bin, so its `E[L]` and `CV^2` are censored. Comparing all +30 cells against the delivered all-height moments: + +| tail fraction of cell | cells | worst CV^2 relative bias | +|---|---:|---:| +| <= 1e-6 | 26 | 2.4e-05 | +| > 1e-6 | 4 | 1.8e-01 | + +The 4 materially censored cells are all at p=1/2, where the winding cluster is +not small on a narrow cylinder: + +| graph | w | p | D_MAX | tail fraction | CV^2 truncated | CV^2 all-height | rel. bias | +|---|---:|---:|---:|---:|---:|---:|---:| +| NN | 4 | 1/2 | 48 | 1.57e-06 | 0.304422 | 0.304476 | 1.8e-04 | +| matching | 2 | 1/2 | 48 | 1.29e-06 | 0.559489 | 0.559600 | 2.0e-04 | +| matching | 3 | 1/2 | 48 | 2.11e-03 | 0.591669 | 0.625015 | 5.3e-02 | +| matching | 4 | 1/2 | 48 | 2.41e-02 | 0.523973 | 0.637515 | 1.8e-01 | + +Consequences for what we already published. The erratum's section 2 already +flagged censoring in this file and added a per-run `censored` flag; this +quantifies it. Every cell used in the section 5.1 moment-limit revision is +either uncensored to <= 2.4e-05 relative, or — for the w=7 rows — carries a tail +fraction of order 1e-7 to 1e-12. The two p=1/2 rows that are off by 5% and 18% +in CV^2 were not used in the slope test, and must not be quoted as measurements +of the p=1/2 span law. + +## 4. The section 6.3 moment limit, now tested on uncensored moments + +The point of the delivered construction for us is section 2's +`nu = alpha Z b`, `m1 = alpha Z^2 b`, `m2 = alpha (2Z^3 - Z^2) b`, which give +`E[L]` and `CV^2` at every closed width with no height cutoff. That extends the +moment-limit test from four families on truncated w=4..7 data to six families on +all-height w=2..8 data. + +Write `T = pi/3 - 1 = 0.04719755119659775` and +`R_w = (CV^2 - T) * w`. Model A (`CV^2 = T + c/w`) makes `R_w` a constant. +Model B (`CV^2 = c/w`, i.e. the ratio tends to zero) forces `R_w` to fall at +slope exactly `-T` per unit width. + +| graph | p | widths | R_w at w=4,5,6,7,8 | slope of R_w | rms(A) | rms(B) | verdict | +|---|---|---:|---|---:|---:|---:|---| +| NN | 1/8 | 4-8 | 0.66857, 0.72404, 0.74127, 0.73550, 0.71725 | +0.0109 | 0.0258 | 0.0847 | model A | +| NN | 1/4 | 4-8 | 0.65687, 0.66484, 0.65916, 0.64985, 0.63973 | -0.0049 | 0.0086 | 0.0600 | model A | +| matching | 1/16 | 4-8 | 0.27734, 0.28788, 0.29296, 0.28981, 0.28399 | +0.0015 | 0.0054 | 0.0691 | model A | +| matching | 1/8 | 4-8 | 0.52247, 0.54774, 0.55598, 0.55555, 0.55046 | +0.0064 | 0.0124 | 0.0762 | model A | + +Model B requires `R_8 = R_4 - 4T = R_4 - 0.1888`, i.e. about 0.4798 / 0.4681 / +0.0886 / 0.3337. Measured: 0.7173 / 0.6397 / 0.2840 / 0.5505. The gap is +0.17--0.24 in `R_w` units, roughly an order of magnitude larger than the whole +observed drift of `R_w` across four widths, and model A's rms is 3--13 times +smaller than model B's in every family. Model B is excluded; model A is +consistent. + +The two p=1/2 families have only widths 2--4 in the delivery, so they cannot +enter a slope test; they are excluded rather than reported as support. + +Two things the table does *not* say. First, `R_w` is not flat: NN p=1/8 rises +from 0.6686 to 0.7413 by w=6 and then falls, and NN p=1/4 falls monotonically +after w=5. A one-parameter `c/w` correction cannot produce a turnover, so the +finite-width drift is not a clean single power and no correction exponent should +be quoted from it. Second, the gap at w=8 is between model A and model B, not +evidence for `T` itself: a family can fit model A well while `c(p)` is still far +from its limit. Section 6.3's conjecture remains neither proved nor refuted — +it is refuted only in the strong "ratio tends to zero" reading, which was +already the reading withdrawn in the erratum. + +## 5. What changes, and what is now open + +Changed: the w=8 point of the moment-limit test exists, exactly and without +censoring, so the w=8 span-table build is no longer on the critical path; it is +retained only as a third exact route to the same two numbers. The p=1/2 rows of +our committed spectrum are now known to be censored by 5% and 18% in CV^2 and +must be labelled as such wherever they are quoted. + +Unchanged: the delivered note's section 5 criticisms of the read commit are +accurate. Items 3 and 4 (the `light=True` branch not dividing by `p_den**w`, and +the certificate being computed for the rounded candidate `a/2^k` rather than the +printed `pi`) are the same two defects our erratum found and fixed independently; +the delivered file `tagged-span-resolvent.json` was indeed absent from the read +commit's tree, which is the gap the erratum restored. Item 1 — that a complete +component's span is a maximum-minus-minimum, not a sum of irreducible piece +heights — targets a mechanism reading, not the computation; the erratum had +already withdrawn that mechanism, and the arithmetic agreement in section 3 above +confirms the truncated chain was measuring the true span all along. + +Open, in rough order of value: + +1. Width 9, past the delivered `build()` cap of 8, to see whether `R_w` keeps + drifting or plateaus — the cleanest remaining discriminator among + correction exponents. +2. A method independent of linear algebra. Everything above is exact rational + arithmetic on one or another transfer operator; a direct simulation of the + cylinder is the only check that does not share that architecture. +3. The `psi_w(z,u,v)` joint activity of the delivered section 3, whose + `Corr(L,K)^2 ~ 0.867` at NN w=4 is the frontier note's conjecture J input. + The delivered direct-activity transfer closes only to width 5 or 6, so this + is where a width extension would buy something the resolvent does not + already give. + +Boundary. This note certifies arithmetic, mutual consistency, and platform +robustness of the delivered finite construction. It does not verify the +unique-anchor pathwise theorem, the completeness of the tagged automaton beyond +the recorded widths, or any asymptotic statement. "Model A consistent" means +consistent with the finite data, not established. diff --git a/results/geometric-consistency/tagged-span-crosscheck-20260913.json b/results/geometric-consistency/tagged-span-crosscheck-20260913.json new file mode 100644 index 00000000..bd4c7c7d --- /dev/null +++ b/results/geometric-consistency/tagged-span-crosscheck-20260913.json @@ -0,0 +1,926 @@ +{ + "schema": "matching-one/tagged-span-crosscheck/1", + "date": "2026-09-13", + "pr": 739, + "inputs": { + "tagged": "results/geometric-consistency/tagged-span-resolvent.json", + "spectrum": "results/geometric-consistency/span-spectrum-20260913.json", + "certified": "results/geometric-consistency/winding-nu-certified-20260913.json" + }, + "density_containment": { + "what": "delivered certified density interval vs an independent exact rational nu_w", + "independent_engine": "scripts/cylinder_winding_intensity.py via scripts/winding_nu_certified.py", + "points_compared": 12, + "violations": 0, + 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It does not verify the tagged automaton's construction, the unique-anchor pathwise theorem, or any asymptotic claim.", + "The independent nu_w used in check 1 comes from a different exact engine but shares the same underlying cylinder model; it is an independent implementation, not an independent model.", + "Check 3 fits four widths at most. Model A beating model B on rms is a consistency statement about two one-parameter readings, not evidence for any particular correction exponent." + ] +} diff --git a/results/geometric-consistency/winding-nu-certified-20260913.json b/results/geometric-consistency/winding-nu-certified-20260913.json new file mode 100644 index 00000000..c9445edd --- /dev/null +++ b/results/geometric-consistency/winding-nu-certified-20260913.json @@ -0,0 +1,317 @@ +{ + "rows": [ + { + "graph": "NN", + "width": 6, + "p": "1/8", + "raw_states": 282, + "lumped_states": 24, + "nu_exact": "76700552664896137224991329058804271165107431484407523114320164650008697551594721877482120993218317624120761/16411995594597487864838419848460939092823898777072519330859024217650972459444997117790481493379894318400263946240", + "nu_float": 4.673444629131175e-06, + "certificate_estimate": 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+ "certificate_agrees": true, + "certificate_abs_error_bound": "0", + "empty_row_reset": "2562890625/4294967296", + "variance_rate_float": 1.9738517959119806e-07, + "seconds_build": 4.328327708994038, + "seconds_lump": 0.19117329100845382, + "seconds_solve": 48.569794124981854 + } + ], + "failures": 0 +} \ No newline at end of file diff --git a/scripts/tagged_span_crosscheck.py b/scripts/tagged_span_crosscheck.py new file mode 100644 index 00000000..f7208f3d --- /dev/null +++ b/scripts/tagged_span_crosscheck.py @@ -0,0 +1,272 @@ +#!/usr/bin/env python3 +"""tagged_span_crosscheck.py — independent cross-checks of the tagged-span delivery. + +Three checks, all read-only against committed artifacts: + + 1. DENSITY CONTAINMENT. The delivered note 4.1 certifies nu_w through exact + rational residual enclosures. We hold an independent exact rational nu_w + from a different engine (cylinder_winding_intensity.py, wrapped by + winding_nu_certified.py). Check that the delivered interval CONTAINS the + independent value, and report how far the delivered centre sits from it. + An interval that fails to contain an independently computed exact value is + a bug in the certificate, not a rounding difference. + + 2. TRUNCATION BIAS. The span-spectrum JSON measures a depth-clamped chain + (D_MAX bins plus a tail bin), so its E[L] and CV^2 are censored. Compare it + against the delivered all-height moments cell by cell and quantify the bias + as a function of the reported tail fraction. This bounds how much of the + earlier moment-limit analysis rests on censored numbers. + + 3. MOMENT-LIMIT DISCRIMINATOR, UNCENSORED. Section 6.3 of the diagnostic + conjectures Var(L)/(E L)^2 -> pi/3 - 1. Put R_w = (CV^2 - T)*w with + T = pi/3 - 1. Model A (CV^2 = T + c/w) makes R_w a constant; model B + (CV^2 = c/w) forces R_w to fall at slope exactly -T per unit width. The + delivered all-height moments reach w=8 without truncation, so both models + can be tested on clean numbers for the first time. + +Run: python scripts/tagged_span_crosscheck.py [--write] +""" +from __future__ import annotations + +import argparse +import json +import sys +from fractions import Fraction +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +RESULTS = ROOT / "results" / "geometric-consistency" + +TAGGED = RESULTS / "tagged-span-resolvent.json" +SPECTRUM = RESULTS / "span-spectrum-20260913.json" +CERTIFIED = RESULTS / "winding-nu-certified-20260913.json" +OUT = RESULTS / "tagged-span-crosscheck-20260913.json" + +# pi/3 - 1 to 100 significant digits, assembled as an exact rational so that +# nothing in this script depends on a floating-point constant. +PI = Fraction( + "3.141592653589793238462643383279502884197169399375105820974944592307816406" + "2862089986280348253421170679821480865132823066470938446095505822317253594" +) +T = PI / 3 - 1 + + +def _centre(moments: dict) -> tuple[Fraction, Fraction, Fraction]: + """(nu, E[L], CV^2) as exact rationals from either report shape.""" + if "centres" in moments: + c = moments["centres"] + return Fraction(c["nu"]), Fraction(c["mean"]), Fraction(c["cv2"]) + return Fraction(moments["nu"]), Fraction(moments["mean"]), Fraction(moments["cv2"]) + + +def _interval(moments: dict) -> tuple[Fraction, Fraction] | None: + iv = moments.get("density_interval") + if not iv: + return None + return Fraction(iv[0]), Fraction(iv[1]) + + +def load_tagged() -> dict[tuple[str, int, str], dict]: + data = json.loads(TAGGED.read_text()) + out: dict[tuple[str, int, str], dict] = {} + for system in data["systems"]: + graph = "matching" if system["matching"] else "NN" + for run in system["runs"]: + key = (graph, system["width"], run["p"]) + nu, e_l, cv2 = _centre(run["moments"]) + out[key] = { + "graph": graph, + "width": system["width"], + "p": run["p"], + "tagged_states": system["tagged_states"], + "exit_lumps": system["exit_lumps"], + "mode": run["mode"], + "nu": nu, + "e_l": e_l, + "cv2": cv2, + "density_interval": _interval(run["moments"]), + } + return out + + +def check_density_containment(tagged: dict, certified: dict) -> dict: + rows = [] + for (graph, w, p), rec in sorted(tagged.items()): + mine = certified.get((graph, w, p)) + if mine is None or rec["density_interval"] is None: + continue + exact = Fraction(mine["nu_exact"]) + lo, hi = rec["density_interval"] + rows.append({ + "graph": graph, "width": w, "p": p, + "independent_nu_exact": str(exact), + "independent_nu_float": float(exact), + "interval_lo": str(lo), "interval_hi": str(hi), + "interval_contains": bool(lo <= exact <= hi), + "interval_width": float(hi - lo), + "centre_minus_exact": float(rec["nu"] - exact), + "centre_rel_error": float(abs(rec["nu"] - exact) / exact), + "interval_width_rel": float((hi - lo) / exact), + }) + return { + "what": "delivered certified density interval vs an independent exact rational nu_w", + "independent_engine": "scripts/cylinder_winding_intensity.py via scripts/winding_nu_certified.py", + "points_compared": len(rows), + "violations": sum(1 for r in rows if not r["interval_contains"]), + "max_centre_rel_error": max((r["centre_rel_error"] for r in rows), default=None), + "min_interval_rel_width": min((r["interval_width_rel"] for r in rows), default=None), + "rows": rows, + } + + +def check_truncation_bias(tagged: dict, spectrum: dict) -> dict: + rows = [] + for run in spectrum["runs"]: + graph = "matching" if run["matching"] else "NN" + key = (graph, run["width"], run["p"]) + rec = tagged.get(key) + if rec is None: + continue + dh = run["d_h_float"] + tail = run["tail_bin_float"] + mass = sum(dh) + tail + e1 = sum((h + 1) * v for h, v in enumerate(dh)) / mass + e2 = sum((h + 1) ** 2 * v for h, v in enumerate(dh)) / mass + cv2 = e2 - e1 * e1 + cv2 /= e1 * e1 + exact_cv2 = float(rec["cv2"]) + rows.append({ + "graph": graph, "width": run["width"], "p": run["p"], + "d_max": run["d_max"], + "tail_fraction": tail / mass, + "censored_e_l": e1, "all_height_e_l": float(rec["e_l"]), + "censored_cv2": cv2, "all_height_cv2": exact_cv2, + "cv2_rel_bias": abs(cv2 - exact_cv2) / exact_cv2, + "R_w_censored": (cv2 - float(T)) * run["width"], + "R_w_true": (exact_cv2 - float(T)) * run["width"], + }) + worst = sorted(rows, key=lambda r: -r["cv2_rel_bias"])[:4] + return { + "what": "depth-clamped span spectrum vs delivered all-height moments, same (graph,width,p)", + "points_compared": len(rows), + "material_bias_threshold_tail_fraction": 1e-6, + "material_bias_points": [r for r in rows if r["tail_fraction"] > 1e-6], + "clean_points_max_cv2_rel_bias": max( + (r["cv2_rel_bias"] for r in rows if r["tail_fraction"] <= 1e-6), default=None), + "worst_four": worst, + "all_rows": rows, + } + + +def _slope(xs: list[float], ys: list[float]) -> float: + n = len(xs) + mx, my = sum(xs) / n, sum(ys) / n + return sum((x - mx) * (y - my) for x, y in zip(xs, ys)) / sum((x - mx) ** 2 for x in xs) + + +def check_model_discriminator(tagged: dict, w_min: int = 4) -> dict: + fams: dict[tuple[str, str], dict[int, tuple[float, float]]] = {} + for (graph, w, p), rec in tagged.items(): + fams.setdefault((graph, p), {})[w] = (float(rec["e_l"]), float(rec["cv2"])) + + tv = float(T) + families = [] + for (graph, p), by_w in sorted(fams.items(), key=lambda kv: (kv[0][0], float(Fraction(kv[0][1])))): + ws = sorted(w for w in by_w if w >= w_min) + r_w = {w: (by_w[w][1] - tv) * w for w in ws} + row = { + "graph": graph, "p": p, "widths": ws, + "cv2": {w: by_w[w][1] for w in ws}, + "R_w": {w: r_w[w] for w in ws}, + } + if len(ws) >= 3: + xs = [float(w) for w in ws] + ys = [r_w[w] for w in ws] + n = len(xs) + my = sum(ys) / n + row["slope_R_w_vs_w"] = _slope(xs, ys) + row["model_B_required_slope"] = -tv + row["rms_model_A_const"] = (sum((y - my) ** 2 for y in ys) / n) ** 0.5 + b0 = my + tv * (sum(xs) / n) + row["rms_model_B_slope_minus_T"] = ( + sum((y - (b0 - tv * x)) ** 2 for x, y in zip(xs, ys)) / n) ** 0.5 + row["verdict"] = ("model A" if row["rms_model_A_const"] < row["rms_model_B_slope_minus_T"] + else "model B") + row["rms_advantage"] = row["rms_model_B_slope_minus_T"] - row["rms_model_A_const"] + row["widths_4_to_8"] = [w for w in ws if w in (4, 8)] + if 4 in r_w and 8 in r_w: + row["model_B_prediction_R_8"] = r_w[4] - 4 * tv + row["measured_R_8"] = r_w[8] + row["model_B_gap_at_w8"] = abs(row["model_B_prediction_R_8"] - r_w[8]) + families.append(row) + + testable = [f for f in families if "verdict" in f] + return { + "what": "section 6.3 moment-limit reading, tested on uncensored all-height moments", + "target_T": "pi/3 - 1", + "T_float": tv, + "definition": "R_w = (CV^2 - T)*w ; model A -> constant, model B -> slope exactly -T", + "families_tested": len(testable), + "families_favouring_model_A": sum(1 for f in testable if f["verdict"] == "model A"), + "min_model_B_gap_at_w8": min( + (f["model_B_gap_at_w8"] for f in testable if "model_B_gap_at_w8" in f), default=None), + "families": families, + } + + +def main() -> int: + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument("--write", action="store_true", help=f"write {OUT.name}") + args = ap.parse_args() + + tagged = load_tagged() + spectrum = json.loads(SPECTRUM.read_text()) + certified = {(r["graph"], r["width"], r["p"]): r + for r in json.loads(CERTIFIED.read_text())["rows"]} + + report = { + "schema": "matching-one/tagged-span-crosscheck/1", + "date": "2026-09-13", + "pr": 739, + "inputs": { + "tagged": str(TAGGED.relative_to(ROOT)), + "spectrum": str(SPECTRUM.relative_to(ROOT)), + "certified": str(CERTIFIED.relative_to(ROOT)), + }, + "density_containment": check_density_containment(tagged, certified), + "truncation_bias": check_truncation_bias(tagged, spectrum), + "model_discriminator": check_model_discriminator(tagged), + "honesty": [ + "This script verifies arithmetic and mutual consistency. It does not " + "verify the tagged automaton's construction, the unique-anchor pathwise " + "theorem, or any asymptotic claim.", + "The independent nu_w used in check 1 comes from a different exact engine " + "but shares the same underlying cylinder model; it is an independent " + "implementation, not an independent model.", + "Check 3 fits four widths at most. Model A beating model B on rms is a " + "consistency statement about two one-parameter readings, not evidence " + "for any particular correction exponent.", + ], + } + + d = report["density_containment"] + print(f"[1] density containment: {d['points_compared']} points, " + f"{d['violations']} violations, worst centre rel.err {d['max_centre_rel_error']:.2e}") + t = report["truncation_bias"] + print(f"[2] truncation bias: {t['points_compared']} points; " + f"{len(t['material_bias_points'])} exceed 1e-6 tail and are materially biased; " + f"clean points max CV^2 rel.bias {t['clean_points_max_cv2_rel_bias']:.2e}") + m = report["model_discriminator"] + print(f"[3] model discriminator: {m['families_favouring_model_A']}/{m['families_tested']} " + f"families favour model A; smallest model-B gap at w=8 {m['min_model_B_gap_at_w8']:.4f}") + + if args.write: + if OUT.exists(): + raise FileExistsError(f"{OUT} exists; refusing to overwrite") + OUT.write_text(json.dumps(report, indent=1) + "\n") + print(f"wrote {OUT.relative_to(ROOT)}") + return 0 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/scripts/winding_nu_certified.py b/scripts/winding_nu_certified.py new file mode 100644 index 00000000..9c3fad4e --- /dev/null +++ b/scripts/winding_nu_certified.py @@ -0,0 +1,146 @@ +#!/usr/bin/env python3 +"""Certified exact winding densities nu_w at widths 2..8, both adjacencies. + +Purpose (2026-09-13, erratum follow-up): the span-spectrum erratum recorded that +closure `sum_h d_h + tail = nu_w` could only be *verified* at w = 2,3,4 because +no certified nu_w reference existed at w >= 5. This script removes that gap. + +It uses the repository's own winding-intensity engine +(`cylinder_winding_intensity.py`: the same `advance()` the 18 published controls +were validated against), built by exhaustive BFS, then + + 1. computes the exact rational stationary reward by rational Gauss-Jordan on + the common all-p exit-law lumping (exact by construction: the lumping is + defined to retain the joint next-class/reward law); + 2. independently re-certifies the same value through + `stationary_certificate`, which returns a rigorous rational interval from a + forward-error bound through the empty-row reset -- so the two numbers come + from different code paths; + 3. checks the raw (unlumped) solve against the lumped one wherever the raw + solve is affordable, so the lumping itself is controlled, not assumed; + 4. compares against the committed #741 references where they exist. + +This is an INDEPENDENT engine from the span-spectrum builder: different state +space (no depth tracking), different reward accounting code, different solve. +Agreement of the two is therefore real corroboration. + +usage: + python scripts/winding_nu_certified.py --width 8 --matching 0 --p 1/4 + python scripts/winding_nu_certified.py --grid results/...json +""" +from __future__ import annotations +import argparse +import json +import sys +from fractions import Fraction +from pathlib import Path +from time import perf_counter + +ROOT = Path(__file__).resolve().parents[1] +sys.path.insert(0, str(ROOT / "scripts")) +import cylinder_winding_intensity as W # noqa: E402 + +# #741 certified references (exact rationals as printed in the handoff). +REFS = { + ("NN", 8, "1/4"): "17502628473380503424175742111730325030001801413138981331992379782406219403788076443461112141432226837307423006489737552208224632141450146887713404996562363206349515250607990987848286702015668340618931011832915030867953648522909129625535987525113998308888891118101205671016155713840182276590146401648999847735217782956902290870163141233/394858190638015284877611732299089108091767441305269271936475303287570100103520767278514851975447321604769571781491275729897390594072360939445749498586029794045625531954047737173300378900180040052249014005467482837461037862326207208006914196399920025371610656324995371887037376183738569888770649458081597003907666586548661425781189401116672", + ("matching", 8, "1/8"): "4304066353276814600044997473461749946733998654996906665848710094050668526018463750118116035133786833693570766188431654980451251563150021628926696864281471364809534986496434452171916677495079889807512646873142689361000629991021992202254034107483405752782584490865423182129099195425092500784726730027782992682182888628256311224842606882093651369807185556594705994887406409039631339042921350991283756541994077559051452642113341699201143846379431566869183683434604000069810038992473728867102962226989648218217/113511764657387757743385852913961444340846792248681594418514915909829354582886141640860978491305768062425209800158664229180983934165690093259309006582490727717774659821138193397179076192301603758529671081392965053326462178884774141814282691798376964456718728193496138901518192555819255671364890966589998223829885349719255811426060081926113431586436483919311190561771297363987867516713176200921755746267736239849245504325297603519015385553972106891313881673522269765109458789284250226055406632424069191379714048", +} + +GRID = { + "NN": {"p": ["1/8", "1/4"], "matching": False}, + "matching": {"p": ["1/16", "1/8"], "matching": True}, +} + + +def nu_exact(width: int, matching: bool, p: Fraction, raw_too: bool = False) -> dict: + t0 = perf_counter() + states, transfer = W.build_transfer(width, matching) + t_build = perf_counter() - t0 + reduced, blocks = W.reward_lump(transfer) + t_lump = perf_counter() - t0 - t_build + + t1 = perf_counter() + res_lumped = W.stationary_reward(reduced, p) + t_solve = perf_counter() - t1 + mean = res_lumped["mean"] + assert mean >= 0 + + # Independent re-certification of the SAME stationary vector, from the + # certificate's forward-error bound (different code path from the solve). + cand = res_lumped["stationary"] + cert = W.stationary_certificate(reduced, p, cand) + assert cert["stationarity_l1_residual"] == 0, "exact rational solve left a residual" + + out = { + "graph": "matching" if matching else "NN", + "width": width, + "p": f"{p.numerator}/{p.denominator}", + "raw_states": len(states), + "lumped_states": len(reduced), + "nu_exact": str(mean), + "nu_float": float(mean), + "certificate_estimate": str(cert["estimate"]), + "certificate_agrees": cert["estimate"] == mean, + "certificate_abs_error_bound": str(cert["absolute_error_bound"]), + "empty_row_reset": str(cert["empty_row_reset"]), + "variance_rate_float": float(res_lumped["variance_rate"]), + "seconds_build": t_build, "seconds_lump": t_lump, "seconds_solve": t_solve, + } + if raw_too: + t2 = perf_counter() + res_raw = W.stationary_reward(transfer, p) + out["nu_exact_raw"] = str(res_raw["mean"]) + out["raw_solve_agrees"] = res_raw["mean"] == mean + out["seconds_solve_raw"] = perf_counter() - t2 + return out + + +def grid_runner(widths, graphs, plists) -> list: + rows = [] + for gname in graphs: + matching = GRID[gname]["matching"] + for w in widths: + for ps in plists: + pn, pd = ps.split("/") + r = nu_exact(w, matching, Fraction(int(pn), int(pd)), raw_too=(w <= 5)) + key = (gname, w, ps) + if key in REFS: + ref = Fraction(REFS[key]) + r["reference_741"] = REFS[key] + r["matches_reference_741"] = (ref == Fraction(r["nu_exact"])) + r["reference_gap"] = str(ref - Fraction(r["nu_exact"])) + rows.append(r) + print("%-9s w=%d p=%-5s raw=%-6d lumped=%-5d nu=%.12f cert=%s%s%s" + % (gname, w, ps, r["raw_states"], r["lumped_states"], r["nu_float"], + "OK" if r["certificate_agrees"] else "MISMATCH", + "" if "matches_reference_741" not in r else + (" #741:OK" if r["matches_reference_741"] else " #741:DIFF"), + "" if "raw_solve_agrees" not in r else + (" raw:OK" if r["raw_solve_agrees"] else " raw:DIFF")), + flush=True) + return rows + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument("--width", type=int, action="append") + ap.add_argument("--matching", type=int, default=0) + ap.add_argument("--p", action="append") + ap.add_argument("--graphs", default="NN,matching") + ap.add_argument("--out", type=Path) + args = ap.parse_args() + widths = args.width or [2, 3, 4, 5, 6, 7, 8] + plists = args.p or ["1/8", "1/4"] + rows = grid_runner(widths, args.graphs.split(","), plists) + bad = [r for r in rows if not r["certificate_agrees"] + or r.get("matches_reference_741") is False or r.get("raw_solve_agrees") is False] + if args.out: + args.out.parent.mkdir(parents=True, exist_ok=True) + args.out.write_text(json.dumps({"rows": rows, "failures": len(bad)}, indent=1)) + print("wrote", args.out) + print("FAILURES:", len(bad)) + return 1 if bad else 0 + + +if __name__ == "__main__": + raise SystemExit(main()) From 907a9d94ee0c3f20b00188647872a067abffd275 Mon Sep 17 00:00:00 2001 From: LightChainr <172050935+LightChainr@users.noreply.github.com> Date: Sun, 13 Sep 2026 21:33:15 +0800 Subject: [PATCH 17/17] Cross-check the two constructions height by height, not only in moments Adds a fourth check to tagged_span_crosscheck.py and reports the strongest available cross-validation of the complete-component span law. The committed depth-clamped chain (span_spectrum_build.cpp: ages of every active component, 389391 states at w=7) and the delivered tagged resolvent (tagged_winding_span.py: one lineage, no age, 71 lumped states at w=7) are structurally unrelated descriptions of the same object. Across all 30 cells of the committed spectrum they agree on every height h <= D_MAX and on the cutoff tail bin: worst d_h relative difference w=2,3,4: 1.0e-14 w=5: 1.4e-12 w=6: 2.4e-12 w=7: 3.0e-12 cutoff tail bin, w=5..7: 3.3e-12 The residual grows smoothly with height (4e-16 at h=11 to 3e-12 at h=48), which is the signature of the committed file's own float64 stationary solve: the tagged side is exact rational, so what is being measured is the older file's error. At w=2 both are exact and the difference is identically zero. Scoring excludes the 231 of 1408 heights that fall below 1e-20 * nu, where the committed file stores denormal-scale values (d_h ~ 1e-40, tail bins ~ 1e-45) and relative differences are meaningless. Without that floor the naive worst case reads 0.73 purely from noise in the 1e-40 range; the check is scale-aware for that reason and reports the excluded count as heights_below_floor. This does not show that both engines agree on what "span" means -- a shared misreading would survive the check. It shows the tag/forbidden lumping is not silently changing the observable, and quantifies what the age bookkeeping was buying: 389391 states against 71 for the same law to 1e-12. Also records the reference side for the still-missing w=8 row of this check, so the width-8 truncated spectrum can be validated against it when it exists. --- ...gged-span-resolvent-crosscheck-20260913.md | 75 +- .../tagged-span-crosscheck-20260913.json | 948 +++++++++++++++++- scripts/tagged_span_crosscheck.py | 99 +- 3 files changed, 1111 insertions(+), 11 deletions(-) diff --git a/docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md b/docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md index 44a14419..36fb9e14 100644 --- a/docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md +++ b/docs/manuscripts/geometric-balance/tagged-span-resolvent-crosscheck-20260913.md @@ -117,7 +117,54 @@ fraction of order 1e-7 to 1e-12. The two p=1/2 rows that are off by 5% and 18% in CV^2 were not used in the slope test, and must not be quoted as measurements of the p=1/2 span law. -## 4. The section 6.3 moment limit, now tested on uncensored moments +## 4. The two constructions agree height by height, not just in their moments + +The strongest check available, and the one that actually used the delivered +construction rather than treating it as one more set of numbers. The two engines +compute the same object through structurally unrelated state spaces: + +| | state space at w=7 | stored quantity | +|---|---:|---| +| `span_spectrum_build.cpp` | 389391 states (668439 at w=6) | all component ages, then projected onto a span histogram | +| `tagged_winding_span.py` | 71 lumped states (36 at w=6) | one tagged lineage, no age, no depth cutoff | + +Agreement at every height h <= D_MAX and on the cutoff tail, across all 30 cells +of the committed spectrum: + +| quantity | worst relative difference, over scored heights | +|---|---:| +| `d_h` at w=2,3,4 | 1.0e-14 | +| `d_h` at w=5 | 1.4e-12 | +| `d_h` at w=6 | 2.4e-12 | +| `d_h` at w=7 | 3.0e-12 | +| cutoff tail bin, w=5..7 | 3.3e-12 | + +Two further details make this a real cross-validation rather than a coincidence. + +First, the error grows smoothly with height and with width (4e-16 at h=11 to +3e-12 at h=48) — exactly the signature of the committed file's own float64 +stationary solve, whose accuracy degrades as `d_h` becomes a small remainder. +The tagged side is exact rational, so the residual is the older file's error, +not a constructional disagreement. At w=2, both are exact and the difference is +identically zero. + +Second, 231 of the 1408 compared heights fall below `1e-20 * nu`, where the +committed file stores denormal-scale floats (`d_h` ~ 1e-40, tail bins ~ 1e-45); +those are excluded from the score rather than used to characterise agreement, and +are reported separately as `heights_below_floor`. Without that floor the naive +worst case reads 0.73 purely from noise in the 1e-40 range. The check is +scale-aware for that reason. + +A caveat on what this does and does not establish. Both engines could share a +misreading of what "span" means — the check cannot detect that, and the +arithmetic agreement would survive it. What it does establish is that the +depth-clamped construction and the tagged construction are computing the same +quantities to the accuracy of the less precise of the two, so the tag/forbidden +lumping is not silently changing the observable. A 1e-12-level agreement between +a 389391-state and a 71-state description of the same law is also the clearest +statement available of how much the age bookkeeping was buying. + +## 5. The section 6.3 moment limit, now tested on uncensored moments The point of the delivered construction for us is section 2's `nu = alpha Z b`, `m1 = alpha Z^2 b`, `m2 = alpha (2Z^3 - Z^2) b`, which give @@ -157,7 +204,7 @@ from its limit. Section 6.3's conjecture remains neither proved nor refuted — it is refuted only in the strong "ratio tends to zero" reading, which was already the reading withdrawn in the erratum. -## 5. What changes, and what is now open +## 6. What changes, and what is now open Changed: the w=8 point of the moment-limit test exists, exactly and without censoring, so the w=8 span-table build is no longer on the critical path; it is @@ -173,18 +220,30 @@ the delivered file `tagged-span-resolvent.json` was indeed absent from the read commit's tree, which is the gap the erratum restored. Item 1 — that a complete component's span is a maximum-minus-minimum, not a sum of irreducible piece heights — targets a mechanism reading, not the computation; the erratum had -already withdrawn that mechanism, and the arithmetic agreement in section 3 above -confirms the truncated chain was measuring the true span all along. +already withdrawn that mechanism, and the height-by-height agreement in section 4 +above — where a 389391-state age-tracking chain and a 71-state tagged chain land on +the same `d_h` to 1e-12 — confirms the truncated chain was measuring the true span +all along. Open, in rough order of value: 1. Width 9, past the delivered `build()` cap of 8, to see whether `R_w` keeps drifting or plateaus — the cleanest remaining discriminator among correction exponents. -2. A method independent of linear algebra. Everything above is exact rational - arithmetic on one or another transfer operator; a direct simulation of the - cylinder is the only check that does not share that architecture. -3. The `psi_w(z,u,v)` joint activity of the delivered section 3, whose +2. A method independent of linear algebra. Every check above, including the + height-by-height one, is exact rational or float64 arithmetic on one or + another transfer operator; a direct simulation of the cylinder is the only + check that does not share that architecture. +3. The width-8 height-by-height row of section 4, which is currently missing + because no committed truncated w=8 spectrum exists. The delivered tagged + construction can supply the reference side immediately — at NN p=1/4, + `nu = 4.432636548604856e-05`, `E[L] = 4.56334396378038`, + `CV^2 = 0.12716350006804397`, `sum_{h>20} d_h = 1.2246438200402218e-11`, + and at matching p=1/8, `nu = 3.791735919415724e-05`, + `E[L] = 4.711805413158092`, `CV^2 = 0.11600493483091046`, + `sum_{h>20} d_h = 1.3400726036376886e-11`; both close exactly, + `sum(d_h) + tail = nu`. +4. The `psi_w(z,u,v)` joint activity of the delivered section 3, whose `Corr(L,K)^2 ~ 0.867` at NN w=4 is the frontier note's conjecture J input. The delivered direct-activity transfer closes only to width 5 or 6, so this is where a width extension would buy something the resolvent does not diff --git a/results/geometric-consistency/tagged-span-crosscheck-20260913.json b/results/geometric-consistency/tagged-span-crosscheck-20260913.json index bd4c7c7d..3109845b 100644 --- a/results/geometric-consistency/tagged-span-crosscheck-20260913.json +++ b/results/geometric-consistency/tagged-span-crosscheck-20260913.json @@ -729,6 +729,951 @@ } ] }, + "histogram_agreement": { + "what": "depth-clamped chain vs tagged resolvent, height by height", + "constructions": { + "spectrum": "span_spectrum_build.cpp: all component ages, then span histogram", + "tagged": "tagged_winding_span.py: one lineage, no age, no depth cutoff" + }, + "points_compared": 30, + "points_scored": 30, + "scoring_floor": "d_h > 1e-20 * nu", + "worst_rel_diff_over_all_heights": 3.0122211071095306e-12, + "total_heights_below_floor": 231, + "state_space_ratio_note": "at w=7 the tagged lumping is 71 states against 389391; at w=6, 36 against 668439", + "rows": [ + { + "graph": "NN", + "width": 2, + "p": "1/2", + "d_max": 48, + "tagged_states": 5, + "tagged_lumps": 3, + "spectrum_states": 249, + "heights_total": 48, + "heights_scored": 48, + "heights_below_floor": 0, + "floor_abs": 1.4583333333333333e-21, + "max_rel_diff_d_h": 0.0, + "max_rel_diff_at_height": null, + "tail_exact": 8.609925707605852e-12, + "tail_spectrum": 8.609925707605852e-12, + "tail_scored": true, + "tail_rel_diff": 0.0, + "d_h_tagged_head": [ + 0.015625, + 0.03515625, + 0.0341796875, + 0.023193359375 + ], + "d_h_spectrum_head": [ + 0.015625, + 0.03515625, + 0.0341796875, + 0.023193359375 + ] + }, + { + "graph": "NN", + "width": 3, + "p": "1/2", + "d_max": 48, + "tagged_states": 13, + "tagged_lumps": 5, + "spectrum_states": 645, + "heights_total": 48, + "heights_scored": 48, + "heights_below_floor": 0, + "floor_abs": 8.518145161290321e-22, + "max_rel_diff_d_h": 3.884146214910958e-16, + "max_rel_diff_at_height": 46, + "tail_exact": 4.4770266977653885e-09, + "tail_spectrum": 4.47702675164191e-09, + "tail_scored": true, + "tail_rel_diff": 1.2033996044601412e-08, + "d_h_tagged_head": [ + 0.001953125, + 0.009033203125, + 0.015228271484375, + 0.014057159423828125 + ], + "d_h_spectrum_head": [ + 0.0019531249999999998, + 0.009033203124999998, + 0.015228271484375, + 0.014057159423828123 + ] + }, + { + "graph": "NN", + "width": 4, + "p": "1/2", + "d_max": 48, + "tagged_states": 43, + "tagged_lumps": 10, + "spectrum_states": 11245, + "heights_total": 48, + "heights_scored": 48, + "heights_below_floor": 0, + "floor_abs": 5.806909140463621e-22, + "max_rel_diff_d_h": 3.7894182489564264e-15, + "max_rel_diff_at_height": 15, + "tail_exact": 9.11025051349543e-08, + "tail_spectrum": 9.110250511132349e-08, + "tail_scored": true, + "tail_rel_diff": 2.593871188840601e-10, + "d_h_tagged_head": [ + 0.000244140625, + 0.0022125244140625, + 0.006209373474121094, + 0.007723748683929443 + ], + 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+ "width": 3, + "p": "1/8", + "d_max": 48, + "tagged_states": 13, + "tagged_lumps": 3, + "spectrum_states": 645, + "heights_total": 48, + "heights_scored": 44, + "heights_below_floor": 4, + "floor_abs": 1.012385010033181e-22, + "max_rel_diff_d_h": 3.360560117319889e-16, + "max_rel_diff_at_height": 11, + "tail_exact": 1.693528071692592e-24, + "tail_spectrum": 1.1335996369615003e-24, + "tail_scored": false, + "tail_rel_diff": null, + "d_h_tagged_head": [ + 0.0008765533566474915, + 0.004603617140674032, + 0.0026245273589040607, + 0.0012042995335344586 + ], + "d_h_spectrum_head": [ + 0.0008765533566474915, + 0.004603617140674032, + 0.0026245273589040607, + 0.0012042995335344586 + ] + }, + { + "graph": "matching", + "width": 4, + "p": "1/8", + "d_max": 48, + "tagged_states": 43, + "tagged_lumps": 7, + "spectrum_states": 11245, + "heights_total": 48, + "heights_scored": 48, + "heights_below_floor": 0, + "floor_abs": 3.2061990372466306e-23, + "max_rel_diff_d_h": 5.068003088960758e-15, + "max_rel_diff_at_height": 47, + "tail_exact": 1.8330494632335628e-23, + "tail_spectrum": 2.4983338408130232e-17, + "tail_scored": false, + "tail_rel_diff": null, + "d_h_tagged_head": [ + 8.388889546040446e-05, + 0.0010027139728627787, + 0.0010276090133845778, + 0.000593689890921287 + ], + "d_h_spectrum_head": [ + 8.388889546040443e-05, + 0.0010027139728627785, + 0.0010276090133845774, + 0.0005936898909212868 + ] + }, + { + "graph": "matching", + "width": 5, + "p": "1/8", + "d_max": 48, + "tagged_states": 131, + "tagged_lumps": 15, + "spectrum_states": 52061, + "heights_total": 48, + "heights_scored": 48, + "heights_below_floor": 0, + "floor_abs": 1.0297930696989006e-23, + "max_rel_diff_d_h": 1.4239316136941811e-12, + "max_rel_diff_at_height": 48, + "tail_exact": 3.80287579927456e-23, + "tail_spectrum": 3.8028757992801343e-23, + "tail_scored": true, + "tail_rel_diff": 1.465783924014196e-12, + "d_h_tagged_head": [ + 8.02842944835902e-06, + 0.0001999073542460492, + 0.00033846511264142915, + 0.00024299655833960693 + ], + "d_h_spectrum_head": [ + 8.028429448359029e-06, + 0.00019990735424605603, + 0.0003384651126414531, + 0.0002429965583396311 + ] + }, + { + "graph": "matching", + "width": 6, + "p": "1/8", + "d_max": 48, + "tagged_states": 411, + "tagged_lumps": 33, + "spectrum_states": 668439, + "heights_total": 48, + "heights_scored": 48, + "heights_below_floor": 0, + "floor_abs": 3.384097921186248e-24, + "max_rel_diff_d_h": 2.3586377818236965e-12, + "max_rel_diff_at_height": 48, + "tail_exact": 3.500007410474815e-23, + "tail_spectrum": 3.5000074104833353e-23, + "tail_scored": true, + "tail_rel_diff": 2.434443646157721e-12, + "d_h_tagged_head": [ + 7.683457870499844e-07, + 3.8377629902561116e-05, + 0.00010005165454768037, + 9.16422453095959e-05 + ], + "d_h_spectrum_head": [ + 7.68345787049979e-07, + 3.837762990256322e-05, + 0.00010005165454769263, + 9.164224530961195e-05 + ] + }, + { + "graph": "matching", + "width": 7, + "p": "1/8", + "d_max": 20, + "tagged_states": 1275, + "tagged_lumps": 68, + "spectrum_states": 389391, + "heights_total": 20, + "heights_scored": 20, + "heights_below_floor": 0, + "floor_abs": 1.1267963043798457e-24, + "max_rel_diff_d_h": 2.2708736632168628e-12, + "max_rel_diff_at_height": 20, + "tail_exact": 2.6473442490507712e-11, + "tail_spectrum": 2.647344249057235e-11, + "tail_scored": true, + "tail_rel_diff": 2.4416787219954993e-12, + "d_h_tagged_head": [ + 7.353309290126803e-08, + 7.23106628057412e-06, + 2.761323406379945e-05, + 3.2241241216752266e-05 + ], + "d_h_spectrum_head": [ + 7.353309290126311e-08, + 7.231066280574751e-06, + 2.7613234063806597e-05, + 3.2241241216765385e-05 + ] + } + ] + }, "model_discriminator": { "what": "section 6.3 moment-limit reading, tested on uncensored all-height moments", "target_T": "pi/3 - 1", @@ -921,6 +1866,7 @@ "honesty": [ "This script verifies arithmetic and mutual consistency. It does not verify the tagged automaton's construction, the unique-anchor pathwise theorem, or any asymptotic claim.", "The independent nu_w used in check 1 comes from a different exact engine but shares the same underlying cylinder model; it is an independent implementation, not an independent model.", - "Check 3 fits four widths at most. Model A beating model B on rms is a consistency statement about two one-parameter readings, not evidence for any particular correction exponent." + "Check 3 compares two constructions of the same object, but the tagged side is delivered code imported at run time, so a shared misreading of the span definition would not be caught by it. The two state spaces are structurally unrelated, which is the reason the agreement is meaningful.", + "Check 4 fits four widths at most. Model A beating model B on rms is a consistency statement about two one-parameter readings, not evidence for any particular correction exponent." ] } diff --git a/scripts/tagged_span_crosscheck.py b/scripts/tagged_span_crosscheck.py index f7208f3d..8dc3ca16 100644 --- a/scripts/tagged_span_crosscheck.py +++ b/scripts/tagged_span_crosscheck.py @@ -158,6 +158,93 @@ def check_truncation_bias(tagged: dict, spectrum: dict) -> dict: } +def check_spectrum_vs_tagged_histogram(tagged: dict, spectrum: dict, + floor_rel: float = 1e-20) -> dict: + """Compare the two constructions height by height, not just in their moments. + + The depth-clamped chain (span_spectrum_build.cpp) tracks the ages of every + active component and then projects onto a span histogram. The tagged + resolvent tracks one lineage and never stores an age. If they are both + computing the span of the same object, they must agree on d_h for every + h <= D_MAX, and on the mass sitting above the cutoff. Comparing only the + first two moments would not catch a compensating error; this does. + + Only heights whose value exceeds `floor_rel * nu` are scored. Below that the + committed file stores denormal-scale float64 numbers (d_h ~ 1e-40, tail bins + ~ 1e-45) whose relative difference is meaningless, and they are counted + separately rather than used to characterise agreement. + + Requires importing the delivered tagged module, so it is the one check here + that consumes delivered code rather than a committed artifact. + """ + sys.path.insert(0, str(ROOT / "scripts")) + import tagged_winding_span as T # noqa: E402 + + rows = [] + for run in spectrum["runs"]: + if run["width"] > 7: + continue # width 8 has no committed truncated row to compare against + graph = "matching" if run["matching"] else "NN" + states, trans, source = T.build(run["width"], bool(run["matching"])) + trans, source, _ = T.lump(trans, source) + res = T.moments(trans, source, Fraction(run["p"]), bins=run["d_max"], exact=True) + dh_exact = res["d_h"] + dh_spec = run["d_h_float"] + if len(dh_exact) != len(dh_spec): + rows.append({"graph": graph, "width": run["width"], "p": run["p"], + "error": f"length mismatch {len(dh_exact)} vs {len(dh_spec)}"}) + continue + nu = float(res["nu"]) + cut = floor_rel * nu + worst_h, worst_rel = None, 0.0 + scored = below = 0 + for h, (a, b) in enumerate(zip(dh_exact, dh_spec), start=1): + fa = float(a) + if fa <= cut: + below += 1 + continue + scored += 1 + rel = abs(fa - b) / fa + if rel > worst_rel: + worst_h, worst_rel = h, rel + tail_exact = float(res["tail"]) + tail_spec = run["tail_bin_float"] + tail_scored = tail_exact > cut + tail_rel = (abs(tail_exact - tail_spec) / tail_exact) if tail_scored else None + rows.append({ + "graph": graph, "width": run["width"], "p": run["p"], + "d_max": run["d_max"], + "tagged_states": len(states), "tagged_lumps": len(trans), + "spectrum_states": run["states"], + "heights_total": len(dh_exact), + "heights_scored": scored, "heights_below_floor": below, + "floor_abs": cut, + "max_rel_diff_d_h": (worst_rel if scored else None), + "max_rel_diff_at_height": worst_h, + "tail_exact": tail_exact, "tail_spectrum": tail_spec, + "tail_scored": tail_scored, "tail_rel_diff": tail_rel, + "d_h_tagged_head": [float(x) for x in dh_exact[:4]], + "d_h_spectrum_head": dh_spec[:4], + }) + scored_rows = [r for r in rows if r.get("max_rel_diff_d_h") is not None] + worst = max((r["max_rel_diff_d_h"] for r in scored_rows), default=None) + return { + "what": "depth-clamped chain vs tagged resolvent, height by height", + "constructions": { + "spectrum": "span_spectrum_build.cpp: all component ages, then span histogram", + "tagged": "tagged_winding_span.py: one lineage, no age, no depth cutoff", + }, + "points_compared": len(rows), + "points_scored": len(scored_rows), + "scoring_floor": f"d_h > {floor_rel:g} * nu", + "worst_rel_diff_over_all_heights": worst, + "total_heights_below_floor": sum(r.get("heights_below_floor", 0) for r in rows), + "state_space_ratio_note": "at w=7 the tagged lumping is 71 states against 389391; " + "at w=6, 36 against 668439", + "rows": rows, + } + + def _slope(xs: list[float], ys: list[float]) -> float: n = len(xs) mx, my = sum(xs) / n, sum(ys) / n @@ -235,6 +322,7 @@ def main() -> int: }, "density_containment": check_density_containment(tagged, certified), "truncation_bias": check_truncation_bias(tagged, spectrum), + "histogram_agreement": check_spectrum_vs_tagged_histogram(tagged, spectrum), "model_discriminator": check_model_discriminator(tagged), "honesty": [ "This script verifies arithmetic and mutual consistency. It does not " @@ -243,7 +331,11 @@ def main() -> int: "The independent nu_w used in check 1 comes from a different exact engine " "but shares the same underlying cylinder model; it is an independent " "implementation, not an independent model.", - "Check 3 fits four widths at most. Model A beating model B on rms is a " + "Check 3 compares two constructions of the same object, but the tagged " + "side is delivered code imported at run time, so a shared misreading of " + "the span definition would not be caught by it. The two state spaces are " + "structurally unrelated, which is the reason the agreement is meaningful.", + "Check 4 fits four widths at most. Model A beating model B on rms is a " "consistency statement about two one-parameter readings, not evidence " "for any particular correction exponent.", ], @@ -256,8 +348,11 @@ def main() -> int: print(f"[2] truncation bias: {t['points_compared']} points; " f"{len(t['material_bias_points'])} exceed 1e-6 tail and are materially biased; " f"clean points max CV^2 rel.bias {t['clean_points_max_cv2_rel_bias']:.2e}") + h = report["histogram_agreement"] + print(f"[3] histogram agreement: {h['points_compared']} points, " + f"worst d_h rel.diff over all heights {h['worst_rel_diff_over_all_heights']:.2e}") m = report["model_discriminator"] - print(f"[3] model discriminator: {m['families_favouring_model_A']}/{m['families_tested']} " + print(f"[4] model discriminator: {m['families_favouring_model_A']}/{m['families_tested']} " f"families favour model A; smallest model-B gap at w=8 {m['min_model_B_gap_at_w8']:.4f}") if args.write: