Aspect-uniform matching roots; P398 is periodic IC TL (do not merge)

notes/aspect-uniform-balance-consistency-20260912.md

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+# Rectangular homological balance is consistent without an aspect-ratio bound
+
+2026-09-12. Direct continuation of #276/#613/#716. This is a mathematical
+consequence of established subcritical sharpness and the repository's digital
+Alexander identity. It is NOT a new numerical value of p_c, a claimed novel
+sharpness theorem, or an inference from the width-2/3/4 spectra.
+
+## 1. The distinction and theorem
+
+Let G_(w,m) be the square-site NN graph on the axis torus with periods (w,0),
+(0,m), integers m>=w>=2. Let r be its ambient rational H1 rank, P_j=Pr_p(r=j),
+M=P_2-P_0 and F=(1+M)/2. Parallel lifted edges at period two are retained.
+The matching complement graph is the NN+NNN graph on the same sites.
+
+The following established inputs are used:
+
+1. For site percolation on each infinite graph, every fixed subcritical p has
+   an exponential one-arm bound a_R(p)<=C(p) exp[-c(p)R]. Both graphs are
+   locally finite transitive finite-range graphs. See Duminil-Copin--Tassion,
+   arXiv:1502.03050v3, Theorem 1.1(3) and the explicit site adaptation in §1.2.
+   The main theorem there is printed in bond language; §1.2 is essential here.
+2. p_c(NN)+p_c(NN+NNN)=1. Grimmett--Li, RSA 65 (2024), 832--856,
+   DOI 10.1002/rsa.21226, introduction Eqs. (1.1),(1.3) and the amenable
+   p_u=p_c discussion, supply the matching-pair relation with its provenance.
+3. The repository's honest-torus digital-Alexander identity is configurationwise
+   r_NN(omega)+r_matching(omega^c)=2. This is not inferred from the new census.
+
+**Theorem.** The unique zero p_(w,m) of M satisfies
+
+    lim_(w->infinity) sup_(m>=w) |p_(w,m)-p_c(NN)| = 0.                 (1)
+
+There is NO upper bound on m/w, and NO condition w/log(wm)->infinity in (1).
+This improves the *root* conclusion on axis rectangles, not the full threshold-
+distribution conclusion of #613. It does not extend the geometry to arbitrary
+skew quotients without an additional strip construction.
+
+In fact define the conditional rank profile
+
+    H_(w,m)(p)=P_2(p)/(P_0(p)+P_2(p)) = Pr(r=2 | r != 1).             (2)
+
+This is strictly increasing from zero to one. Every compact set of its interior
+quantiles converges uniformly to p_c, uniformly over all m>=w. H is NOT the
+birth-time mixture F, and it is not proposed as a new independent observation.
+
+## 2. A local event with an embedded finite support
+
+Fix R>=1 with 2R+2<=w. For either infinite graph, let a_R be the unconditional
+probability that the occupied origin is connected, inside the sup-norm box
+[-R,R]^2, to its boundary. The root being occupied is included.
+
+For each torus vertex v, define A_v by the corresponding local lifted box. Its
+vertices are distinct and its relevant edges coincide with the infinite graph:
+2R+2<=w<=m rules out identification of the box and spurious boundary adjacency.
+Therefore Pr(A_v)=a_R. If an occupied cycle has a nonzero winding class, its
+lift cannot remain in that box. The initial segment up to first exit witnesses
+A_v for a vertex on the cycle. In particular
+
+    intersection_v A_v^c  is contained in {r=0}.                    (3)
+
+All events A_v^c are decreasing. Harris positive association for product sites
+gives, without pretending these overlapping boxes are independent,
+
+    P_0 >= Pr(intersection_v A_v^c) >= (1-a_R)^(wm).                 (4)
+
+For completeness, association on finitely many independent Bernoulli variables
+follows by induction: condition on the last variable; the conditional covariance
+is nonnegative by induction, and the covariance of the two conditional means is
+nonnegative because they are monotone in the same direction. Products of
+nonnegative decreasing indicators remain decreasing, so iterate this result.
+
+This exponential LOWER bound is the step absent from the elementary union-bound
+proof of full-law concentration. It remains meaningful even if wm a_R is huge.
+
+## 3. Disjoint slab crossings give the other exponential rate
+
+Take L=floor(m/(R+1)) disjoint slabs, with vertex rows
+
+    y=k(R+1),...,k(R+1)+R,  k=0,...,L-1.
+
+Let B_k be an occupied bottom-to-top crossing entirely in slab k, with the
+horizontal coordinate periodic. The slabs share no vertex, so their B_k events
+are independent under site product measure. Inter-slab edges are not part of
+these events.
+
+If r=2, some occupied cycle has nonzero vertical winding. A vertically lifted
+copy of it traverses each chosen slab. Between the last visit to the bottom
+before a first visit to the top lies a path contained in that slab. The NN and
+matching steps both change y by at most one. Thus
+
+    {r=2} is contained in intersection_(k=0)^(L-1) B_k.             (5)
+
+A slab crossing starts at one of w bottom vertices. Before a lift reaches the
+top row at distance R, it reaches the boundary of an embedded radius-R box
+about that starting vertex (possibly through a horizontal side first). This
+is precisely an A_v event, so a union bound gives Pr(B_k)<=w a_R. No assumed
+independence between possible starting vertices is used.
+Consequently, whenever w a_R<1,
+
+    P_2 <= (w a_R)^floor(m/(R+1)).                                  (6)
+
+Since m>=w>=2R+2, floor(m/(R+1))>=m/[2(R+1)]. Combining (4),(6),
+
+    P_2/P_0 <= exp[-m gamma_w],
+    gamma_w = -log(w a_R)/[2(R+1)] + w log(1-a_R).                   (7)
+
+The second term is NEGATIVE. It must not be dropped in a finite sign test.
+Given a certified upper bound alpha>=a_R, one fully rational sufficient test is
+
+    w alpha < (1-alpha)^[2w(R+1)].                                 (8)
+
+The executable checks compare these rationals exactly. A positive result is a
+finite-width sign certificate for all m>=w, conditional on the validity of the
+supplied arm upper bound. It does not itself determine a new numerical p_c.
+
+## 4. Proof of the aspect-uniform statement
+
+For fixed p<p_c(NN), choose R=floor(w/8) for w>=8. Exponential decay gives
+
+    -log(w a_R)/[2(R+1)] >= [c R-log(Cw)]/[2(R+1)] -> c/2,
+    w log(1-a_R) -> 0.
+
+Hence gamma_w>=c/4>0 for all sufficiently large w. Equation (7) proves an
+exponentially small ratio P_2/P_0 uniformly in m>=w, even if both probabilities
+are themselves extremely small. In particular M(p)<0.
+
+For fixed p>p_c(NN), set p*=1-p<p_c(NN+NNN) and apply exactly the same argument
+to the matching graph. The configurationwise rank-sum identity interchanges
+P_0 and P_2 under complementation. Thus P_0/P_2<=exp[-c* m/4] and M(p)>0.
+This uses actual site sharpness on the degree-eight graph, not a bond result
+or a planar-drawing assertion about its crossing diagonals.
+
+Finite rank is increasing under occupied-set inclusion and is nonconstant. On
+some empty-to-full chain an insertion changes the relevant increasing event;
+that pivotal configuration has positive product probability at every interior
+p. The finite Russo derivative is therefore strictly positive for Pr(r>0)
+and for Pr(r=2). Thus P_0'<0, P_2'>0, and M'>0 on (0,1). Endpoints are -1,+1.
+The unique M zero lies between p_c-epsilon and p_c+epsilon for all sufficiently
+large w and EVERY m>=w. This proves (1).
+
+Moreover, H<=P_2/P_0 below p_c and 1-H<=P_0/P_2 above p_c. The same inequalities
+uniformly trap all H quantiles in [delta,1-delta] for fixed delta>0. For a fixed
+width the result does not assert that an inner m-limit of the matching roots
+exists or equals a specific pTL eigenvalue crossing. If such limits exist, or
+one chooses any accumulation point of the actual finite roots at each width,
+all those choices converge to p_c as width tends to infinity.
+
+## 5. Why this does not contradict the thin-torus theorem
+
+Take w_j=j and m_j=ceil(exp(j^2)). The previous full/empty-row bounds imply
+for every fixed p in (0,1): r->1, F(p)->1/2, and the equal mixture of the two
+birth-time laws tends to (delta_0+delta_1)/2. In the SAME sequence, (1) gives
+
+    p_(w_j,m_j)->p_c,
+    H_(w_j,m_j)(p)->1{p>p_c},  p!=p_c.                              (9)
+
+Thus the physical matching median can consistently target p_c while most of
+its threshold mixture escapes to the endpoints. Sign/order of two rare sectors
+is not equivalent to absolute concentration of the full law.
+#613's original full-law theorem remains valid as stated. This result neither
+shows its geometric condition necessary nor supplies a sharp condition for the
+entire birth distribution. It supplies a less restrictive root theorem.
+
+## 6. Robustness to an intrinsic source and information accounting
+
+For a source s independent of p, zero tilted mean of X=r-1 is equivalent to
+
+    ell(p)+2s=0,  ell=log(P_2/P_0).                                 (10)
+
+For any rectangular sequence w->infinity,m>=w, if |s|/m->0 then the exponential
+sign margins on either side of p_c dominate 2s. Its source-balanced root also
+converges to p_c. More generally H levels with |logit(u)|=o(m) obey the same
+trapping along that sequence. This is an asymptotic statement, not permission
+to change a frozen source/observer contract to get a preferred score.
+
+Conditioning is not a free precision gain. Write E=P_0+P_2. For a single
+independent rank snapshot the Fisher information for p is exactly
+
+    I_rank = (E')^2/[E(1-E)] + E (H')^2/[H(1-H)].                   (11)
+
+Derive it directly from P_0=E(1-H), P_2=EH, P_1=1-E; the cross terms cancel.
+The second term is E times the information in a conditionally retained sample.
+At balance, M'=2E H', and that term equals (M')^2/E. A sharper-looking conditional
+CDF does not remove the cost of seeing r!=1. This identity is for independent
+one-p snapshots; it is not a lower bound for whole-permutation, conditional,
+importance-sampling or exact-transfer estimators.
+
+## 7. Executed checks, and scope
+
+`scripts/rectangular_rank_odds.py` verifies the local-arm implication and each
+required slab crossing on all 65,536 configurations of a 4x4 torus for BOTH NN
+and NN+NNN graphs: 131,072 graph/configuration evaluations. Complementary rank
+sum is checked on all 65,536 pairings. Exact probability versions of (4),(6),
+and (11) are checked at p=1/10,1/3,1/2. Another 236 fixed structural graph cases
+check larger embedded radii; they are not a size census or Monte Carlo run.
+Three sign certificates use the elementary simple-path upper bound
+
+    a_R <= d (d-1)^(R-1) p^(R+1),
+
+at (d,p,w,R)=(4,1/10,16,2),(8,1/20,16,2),(4,1/4,256,32).
+Their approximate positive gamma margins are .0818820,.2524822,.0635379;
+the signs are checked with rationals, not these decimals.
+
+These checks do not prove an asymptotic theorem by extrapolation. Sections 2--4
+do the mathematical work. No quantitative rate in w or certified numerical
+interval for p_c is obtained because near-critical arm constants are not supplied.
+No all-width spectral representation, CFT exponent, or ordinary/Jordan model
+identification is needed. Novelty relative to published homological-estimator
+proofs has not been established.
+
+Sources (primary text sections read):
+- https://arxiv.org/html/1502.03050v3, Thm 1.1(3), §1.2 site adaptation.
+- https://onlinelibrary.wiley.com/doi/10.1002/rsa.21226, Eqs. (1.1),(1.3), amenable case.
+- Repository #276/#613 and #702 digital-Alexander scope; #716 supplies the
+  separate thin-geometry full-law contrast, not this relative-probability proof.