Geometric balance manuscript: sharp full-law criterion for arbitrary integer periods

[LightChainr]

One probability manuscript, with the missing oblique necessity proved

Owner-delegated continuation after the #738 navigation reset. This is not another analysis queue. Five new files consolidate the probability part of #735, incorporate #736, and finish the arbitrary-orientation winding-corridor lemma that those notes left open. Base main d31fa5fe; head 758800f9. No existing data, freeze, research code, navigation, or old PR is overwritten or merged.

Main result

For independent NN square-site percolation on any sequence of honest integer-period tori with N=[Z²:Lambda] -> infinity and genuine shortest period ell,

all fixed interior quantiles Q_n(u) -> p_c
  iff convergence is uniform on compact u-intervals
  iff both rank birth times concentrate at p_c
  iff F_n=E[r]/2 tends to the threshold step
  iff log(N_n)/ell_n -> 0.

The root theorem consolidated from #735 needs only ell -> infinity, with no aspect, area, or shear restriction. Necessity of that weaker condition for the median is NOT claimed. Rank, directional wrapping, F, and conditional odds H are kept distinct.

New proof, rather than a new width census

1. Approximate the actual shortest integer period by a bounded-step staircase. Axis-aligned 2s--3s crossing rectangles and two crossing directions in every hub make a connected ring. The last hub is explicitly the translate of the first, so the lift joins z to z+u. No rotation of the NN interaction, and no ambient-primitivity assumption.
2. Critical square-site RSW plus finite-product continuity give, for every eta>0, a FIXED s_eta and FIXED p_eta<p_c with ring probability >=exp(-eta*ell), uniformly in orientation. Harris is used within the overlapping ring, not independence.
3. The lifted support lies in a 4s tube. Its difference set has at most 64sell sites after projection. A maximal disjoint-translate packing in Z²/Lambda has at least N/(64sell) members. This avoids a separate tilted-coordinate packing hypothesis.
4. Independence of those disjoint site supports gives

P0(p_eta) <= exp[-N/(64s_eta*ell) * exp(-eta*ell)].

This proves full-law necessity. Sufficiency is the existing one-arm union bound. A forced-path version also proves endpoint splitting when log N/ell -> infinity on arbitrary period shapes, not only axial strips.

Single readable unit

• docs/manuscripts/geometric-balance/manuscript.md: complete model, finite duality explanation, imported inputs, root proof, new corridor/packing proof, six-way equivalence, explicit oblique examples, and closest-source scope comparison.
• docs/manuscripts/geometric-balance/README.md: entrypoint and two commands.
• One standalone stdlib script, five mathematical tests, and one deterministic result JSON.

The external critical inputs are named explicitly: site sharpness (including DC--T's site adaptation), critical square-site RSW (Zeng Theorem 1.1), and the amenable matching relation (Grimmett--Li Theorem 5.5). Mertens--Ziff/DKS establish the prior context, not an originality certificate. No Jordan, high-order local-source, or sampler work is a dependency.

Executed scope

Five local tests passed. Three tiny injecting-box tori exhaust 135,168 configurations, with zero ring-without-winding failures; six larger reduced oblique bases check geometry, including ambient-nonprimitive shortest vectors; finite-group packing controls pass. The result regenerates exactly. Tiny controls test gluing, not the asymptotic constants or RSW.

Full repository CI has not been run. This is an author-supplied proof, not a newly independent publication acceptance; literature novelty is not certified. No Monte Carlo, GPU, external compute, new numerical p_c, L^-4 law, continuum identity, or full-law prediction at finite positive log N/ell is asserted.

Return channel: #735, with #613/#650 as the existing coordination. No new issue or automatic parallel verification round requested.

[LightChainr]

Completed continuation of this SAME manuscript; no new issue, width ladder, or acquisition request. The owner handoff contains four additive files under the existing geometric-balance manuscript, a standalone width-two control script, its tests, and one result JSON. These files are not yet committed to this branch.

The next analytical reduction is now explicit: two sharp births, not an unrestricted broad law. Let f1=Pr(r>=1), f2=Pr(r=2), and let a_N,b_N be their medians. Each event is monotone and invariant under the transitive translation action on the N independent sites. Friedgut–Kalai (1996), in the publisher's theorem statement and DKS Theorem 6, therefore gives a universal C such that

Pr(|T_j-theta_j|>x) <= exp[-x log(N)/C], theta=(a_N,b_N).

Thus E|T_j-theta_j|^k <= k! (C/log N)^k, and
W1(Law(T), (delta_a_N+delta_b_N)/2) <= C/log N.
Every subsequential unscaled law has at most two equally weighted atoms. This is an explicit COROLLARY of a published theorem, not a new general sharp-threshold result. It requires neither RSW nor a bound on aspect; it does not synchronize the two centers, assume independent births, give a calibrated small-N constant, or close a standardized finite-size profile.

Existing-data link: G_N=integral_0^1 P1(p)dp=E(K2-K1)/(N+1) exactly, and |G_N-(b_N-a_N)|<=2C/log N. Moreover Delta_N <= IQR(F_N) <= Delta_N+2C log(2)/log N, and a_N<=q_N<=b_N. Combined with this manuscript's geometry theorems, Delta_N, G_N, and IQR vanish iff log N/ell->0. This combination is not an independent validation of the corridor proof. The median is not the midpoint of the two centers: the limiting plateau forgets the rare-sector odds selecting q_N.

A completed quantitative start for the remaining regime: on axial w-by-m tori with w->infinity and log(m)/w->d in (0,infinity), simple nonbacktracking cycle counts and independent full rows show that every subsequential center pair obeys
e^(-d)/3 <= a <= min(pc,e^(-d)),
max(pc,1-e^(-d)) <= b <= 1-e^(-d)/7.
At d=log 4 these imply a in [1/12,1/4], b in [3/4,27/28]. They are broad rigorous brackets, not exact center predictions or measured inverse correlation lengths. Existence/uniqueness of a limiting pair for fixed d is not yet established here.

Allocation: the one next question is to locate these two centers in exponentially elongated geometry via microscopic winding costs. Keep the work in #739. No more generic source-order/Jordan examples are needed for this question. Correctly typed old birth-rank archives already contain G_N; directional wrapping times cannot be substituted.

Executed: 336 independent physical configurations at 2x2/2x3/2x4, all 720 orders on 2x3 (exact gap 3/14, covariance 1123/58800), seven local tests; existing #705 width-two formulas evaluated at m=2,4,8,16,32,128. Numerical roots and integrals at 80 digits, m=4 and 128 recomputed at 110 digits and agreeing at all 24 reported digits. No fitted sharpness constant, no Monte Carlo, no new pc, no novelty claim. Full repository CI has not been run.

Sources read: https://www.ams.org/journals/proc/1996-124-10/S0002-9939-96-03732-X/ (publisher theorem statement; original PDF fetch failed), https://arxiv.org/html/2011.11903v4 (Theorem 6 and the distinction between site transitivity and homology point-group symmetry).

[LightChainr]

Completed continuation of the SAME geometric-balance manuscript, in four additive files in the current owner handoff (not yet committed to this branch). This addresses the previous comment's missing axial centre locations; no new issue or compute request.

Result. For w->infinity, m>=w and log(m)/w->d in (0,infinity), let kappa_G(p)=lim_n -log Pr_p^G(0<->n e1)/n for independent SITE percolation on G=NN or matching NN+NNN. The two rank births converge in probability to the unique values

a(d)=kappa_NN^{-1}(d), b(d)=1-kappa_matching^{-1}(d).

They satisfy 0<a(d)<pc<b(d)<1; the whole sequence, not just a selected subsequence, has birth mixture (delta_a+delta_b)/2. Every fixed nonmedian quantile goes to the corresponding endpoint. The separate #735/#739 root theorem still places the finite mixture median at pc, not necessarily (a+b)/2.

The new microscopic bridge is proved explicitly. Normalize s_p(x)=Pr(0<->x)/p. Conditional Harris at a common occupied endpoint gives s(x+y)>=s(x)s(y); Fekete and reflection give tau_p(x,y)<=p exp[-kappa max(|x|,|y|)]. A lifted nonzero-winding walk, stopped when its first coordinate RANGE reaches w-1, lies in a single injecting w-by-w vertex square. Only its planar edges are used. Therefore

f_G(w,m;p)=Pr(r_G>0) <= 2p m w^3 exp[-(w-1)kappa_G(p)].

Conversely finite-box exhaustion chooses one fixed conditional connection seed close to kappa. Concatenate its translates and explicitly close the final periodic seam. Conditional Harris, not independence, gives a winding ring of probability >=exp[-(kappa+epsilon)w] in a fixed D-row band. Disjoint bands give

1-f_G <= exp[-floor(m/D) exp[-(kappa+epsilon)w]].

Hence log(f_G)/w -> -(kappa_G(p)-d)_+, and f_G->1 when kappa<d. This does not assume an Ornstein-Zernike prefactor or silently transfer a bond crossing theorem to sites.

Inversion is not an unproved regularity assumption. Subcritical continuity follows from finite-seed upper semicontinuity and an explicit finite-cluster likelihood comparison plus the site CLUSTER-VOLUME exponential tail. Strict decrease follows by combining the preceding rare-event exponent with the Friedgut-Kalai transitive threshold theorem: a plateau of kappa at two parameters would force f at the upper parameter simultaneously to zero and one on a suitably chosen exponential torus sequence. Susceptibility divergence and the reflection bound give kappa->0 at pc; path counting gives kappa->infinity at p->0. The order of these steps avoids circularity.

Small quantitative gain. Conditional 3x3 seed probabilities are exactly
q_NN=p^2+2p^4-2p^5-p^7+p^8,
q_matching=3p^2-3p^3+p^4.
For d=log4, their q=1/16 roots are alpha=0.239805566880062101... and beta=0.156394361442176291.... Thus
a in [1/12,alpha], b in [1-beta,27/28].
These are certified one-sided bounds, NOT numerical estimates of the actual infinite centres or of kappa.

Executed: 140,288 independent graph/configuration checks on 3x3,3x4,4x4 across BOTH adjacencies; 91,668 lifted first-span witnesses checked against a separate planar-box BFS; exact seed polynomial/root enclosures, conditional-Harris ring bounds and finite-cluster likelihood controls; nine local tests. One deterministic JSON regenerated identically. Full repository CI not run. No Monte Carlo, pc decimal, fitted exponent, all-oblique centre formula, Gumbel law or finite-width shift coefficient is claimed.

Files: docs/manuscripts/geometric-balance/exponential-birth-centres.md; scripts/winding_rate_centres.py; tests/test_winding_rate_centres.py; results/geometric-consistency/winding-rate-centres.json.

Primary inputs read: Antunovic-Veselic https://arxiv.org/html/0707.1089v3 (Theorems 2-3, Proposition 5, site section); Friedgut-Kalai as stated in DKS https://arxiv.org/html/2011.11903v4 (Theorem 6). Closest mechanism: Damron-Lam https://arxiv.org/html/2502.18235v1 (1.1.2 and Section 2, BOND wedges/rectangles). The entropy-versus-connection-cost mechanism has prior art; no novelty certification or merge request.

[LightChainr]

Completed a focused continuation of the SAME probability manuscript in four additive files in the owner handoff. This is an author-supplied proof, not independent publication acceptance, and the files are not yet committed to this branch. No new issue, width ladder, Monte Carlo, or merge request.

1. A locally defined once-per-COMPONENT winding intensity nu_w^G(p) on the infinite cylinder replaces an unspecified prefactor. Anchor each full winding component at its lowest row and one tie-broken site; do not count paths or every winding vertex. A window of H=w^2 rows with two guard rows decides the truncated anchor. Uniform subcritical strip estimates give nu_w-nu_{w,H} <= C w exp(-cH), and the previous finite-seed/first-span proof gives -log(nu_w)/w -> kappa_G(p).

2. Local dependence + BK on DISJOINT INCREASING WINDING WITNESSES gives Poisson process approximation near each centre, uniformly on compact parameter intervals with 2 inf kappa>d. In particular Pr(r_G=0)=exp[-m nu_w^G(p)]+o(1). The anchors themselves are nonmonotone and can have positive short-range covariance because they share closed guard sites; the proof does not apply BK directly to them. On the SAME uniform labels, black winding at the lower window and white matching winding at the upper window have an independent two-type Poisson limit. Opposite Harris is applied to the enclosing monotone colour events, not to anchors.

3. Stronger than a conditional intensity-clock statement: at all d outside an at-most-countable possible exceptional set, the actual finite-median-centred 1/w fluctuations are jointly independent Gumbels. Let a_m,b_m be the true marginal medians, a=kappa_4^{-1}(d), c=kappa_8^{-1}(d), and v4=-kappa_4'(a), v8=-kappa_8'(c). Then

Pr(v4w(T1-a_m)<=x) -> 1-2^(-exp(x)),
Pr(v8w(T2-b_m)<=y) -> 2^(-exp(-y)),

with joint factorization. No Ornstein-Zernike prefactor is assumed for this statement. The extra proof is a uniform CYLINDER cluster-volume tail from coarse blocks, then the component activity identity nu=sum_C p^|C|(1-p)^|boundary C|. It gives (log nu)''=Var(score)-E[n/p^2+b/(1-p)^2] and E(n+b)=O(w), hence uniform semiconvexity of log(nu_{w,w^2})/w. Moving-point derivative convergence at mass differentiability points follows from convex secants. The earlier FK argument sharpens to kappa(p)-kappa(q)>=(q-p)kappa(q)/rho, so the limiting slopes are positive. Mass corners form at most a countable set. At such possible corners the note proves a convex-log-intensity subsequential classification, not an unsupported unique affine law.

4. With h0=EulerGamma+log(log 2), regular-d means and variances are explicit: E T1=a_m-h0/(v4w)+o(1/w), E T2=b_m+h0/(v8w)+o(1/w), Var(Tj)=pi^2/(6vj^2w^2)+o(w^-2). Cov(T1,T2)=o(w^-2) holds for every fixed d>0 by joint factorization and FK moment bounds. A scale-free archive-facing consequence at regular d is

[E(T2-T1)-IQR(mixture)]/[IQR(T1)+IQR(T2)]
-> [EulerGamma+log(log 2)]/log[log(4)/log(4/3)]
= 0.133989344651100063543034...

No numerical mass slope is needed for this ratio. It is NOT a prediction for fixed width or fixed aspect (d=0).

5. The boundary CDF is genuinely not determined by d alone. At fixed p0<pc and d=kappa(p0), m_w=floor(t/nu_w(p0)) still has log(m_w)/w->d, but Pr_{p0}(r_G>0)->1-exp(-t). Endpoints 0/1 follow with subexponential t_w. Thus the same actual site model realizes all boundary probabilities with the same exponential geometry rate.

6. The remaining displacement FROM infinite a(d),b(d) is kept separate: IF nu_w(p)=A(p)w^(-beta)e^(-wkappa(p))(1+o(1)) locally uniformly and log m=dw+gamma log w+c0+o(1), the first-birth u-quantile shift is [(beta-gamma)log w-c0-log A(a)+log(-log(1-u))]/(v4w)+o(1/w). Neither A,beta nor existence of that prefactor expansion has been proved here. A planar two-point w^(-1/2) cannot simply be copied into periodic component counting. No all-oblique fluctuation formula or d->0 uniform crossover is claimed.

Executed: 75,776 physical graph/configuration comparisons on both adjacencies; 6,561 exact shared-label categorical assignments; 8,192 fixed full/empty-row masks; 18 exact component-activity/full-window derivative comparisons; 16 local tests. Patch applied in an isolated minimal Git tree, all four files byte-identical, deterministic JSON regenerated byte-identically. Small H=1/2 controls explicitly retain void/rank discrepancies and nonzero finite covariance. They do not prove the asymptotic theorem. Full repository CI not run.

Files: docs/manuscripts/geometric-balance/poisson-birth-windows.md; scripts/winding_poisson_controls.py; tests/test_winding_poisson_controls.py; results/geometric-consistency/winding-poisson-controls.json.

Primary tools read: Arratia-Goldstein-Gordon (1989), Theorem 2, author PDF printed pp10-11 rendered, https://dornsife.usc.edu/larry-goldstein/wp-content/uploads/sites/221/2023/06/AGG-1.pdf (TV convention converted); Antunovic-Veselic https://arxiv.org/html/0707.1089v3 (site Harris/BK/sharpness); FK as stated in DKS Theorem 6 https://arxiv.org/html/2011.11903v4. Declumping and Poisson/extreme-value methods have prior art; no novelty certification is asserted.

[LightChainr]

Two handoffs committed to this branch

Both owner handoffs announced above are now on analysis/geometric-balance-manuscript-20260913.
Pushed as two additive commits on the existing branch; no new PR, no issue, no merge, no
docs/STATUS.md edit, no re-run of any old production.

2f63cbb  Add the two exponential-rectangle birth centres to the geometric-balance manuscript
162db6e  Add winding-cluster intensities, Poisson windows, and the birth fluctuations

758800f..162db6e. The PR is now 13 files, +4872. All eight added files are added; the five
pre-existing files from the original delivery are untouched.

file	commit
docs/manuscripts/geometric-balance/exponential-birth-centres.md	2f63cbb
scripts/winding_rate_centres.py	2f63cbb
tests/test_winding_rate_centres.py	2f63cbb
results/geometric-consistency/winding-rate-centres.json	2f63cbb
docs/manuscripts/geometric-balance/poisson-birth-windows.md	162db6e
scripts/winding_poisson_controls.py	162db6e
tests/test_winding_poisson_controls.py	162db6e
results/geometric-consistency/winding-poisson-controls.json	162db6e

Checks executed on the real repository tree

Not the packaged notes repeated — re-run here after git apply on this branch.

1. git apply --check then git apply for both patches, in order; both clean, no conflict,
   disjoint path sets.
2. All eight applied files byte-identical (sha256) to the packaged repo/ copies.
3. python -m unittest discover -s tests -p 'test_winding_rate_centres.py' — 9 tests, OK.
   python -m unittest discover -s tests -p 'test_winding_poisson_controls.py' — 16 tests, OK.
   25 total, in-tree, ~0.08 s.
4. Both stdlib scripts re-run with --output and their JSON regenerated byte-identically
   (sha256 25017e97… for the rate centres, f0d05f88… for the Poisson controls). The
   overwrite refusal fires as documented.

What this does not establish

Author-supplied proofs, not independent referee acceptance, and not a priority claim.

• No numerical value for either centre; the [1/12, α] / [1−β, 27/28] brackets at d = log 4
  are certified one-sided bounds, not estimates.
• The displacement from the infinite a(d), b(d) — including any log(w)/w term — still needs a
  subexponential prefactor for ν_w, which is not proved. The notes state the expansion
  conditional on that prefactor and do not transplant a planar w^{-1/2} onto periodic
  component counts.
• No Gumbel law claim at the possible countable exceptional d; only a convex-log-intensity
  subsequential classification there. No all-oblique formula, no d → 0 uniform crossover,
  no Monte Carlo, no new numerical p_c.
• The H = 1/2 controls retain void/rank discrepancies by construction and do not prove the
  H = w² asymptotic. Finite checks are not a proof of the imported RSW / sharpness / Harris /
  Friedgut–Kalai / Arratia–Goldstein–Gordon inputs.
• Full repository CI has not been run for either commit. It was not run for the previous
  commit on this branch either, and this PR is still based on main.

One handoff is still missing from this branch

Comment 5650436953 (this thread, 02:53 UTC) also announces a delivery — four additive files under
the manuscript directory, a standalone width-two control script, its tests, and one result JSON —
and states they are "not yet committed to this branch". Those files are not on
analysis/geometric-balance-manuscript-20260913, and the package for them is not available here.
The two commits above do not depend on it at file level, but the branch is therefore still missing
one announced set of files. Flagging rather than reconstructing: rebuilding it from the comment
text would produce new authoring, not the delivered proof.

No merge, no close, no STATUS edit. The manuscript bundle remains unmerged and based on main.