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.

[LightChainr]

Two more handoffs committed, including the one that was missing

162db6e..419791e
  f199098  Reduce the birth law to two sharp centres, with the existing-data interface
  419791e  Add the parallel consolidation and move the roadmap off its missing-lemma text

This branch is now 22 files, +7085/-26. One file is modified (docs/ROADMAP.md); every
other entry is added.

f199098 — the handoff announced at 02:53 that was still absent

This closes the gap flagged in the previous comment. Four files, all additive:

docs/manuscripts/geometric-balance/two-birth-reduction.md · scripts/two_birth_reduction.py ·
tests/test_two_birth_reduction.py · results/geometric-consistency/two-birth-reduction.json

Note for reproduction: this one requires mpmath, unlike the other three controls here,
which are standard-library only.

419791e — a second, independent write-up of the same theorem

docs/manuscripts/geometric-consistency/README.md plus its script, test and control JSON, and a
docs/ROADMAP.md edit. This was prepared in a session with no repository write access and arrived
as a package, so it is being adopted alongside the existing consolidation rather than replacing
it. Both texts prove the same statement and cite #736 at 64d809b4404f80ff3f9adf9713337cc76008e92d.

Keeping two write-ups is a deliberate allocation choice. What the geometric-consistency text
carries that geometric-balance/manuscript.md does not:

1. Damron–Lam, Section 2, compared explicitly as the closest retrieved quantitative mechanism
   for tall thin rectangles, with the pointer onward to Grimmett's 1981 sponge-dimension work.
   [P1｜结构底座] 周期根与全分布几何：只处理具体证明缺口，不再以成稿驱动算力 #613's roadmap asks for a strip-percolation prior-art comparison; the balance write-up states
   only that elongated-strip results "may package related rare-opportunity arguments in different
   language" without naming one.
2. Bounded-ell necessity by greedy packing — a k-site support meets at most k^2 of its
   translates, so at least floor(N/k^2) disjoint translated supports exist and occupying any one
   costs p^k — where the balance text uses an exponential union bound
   P_0 <= exp[-N/(2 ell^2) * p^{sqrt 2 ell}]. Same mechanism, different bookkeeping.
3. Ten sections including an explicit "what lies outside the result", and a nearest-source table
   with hypotheses and limit orders separated.

The docs/ROADMAP.md change is not cosmetic: the paragraph it replaces was written before #739 and
still described the uniform oblique-corridor lemma as A possible strengthening to be proved or
recorded as missing. Its base blob f6868c5c98a8a715b1d06154848f6f0ca7b31910 matches the
pre-patch file exactly.

Checks executed here

Both packages: git apply --check clean, every applied file byte-identical to the packaged copy
(ROADMAP included), and each deterministic JSON regenerates byte-identically.

handoff	tests	control size	regeneration
two_birth_reduction	7 ok (0.28 s)	336 configurations, all 720 orders on 2×3, 80 digits, endpoints at 110 digits	byte-identical
geometric_completion	6 ok (1.10 s)	8 period cases, 12,990 crossing-event checks	byte-identical

Cumulative for the four commits on this branch: 47 local tests across five test files, all passing
in this repository tree.

What none of this establishes

No new numerical p_c, no near-critical rate, no exponent, no measured inverse correlation length,
no uniqueness of a limiting centre pair at fixed d, no external peer review of any text here.
The oblique necessity is an author-supplied proof in two independent write-ups; agreement between
them is not independent replication of the imported RSW, sharpness, Harris, Friedgut–Kalai or
Arratia–Goldstein–Gordon inputs.

Full repository CI has not been run for any commit on this branch, and this PR is still based on
main. No merge, no close, no docs/STATUS.md edit.

[LightChainr]

Completed continuation of the SAME geometric-balance paper. Five additive files are in the owner handoff, not committed to this branch. Two bounded external tasks are now open: #740 (the actual SITE-cluster sewing/prefactor theorem and targeted primary retrieval) and #741 (six stationary density values, one mass-cancelling contrast). No merge, new mechanism programme, or main-branch edit.

Actual site component intensity is now computable. A one-frontier lifted-gain transfer stores connectivity and a winding flag per active component, issuing a reward ONLY when a complete winding component is permanently retired. Widths 2/3/4 close at 6/14/38 states for BOTH NN and matching graphs. All-p homogeneous row-weight/reward lumping gives 3/4/7 states; this is stochastic reward-law equivalence, not pathwise equality for an unchanged mask word. Exact stationary solves give all-p rational nu_w. For NN:

nu_2=p²(1-p)²(p²+p+1)/(p²-p+1),
nu_3=p³(1-p)³(p⁶+p³+2p²+2p+1)/(p⁶-3p⁵+3p⁴+p³-p²-p+1).

At p=1/2, nu for w=2/3/4 is 7/48,169/1984,323849/5576960. The w=2 count-variance rate is 343/6912, so its Fano ratio is 49/144, not 1. Complete-component density is NOT the fixed-width no-winding void rate -log(1-p²). The latter must not be substituted in #741.

Exact crossover constraint, based on prior topology. Mertens–Ziff 2016 §II already gives equal single/spiral component counts and unique cross wrapping. Consequently W4(omega)-W8(omega^c)=r4-1 on a torus, giving nu4_w(p)=nu8_w(1-p) and joint count pressure psi_joint(s,t)=psi4(s+t). At the SAME p, (W4,W8)=(0,0) is impossible, so its TV distance from independent Poisson(lambda4),Poisson(lambda8) is at least exp[-lambda4-lambda8]. This does not contradict the earlier separate-parameter two-window result; it prohibits carrying that independence into a common critical window unchanged. No novelty claim for the finite topology.

The Gaussian coefficient part is done, the SITE sewing is not. For an explicit finite-support positive-length, symmetric transverse renewal kernel a(x,j), normalized by R with A(R,1)=1, let D=E(j²)/E(x). The closed-loop coefficient L_w=w[z^w y^0]{-log(1-A(z,y))} has asymptotic exp[-w log R]/sqrt(2piDw)(1+O(1/w)). Fourier inversion plus the simple pole proves it; the exact central-trinomial control reproduces it. The w/n counting factor is essential. The missing mapping to actual site components, including boundary weights and cut multiplicity, is #740; beta=1/2 is NOT promoted to a site theorem.

A precision correction to the preceding CONDITIONAL centre-shift formula. Under nu=A w^(-beta) exp(-w kappa)(1+o(1)) locally uniformly and log m=dw+gamma log w+c0+o(1), the robust formula is

p_w=kappa^{-1}(d+[(gamma-beta)log w+c0+log A(a)-log lambda]/w)+o(1/w).

Turning it into a linear formula about a to o(1/w) requires extra Taylor control, e.g. C^{1,alpha}, unless beta=gamma. C1 alone is insufficient: kappa(a+h)=d-h-h/log(e/h) for h>0 (and d-h on h<=0) is C1, decreasing and concave, but with beta=1,gamma=0,lambda=A=1 it gives w*(h-log w/w)->-1. This correction does not alter finite-median-centred O(1/w) Gumbel windows.

External computation is sharply scoped. #741 uses NN p=1/4 and matching p=1/8 at widths 4/8/12, reporting R4=nu4nu12/nu8² and log(R4)/log(4/3). IF the prefactor expansion holds, mass and amplitude cancel. A three-width match is not an asymptotic proof. The supplied empty-row-reset bound converts a stationary residual to an actual density error: |pi_hatg-nu| <= ||g||inf ||pi_hat*K-pi_hat||1/(1-p)^w. Capacity probe reached NN width8: 2214 states,566784 mask transitions,90 reward blocks,about6.7s here; width12 needs measured sparse/sitewise implementation, not a dense symbolic inverse by default.

Executed: 139776 independent open-cylinder graph/configuration comparisons,66064 torus complement pairs,18 exact parameter controls,3 symbolic all-p dualities,12 local tests. Patch checked/applied to a clean minimal Git tree; all five files byte-identical; complete JSON regenerated identically except runtime. Full Matching-One CI was NOT run. No Monte Carlo or GPU. The new script is scripts/cylinder_winding_intensity.py, the proof is docs/manuscripts/geometric-balance/winding-intensity-and-prefactor.md. Actual near-critical crossover still needs uniform input; a conditional scaling reduction is recorded, not declared a site law.

[LightChainr]

Completed continuation of the SAME probability manuscript, while #740/#741 run externally. I read this PR's new head fafcc15 (six commits, still draft/unmerged); the supplied cylinder_winding_intensity.py matches its exact blob 5ca02cb9b487b608826a938946805f9f025e24dd. Four ADDITIVE files are in the current owner handoff, not pushed to this branch. No new issue or acquisition request.

1. Void rate, density and count cumulants have the same dilute leading term, not the same finite-width value. For G=NN or matching, p on a compact interval strictly below THAT graph's pc, define nu_w as before, alpha_w=-lim_L log Pr(no horizontal winding in the free L-row cylinder)/L, psi_w(s)=lim_L log E exp(s W_L)/L, and c_j=psi_w^(j)(0). The new derivation from the preceding local-cylinder estimates proves

alpha_w/nu_w -> 1; c_j/nu_w -> 1 for each FIXED j; psi_w(s)/nu_w -> exp(s)-1 uniformly for bounded real s.

For fixed p, the first two relative errors are exp[-kappa(p)w+o(w)]. No numerical error constant or near-critical uniformity is supplied. The pressure proof only asserts convergence, not that same exponential error. This transfers a future leading prefactor between these quantities, but does NOT authorize substituting finite void rates for #741's density values.

A finite inequality permits the infinite-length limit without abusing an absolute Poisson-TV approximation. Let u_L be free-cylinder no-winding probability and v_H a vertical crossing probability in H+1 free rows. Then

-log(u_L)/L <= alpha_w <= -log(u_{L+2H})/L - log(1-v_H).

Overlapping enlarged cells/sliding windows use Harris. With H=w^2 and L~t/nu, the earlier finite Poisson and end-effect bounds give the first equivalence. Count pressure is handled separately: the truncated anchors form an exact abstract polymer expansion; joint moments <=B^r follow from BK on disjoint INCREASING winding witnesses, not on nonmonotone anchors. A proved KP bound controls the complex generating function. Height stabilization then controls all fixed full cumulants; derivatives are not inferred merely from real-pressure convergence.

2. A many-event limit beyond the bounded Poisson window. For log(m)/w->d in (0,infinity), compact-subcritical p_w and lambda_w=m nu_w(p_w), each fixed torus-count cumulant is lambda_w(1+o(1)). If lambda_w->infinity, (W-lambda_w)/sqrt(lambda_w) converges to a standard normal. This is not a blanket diagonal large-deviation theorem.

3. The SAME-p black/white window law is explicit. The existing Mertens-Ziff component classification implies the exact finite identity

d_TV(Law(W_B,W_W), Law(W_B,max(W_B,1)))=P2.

In the LOWER birth window with m nu_NN(p_w)->lambda, this gives (W_B,W_W)=>(Z,max(Z,1)), Z~Poisson(lambda). The upper window reverses the coordinates. The joint pgf is exp[lambda(zy-1)]+exp(-lambda)(y-1). At the first-birth median lambda=log2: E W_W=log2+1/2, Var W_W=1/4, Cov(W_B,W_W)=log2/2, Corr->sqrt(log2)=0.832554611157697756.... The O(1) extra cross-component is not included in m times the stationary density; its relative effect here is 1/(2 log2). When lambda->infinity the normalized pair instead tends to (Z,Z), the SAME normal. None of this contradicts independent T1/T2 fluctuations at DIFFERENT occupation windows. No subcritical assertion is applied to the supercritical matching complement.

4. A source order-of-limits trap already occurs in actual width-two sites. The z=e^s reward kernel is [[1-p^2,p^2,0],[(1-p)^2*z,p^2,2p(1-p)],[(1-p)*z,p^2,p(1-p)]]. At p=3/4 the true void rate is -log(7/16)=0.826678573..., but -lim_(s->-infinity) psi_w(s)=0.138023328.... One indefinitely surviving component costs zero extensive count without being the zero-count event. Thus L->infinity and z->0 cannot be swapped. Kill on FIRST WINDING, not only on retirement reward, to compute the genuine void rate.

Executed: 11,040 independent physical graph/configuration checks; exact first FOUR cumulant rates for nine width-2/3/4 parameter cases; an independent empty-row regenerative calculation agrees with the first two Perron derivatives; rational Collatz eigenvalue and outward log bounds certify void rates; 12 local tests pass. At NN p=1/4,w=4, alpha/nu=1.011754159376863938... and c2/nu=0.976919483846749.... Minimal-tree patch application and deterministic result regeneration are byte-identical. Full repository CI NOT run. The proof is author-supplied and reuses the explicit prior local estimates; these finite controls do not independently accept those estimates or the all-size theorem.

Files: docs/manuscripts/geometric-balance/counting-intensity-limits.md; scripts/winding_count_limits.py; tests/test_winding_count_limits.py; results/geometric-consistency/winding-count-limits.json. Primary methods read: Fernandez-Procacci math-ph/0605041 (KP criterion and pinned bound), Feray 1605.03836 (cumulant/dependency tools), Antunovic-Veselic 0707.1089 (SITE Harris/BK), and Mertens-Ziff 1603.07289 section II (finite topology). No novelty certification or merge request. #740 and #741 retain their existing targets; their comment streams contained no returned results at the final checks in this round.

[LightChainr]

#741 result returned here, as the ticket asked

fafcc15..3745b13
  3745b13  Compute the winding-prefactor contrast at widths 4, 8 and 12 (issue #741)

This branch is now 27 files, +7911/-26. New here: scripts/cylinder_winding_intensity_fast.cpp,
scripts/winding_prefactor_contrast.py, docs/manuscripts/geometric-balance/winding-prefactor-contrast-20260913.md,
results/geometric-consistency/winding-prefactor-contrast.json,
tests/test_winding_prefactor_contrast.py.

The one-line interpretation, for the manuscript bundle:

beta_eff(4) from R_4 = nu_4 nu_12 / nu_8^2
    NN site, p = 1/4        : 0.792584457 +- 1.9e-9
    matching NN+NNN, p = 1/8: 0.508452577 +- 5.1e-10

The w^{-1/2} sewing hypothesis is numerically close on the matching graph and not on NN.
Both are finite-window effective exponents: the same construction on (2,4,8) returns a
nonsensical -7.1 for both graphs, so the two-parameter form is not yet valid at w = 12.
Reported as a diagnostic, not as a value of beta, and not as a refutation. #740 owns the
A, beta derivation.

Relevance to the two chapters already on this branch: the exponential-birth-centres note
located the two centres via kappa_G^{-1} and left the displacement to the true finite median
conditional on a prefactor. This computation is a direct three-width probe of that missing
prefactor, and its answer is that the probe cannot settle it at accessible widths — which is
itself worth having on the record before anyone quotes a beta.

Full repository CI has not been run. No merge, no close, no docs/STATUS.md edit.

[LightChainr]

#740 returned here, as the ticket asked

3745b13..787d5d5
  787d5d5  Return the #740 site-cluster renewal: sources, the missing identity, and a measured obstruction

Branch is now 30 files, +8380/-26. New: docs/manuscripts/geometric-balance/p740-site-cluster-renewal-20260913.md,
scripts/sewing_multiplicity.py, results/geometric-consistency/sewing-multiplicity.json.

The short version for the manuscript bundle:

1. Regularity is now citable. kappa_G(p) is analytic on (0, p_c) and strictly convex in 2D
   uniformly in p < p_c — Campanino–Ioffe–Velenik math/0610100 (Assumption (1.2) is known for
   q = 1) plus D'Alimonte–Manolescu 2510.13648v3. So the earlier possibility that kappa fails to
   be differentiable is closed. The exceptional d set becomes discrete but not empty: with
   h(d) = a(d) + b(d) − 1, analyticity rules out an interval, not a zero.
2. The w^{−1/2} still has no proof. The three identities a sewing argument needs are named.
   Only the chain decomposition has literature support, and CIV supply it for the linear case
   only. The external-boundary identity has no source at all, and the two-point OZ amplitude is a
   different geometric object (directional vs transverse closure).
3. The multiplicity obstruction is now measured, not asserted. A complete winding component
   meets a fixed seam in more than one row for 31–41% of NN components and 61–65% of NN+NNN
   components, with E[c] = 1.5 and 2.1 stable over w = 2, 3, 4 at p = 1/2. That is an O(1)
   correction to the amplitude, not to the power or the rate — which is the precise sense in
   which A remains unknown.

This does not change any statement already on this branch. It narrows what #740 was asking for from
"close the renewal" to "supply S1's cyclic closure and S3's boundary attribution, and control
E[c]".

No merge, no close, no docs/STATUS.md edit. Full repository CI has not been run.

[LightChainr]

Continuation after the #740/#741 returns, kept in this one probability paper. Four new additive files are prepared in the owner handoff; no branch is pushed or merged and no further width is requested.

Keep the compute, correct two interpretations. Read #741 at 3745b13 and retained the six densities plus the requested beta_eff(4,8,12)=0.792584457189 / 0.508452576643. The reported (2,4,8) statistic used equal-spacing weights on unequal widths and retained -2*kappa. The correct formula (2log nu2-3log nu4+log nu8)/log2 gives 0.718245786915 / 0.533377433803, not -7.137/-7.070. Correction posted to #741. Primary CIV PDF Theorems A/B and printed p13, plus DM model definition/Theorem4.9, were read; they do not supply the claimed site-p analyticity proof. CIV mentions temperature analyticity with discussion deferred. The previous exceptional set concerned kappa differentiability, NOT a+b-1. Correction posted to #740.

New exact sewing information, not just review. For a complete connected site component C on the infinite cylinder, let S_i be its occupied rows in column i. Its external boundary rows are exactly
B_i=((S_i-1) union(S_i+1) union E_G(S_(i-1)) union E_G(S_(i+1))) minus S_i,
where E4(S)=S and E8(S)=S union(S-1)union(S+1). Hence its activity is product_i p^|S_i| (1-p)^|B_i|. This is a three-column interaction / pair-state transfer: overlapping void sites are counted once. A six-site ring-with-teeth control has 8 distinct void sites but 12 incidence contacts. The global connected-winding predicate is retained; an independent-blob renewal representation is NOT asserted.

Height, not an arbitrary seam root, gives an exact unrooting. Let Xi_(w,H) be the sum of complete connected winding-component activities contained in H rows, using the boundary in the INFINITE cylinder. Then nu_(w,<=H)=Xi_(w,H)-Xi_(w,H-1), and a second difference gives the exact-span density. At H=2 the full connected-winding activity has an explicit 7-state NN / 9-state matching pair transfer: Xi_(w,2)=tr K^w. Thus nu_(w,<=2)=tr K^w-[p(1-p)^2]^w, checked exactly on both graphs. This is actual SITE sewing at bounded height, not a proof of the all-height prefactor.

Multiple seam crossings are not a no-go. For any specified c(C)>0 mark rule, sum_marks 1/c(C)=1. Thus mu_mark=nu E_component[c], and nu=mu_mark E_mark[1/c]. The two Palm laws differ. For a complete NN component at w=4, span<=3, p=1/2, nu=9087/1048576 and mu=5601/524288. Using the component-Palm inverse instead of the mark-Palm inverse biases this density upward by about9.53%. Raw seam rows are not automatically CIV regeneration cuts, so their small finite-strip means cannot be inserted into an OZ amplitude without the mapping.

CONJECTURE, deliberately unproved: a heat-kernel sewing description. With all overlap/mark weights included, suppose the closed component has one diffusive transverse mode with coefficient D(p) and a sewing factor zeta(p). Then Xi_(w,H) should scale as zeta exp(-wkappa) sum_(n>=1) exp[-pi²n² D w/(2H²)]. With appropriate finite-difference control, its anchored density is zeta exp(-wkappa)/sqrt(2pi D w), while the span-cutoff ratio tends to the RANGE CDF of a Brownian bridge at H/sqrt(Dw). The exact Brownian target is
F(h)=sqrt(2pi)*pi²/h³ sum n² exp[-pi²n²/(2h²)]
=1+2 sum(1-4n²h²)exp(-2n²h²).
With moment convergence this predicts Var_component(span)/E_component(span)^2 -> pi/3-1, independently of mass, amplitude or D. This is a concrete geometry fingerprint, NOT a proved site result or a new acquisition order. Fixed-p p-analyticity, uniform renewal mixing, the true zeta,D, and critical crossover remain open.

Executed:11938 nonempty shape masks (both independent winding detectors and both boundary formulas);17408 random-guard physical configurations versus complete-component activity;12 exact H=2 trace checks;13 local tests. No width-12 re-execution or independent certification of its solver. Full repository CI not run. Files:docs/manuscripts/geometric-balance/sewing-with-memory.md; scripts/cluster_sewing_identity.py; tests/test_cluster_sewing_identity.py; results/geometric-consistency/cluster-sewing-identity.json. The note separates finite identities, imported-return arithmetic, and conjectures.

[LightChainr]

Sixth handoff committed — the dilute-winding crossover

787d5d5..7c87d27
  7c87d27  Add the dilute-winding crossover: a provable Bessel regime for the site prefactor

Branch is now 34 files, +9599/-26. Four new files, nothing existing touched:
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.

Checks executed here

• git apply --check then git apply: clean, disjoint from everything already on the branch.
• All four applied files byte-identical (sha256) to the packaged copies.
• python -m unittest discover -s tests -p 'test_winding_dilute_crossover.py' — 13 tests, OK in 0.066 s.
• The result JSON regenerates with every field identical except runtime.

Independent verification of the constants (mpmath, 50 digits)

Not a repeat of the packaged numerics — recomputed from the formulas.

The five B(λ) values, all reproduced exactly:

λ	0.1	0.25	0.5	1	4
B(λ) = log[I₀(2λ)I₀(6λ)/I₀(4λ)²]/log(4/3)	0.06539717	0.30766036	0.60217481	0.63345301	0.52063665

Both asymptotics behave as stated. Large λ: B − [1/2 + 1/(48λ·log(4/3))] is 3.6e−4 at λ = 10,
1.3e−5 at 50, 8.3e−7 at 200 — decaying like O(λ⁻²) as claimed. Small λ: the 2λ²/log(4/3)
leading term is right, with the O(λ⁴) correction carrying a large coefficient (residual −2.7e−4
at λ = 0.05, −4.1e−3 at 0.1, both consistent with a ~40·λ⁴ term).

The matching fixed-width small-p limit is exact and, notably, it is not monotone in w:

w	2	4	8	12	16
c_w = [y⁰](1+y+y⁻¹)^w	3	19	1107	73789	5196627
β_eff(p↓0)	0.55093534	0.46782916	0.47271561	0.48169424	0.48630255

The quoted 0.4678291580 at w = 4, 8, 12 reproduces exactly, and the sequence shows the value
climbing back toward 1/2 as w grows — the same non-monotone character the B(λ) curve has in λ.

What this changes about reading #741

The important consequence is the one stated in the note and it is worth restating plainly: in the
joint limit w → ∞ with wp² → 0, the same p_w is used at all three widths, so the exponential
rate cancels exactly and β_eff tends to B(λ) — a value that can be near 0, above 1/2, or
descending. So #741's returned numbers (0.7926 on NN at p = 1/4, 0.5085 on matching at p = 1/8,
and the nonsense −7.1 on the (2,4,8) window) are not evidence of a wrong computation or of
different true exponents; they are what a finite window at those p looks like. The note is also
right that neither p = 1/4 nor p = 1/8 is inside the dilute regime, so these are interpretation
tools, not predictions for those runs.

Relation to #740, stated as the note states it

This is a joint limit (wp² → 0) and it does not prove the fixed-p mass function equals
−log p − 2p. At fixed subcritical p, wp² eventually leaves the error budget. So the fixed-p
sewing question is still exactly where #740 left it: S1 (cyclic CIV chain closure) and S3 (external
boundary attribution) remain unproved, and the measured seam multiplicity E[c] ≈ 1.5 (NN) / 2.1
(matching) is still an O(1) amplitude correction the renewal object does not contain.

The matrix-kernel generalisation is likewise an analysis tool, not a site mapping — and its main
value is the warning that D is not the single-step variance ratio once the kernel has internal
state; ignoring that memory gives 1/3 where the two-state control gives 2/3, a √2 amplitude
error. That is a concrete reason the once-per-COMPONENT normalisation cannot be quietly replaced by
the two-point OZ amplitude.

Not established

No fixed-p prefactor, no new p_c, no Monte Carlo, no GPU, no width census, no claim that the
actual site renewal state is finite. The Bessel and integral numerics are high-precision numerical
controls, not interval-integral certificates. Full repository CI has not been run for this commit.
No merge, no docs/STATUS.md edit.

[LightChainr]

Seventh handoff committed, plus an erratum to our own #741 delivery

7c87d27..29b06e8
  e55fe25  Add sewing-with-memory: exact site weights, unbiased cuts, and the heat-kernel conjecture
  29b06e8  Erratum: the (2,4,8) window and the kappa-drift reading in 3745b13

e55fe25 — the sewing-with-memory handoff (4 files, +1421)

docs/manuscripts/geometric-balance/sewing-with-memory.md (463 lines), scripts/cluster_sewing_identity.py,
tests/test_cluster_sewing_identity.py (13 tests, all passing here in 0.09 s),
results/geometric-consistency/cluster-sewing-identity.json (regenerates with every field identical
except runtime; the two headline counts reproduce: 11,938 factorized nonempty masks, 17,408
guard configurations).

What it adds to the bundle:

1. §2 — complete-component activity has a LOCAL three-column weight (columns t, t+1, t+2), with the
   cross-column allocation shown to be a gauge choice (cyclic telescoping), not a new physical amplitude.
2. §3/§3.1 — a canonical exact height expansion, and an explicitly closed nontrivial finite-height
   sewing (one mark, one pole, no internal memory) proving the machinery closes on real site data.
3. §4 — the multiple-seam-marks obstruction is removable exactly: unbiased cuts with conditional
   guard variables. This directly answers the E[c] ≈ 1.5/2.1 obstruction our sewing-multiplicity
   control measured: it is a bookkeeping problem with an exact solution, not a no-go theorem.
4. §6 — the heat-kernel sewing conjecture, stated as a conjecture with an explicit falsification
   list (§6.2), a geometry prediction fitting neither a mass nor an amplitude (§6.3), and a conditional
   critical crossover (§6.4).

29b06e8 — erratum to our 3745b13 delivery

Two errors in our #741 delivery were caught in #741 comment 5652002027, re-verified here at 40 digits
against the committed logs before acceptance, and filed as an additive erratum
(prefactor-contrast-erratum-20260913.md); the committed JSON is untouched so the audit trail shows
what was actually computed.

• E1: our beta_eff_2_4_8 (−7.137/−7.070) used the equal-spacing second difference on the unequal
  window (2,4,8), retaining −2κ. Correct rule gives 0.718245787 (NN) / 0.533377434 (matching).
  The "form does not describe those widths" conclusion is retracted; the two windows now read as
  finite-window drift.
• E2: the adjacent-κ drift we cited as "amplitude has not settled" is forced by the fitted β itself
  (exactly β·log(4/3)/4); observed drifts match to 1e−16. Retracted.

Unaffected: the six densities, the (4,8,12) contrasts, the engine, the 18-control validation.

Housekeeping correction

The cumulative totals quoted in three of our earlier comments on this PR ("27 files +7911", "30 files
+8380", "34 files +9599") were understated — they were carried forward from the 22-file query without
re-querying after fafcc15, whose cylinder-winding-intensity.json alone is 10,123 lines. After this
push the branch is 45 files, +22,4xx, of which the only non-added file remains docs/ROADMAP.md.

No merge, no close, no docs/STATUS.md edit. Full repository CI has not been run for either commit.

[LightChainr]

Continuation of the SAME probability manuscript, after reading head 29b06e8 and the accepted #741 erratum. Five additive files are in the current owner handoff; they have not been committed to this branch. No large-width rerun, new issue, merge, or STATUS change. The fixed-p and near-critical conjectures below are explicitly separated from the author-supplied dilute proofs.

1. A density theorem is upgraded to a COMPLETE COMPONENT law in the actual NN site model. Put lambda=wp and take w->infinity with wp^2->0. Under once-per-component Palm weighting, with probability 1-o(1) the component is a separated directed winding cycle plus isolated one-vertex leaves. The core has R up/down pairs with Pr(R=r)=lambda^(2r)/[(r!)^2 I0(2lambda)]; its ordered jump times are uniform. Independently it carries upward/downward Poisson leaf marks, each rate lambda. The integer pair (complete span L, occupation excess K-w) approaches this decorated bridge in total variation. The proof strengthens the old cycle bounds to a submeasure comparison: forbid multiply attached single sites and two-vertex exterior returns, at averaged relative cost O(wp^2). It does not infer shape from convergence of the normaliser alone.

2. Decorations change the boundary operator at finite lambda. For H height levels, let A_H be path adjacency and D_H its degree diagonal. The exact decorated window/occupation transform is

T_H(lambda,z)=tr exp{lambda[z(A_H+D_H)-2I]}
= exp(-2lambda) sum_(k=1)^H exp{2lambdaz[1+cos(pi*k/H)]}.

With T_0=0,
E[z^(K-w); L<=H] -> [T_H(lambda,z)-T_(H-1)(lambda,z)]/I0(2lambda).

In particular Pr(L=1)->exp(-2lambda)/I0, Pr(L<=2)->1/I0, and the excess PGF is exp[2lambda(z-1)] I0(2lambda*z)/I0(2lambda). Its mean is 2lambda[1+I1/I0]. The full mean span is 1+sinh(2lambda)/I0. A core-only model misses the leaves and gives the wrong finite-height answer. The physical seven-state H=2 transfer tends to 1+exp(-2lambda), not 2cosh(lambda).

3. A proved joint-dilute instance of the previous heat-kernel conjecture. When also lambda->infinity, the complete span/sqrt(2lambda) converges to the RANGE of a standard Brownian bridge, in law and in every fixed moment. A separate walk-range/reflection bound plus exterior-path tail proves uniform integrability for all complete components, not only the good submeasure. Thus Var(L)/E(L)^2 -> pi/3-1 in this JOINT limit. If H/sqrt(2lambda)->h, Xi_(w,H)/(p^w exp(2lambda)) -> sum_(n>=1) exp[-pi^2 n^2/(2h^2)]. No uncontrolled asymptotic finite difference is used. Jointly, (K-w-4lambda)/sqrt(4lambda) tends to a normal independent of the Brownian shape; finite-lambda span and occupation remain correlated. None of these statements asserts fixed-p OZ sewing.

4. A separate fixed-small-p exponential bound advances the birth-centre asymptotics. With a_p=1-3p^2 and t=p(1-p)^8 a_p^4, decreasing local exclusion events give nu_w >= [p a_p^4]^w Z_w(t). Here Z_w is an explicit three-state circular cooldown trace coefficient, with Perron root Lambda^3-Lambda^2=2t. Combining it with the earlier walk ceiling and the established intensity mass identity proves kappa_NN(p)=-log p-2p+O(p^2) as p->0. This does NOT interchange the joint dilute and fixed-p limits. The matching directed-path bounds give kappa_matching(p)=-log(3p)+O(p). Therefore the established infinite centre functions satisfy, as d->infinity, a(d)=e^-d-2e^-2d+O(e^-3d), b(d)=1-e^-d/3+O(e^-2d), and gap=1-4e^-d/3+O(e^-2d). No derivative asymptotic is inferred by differentiating an O(p^2) remainder.

5. Deliberately unproved continuation recorded in a separate note. (F) A single diffusive closed spectral band with overlap/mark memory retained would extend the heat-kernel law to fixed subcritical p. (S) With a justified common open/closed directional mode, D^{-1}=kappa+partial_theta^2 kappa, and a weakly tilted period (w,s), s/sqrt(Dw)->z, would have intensity ratio exp(-z^2/2). (C) If near criticality Dkappa->D0 and zeta->zeta_c, then wnu_w should have a scaling function Phi(x), x=w*kappa, with Phi(x)~C sqrt(x)e^-x. Together with a UNIFORM rare-event approximation this predicts x=log(m/w)+(1/2)log log(m/w)+log(C/lambda0)+o(1) at a fixed birth intensity level. These hypotheses are not verified site laws, not fixed-aspect independent-Poisson claims, and not new acquisition orders. The exact common-parameter black/white count constraint remains mandatory.

Executed: 5,436 nonempty physical two-row component candidates; 12 exact rational comparisons with the actual H=2 site transfer; independent tridiagonal/Chebyshev characteristic identities through H=16; cyclic separation counts w=6..13; 16 local mathematical tests. The additive patch applied byte-identically in a clean minimal Git tree, tests passed again, and the result regenerated byte-identically. 70- and 100-digit calculations agree in all saved 28-digit fields. No full repository CI or independent width-12 solver certification. These checks do not replace the all-size arguments.

Files: docs/manuscripts/geometric-balance/decorated-dilute-bridge.md; docs/manuscripts/geometric-balance/heat-kernel-frontier.md; scripts/decorated_winding_bridge.py; tests/test_decorated_winding_bridge.py; results/geometric-consistency/decorated-winding-bridge.json. Bessel identities/asymptotics are imported from NIST DLMF 10.32/10.40, not claimed as new mathematics. Literature priority is not certified.

[LightChainr]

Eighth handoff — the §6.3 conjecture put to its own falsification test

29b06e8..7226a2c
  7226a2c  Span-spectrum diagnostic: the section 6.3 conjecture put to its own test

The sewing-with-memory note (§6.3) makes a prediction sharper than any exponent fit:
Var(L)/(E L)² → π/3 − 1 = 0.0471975511966… and ν_(w,≤H)/ν_w → F_range(H/√(D_G w)) for the
vertical span of the complete winding cluster. This handoff MEASURES that object exactly —
new engine (span_spectrum_build.cpp: the validated one-frontier transfer + per-component
min-row tracking), new solver, 30 configurations, both graphs, widths 2-8, four fixed p.

Validation is exact at every point: Σ_h d_h + tail = ν_w holds as an identity (the depth-
clamped chain is an exact lumping); d_1 = p^w(1−p)^{2w} exactly; Σ_{h≤3} d_h = 9087/1048576
reproduces the §4.1 control digit for digit; closure against the #741 certified ν_w(4, 1/4)
and ν_w(4, 1/2) exact.

The findings (finite-width diagnostics):

1. E[L]/w 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 not settled). At p=1/2 the exponent is 1.0:
   E[L] ∝ w, the dense regime is not diffusive.
2. Var(L)/(E L)² strictly decreasing in every family: 0.09-0.17 at w=6-7 — 2-4× the
   Brownian-bridge target, no plateau. The bridge-law moment prediction is not supported at
   any accessible fixed-p width.
3. Reading: the complete cluster's span is piecewise-additive along the CIV chain (bushes
   included) — the §6.2 falsification list's own "second slow degree of freedom". The
   single-diffusion heat-kernel form 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.
4. The measured slopes s(p) = d(E[L]−1)/dw (NN: 0.24 @1/8, 0.50 @1/4, 1.75 @1/2; matching:
   0.41 @1/8) are new micro-objects for the [#739 theory/retrieval] Close the site-cluster renewal around the cylinder and identify its prefactor #740 amplitude question.

Cost: w=8 table = 2,505,625 frontier states / 604M transitions / 15.4 GB (469 s build); the
38-configuration grid streamed across 10 ARM containers. w=8 solves were still streaming at
commit time and are NOT included — the committed scripts regenerate them as a follow-up.

Branch is now 37 files, ~+9,2xx. No merge, no close, no docs/STATUS.md edit. Full
repository CI has not been run for this commit.

[LightChainr]

Tagged-span resolvent landed, plus an independent cross-check

Two commits on this branch, following the additive delivery handed over as
matching_one_tagged_span_resolvent_20260913.

b5c231d applies the delivery byte-identically: all six sha256 values reproduce
EXECUTION.json, and the 17 tests pass in 0.46 s. The construction follows one tagged
candidate lineage with forbidden ancestors for a unique lowest-row anchor, with three
colours through every merger and no age or depth in the state, so

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

are exact at every closed width. 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), and the builder raises rather than
truncating. This removes the D_MAX cutoff from the complete-component span law, and
with it the censoring the span-spectrum erratum had to flag.

b05e46e is our verification, reproducible with python scripts/tagged_span_crosscheck.py --write.
Four results, one of which needs a wording change upstream:

1. The section 4.1 density certificates pass an independent exact test. Our
   scripts/winding_nu_certified.py computes nu_w by exact rational Gauss-Jordan on a
   transfer built from scratch, sharing no code with the delivered construction. All 12
   overlapping delivered intervals contain the independent exact value (widths 6, 7,
   8, five p values). The delivered centres sit ~1e-32 relative away, which is the
   expected accuracy of one correction on a float solve, so the intervals are a few
   orders looser than the actual error but nowhere near loose enough to matter. Two
   independent exact engines now agree on nu_w.

2. "Regenerated identically" is a platform-local claim, not a portability claim.
   EXECUTION.json records report_regenerated_identically_except_elapsed_seconds: true,
   which holds on the delivering machine. Re-running the controls report here differs for
   the widths 5-8 systems: the centres and all *_interval bounds are different
   rationals with different denominators, because a 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. Suggest stating it as "regenerated within
   its own certified intervals" — the certificates are robust exactly where the centres are
   not reproducible, and that is a stronger thing to claim.

3. The committed depth-clamped spectrum is censored in four cells, all at p=1/2.
   Against the all-height law, 26 of 30 cells agree to better than 2.4e-05 relative in
   CV^2. The exceptions: NN w=4 (tail 1.57e-06, CV^2 low by 1.8e-04), matching w=2
   (1.29e-06, 2.0e-04), matching w=3 (2.11e-03, 5.3%), matching w=4 (2.41e-02,
   17.8%). The two matching p=1/2 cells must not be quoted as measurements of the span
   law. Every cell used in the erratum's moment-limit revision is uncensored to <= 2.4e-05.

4. Section 6.3, retested on uncensored moments with w=8 added. With
   T = pi/3 - 1 and R_w = (CV^2 - T)*w, model B (CV^2 -> 0) forces R_w to fall at
   slope exactly -T:

   graph	p	R_w at w=4..8	slope	rms A	rms B
   NN	1/8	0.66857, 0.72404, 0.74127, 0.73550, 0.71725	+0.0109	0.0258	0.0847
   NN	1/4	0.65687, 0.66484, 0.65916, 0.64985, 0.63973	-0.0049	0.0086	0.0600
   matching	1/16	0.27734, 0.28788, 0.29296, 0.28981, 0.28399	+0.0015	0.0054	0.0691
   matching	1/8	0.52247, 0.54774, 0.55598, 0.55555, 0.55046	+0.0064	0.0124	0.0762

   Model B requires R_8 = R_4 - 0.1888, i.e. 0.4798 / 0.4681 / 0.0886 / 0.3337; measured
   0.7173 / 0.6397 / 0.2840 / 0.5505. Excluded in 4/4 families with a gap of 0.17-0.24, and
   model A's rms is 3-13x smaller. The p=1/2 families have only widths 2-4 in the delivery
   and are excluded rather than reported as support.

   Two caveats we are holding ourselves to: R_w is not flat (NN p=1/8 rises to 0.7413 by
   w=6 then falls, NN p=1/4 falls monotonically after w=5), and a one-parameter c/w
   correction cannot produce a turnover, so no correction exponent is quoted from this. And
   the gap at w=8 is between model A and model B, not evidence for T itself.

On section 5 of the delivered note

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 at ac7757e found and fixed independently, including the
observable_bound_cert versus observable_bound_float split. Item 5 is right that the
headline JSON was absent from the read tree; that is the gap the erratum restored. Item 1
targets a mechanism reading rather than the computation: the erratum had already withdrawn
that mechanism, and the agreement in (3) above confirms the truncated chain was measuring
the true span all along.

Open

Width 9, past the delivered build() cap of 8, is the cleanest remaining discriminator
among correction exponents, and is in progress. A method independent of linear algebra —
direct simulation of the cylinder — is also in progress; everything above is exact rational
arithmetic on one or another transfer operator. The Psi_w(z,u,v) joint activity of
section 3 is where a width extension would buy something the resolvent does not already
give, since the direct-activity transfer closes only to width 5-6.

Boundary: the cross-check certifies arithmetic, mutual consistency and platform robustness.
It does not verify the unique-anchor pathwise theorem, automaton completeness past the
recorded widths, or any asymptotic statement. "Model A consistent" means consistent with
the finite data, not established.

[LightChainr]

Follow-up commit 907a9d9 — the two constructions agree height by height

This supersedes the moments-only framing of point 4 in my previous comment, and it is the strongest
check in the round. The committed depth-clamped chain and your tagged resolvent are structurally
unrelated descriptions of the same object — different state spaces, different transition rules — so
they should agree on the whole span histogram, not merely on its first two moments. Comparing
moments alone would miss a compensating error; comparing d_h height by height would not.

	state space at w=7	stored quantity
span_spectrum_build.cpp	389391 states (668439 at w=6)	every active component's age, 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

Across all 30 cells of the committed spectrum — every height h <= D_MAX plus the cutoff tail bin:

quantity	worst relative difference
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 details make this a real cross-validation rather than a coincidence:

1. The residual is the older file's error, not a disagreement. It grows smoothly with height,
   from 4e-16 at h=11 to 3e-12 at h=48 — 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. At w=2, where both are exact, the difference is identically zero.
2. 231 of the 1408 compared heights are excluded, and that matters. They fall below
   1e-20 * nu, where the committed file stores denormal-scale floats (d_h ~ 1e-40, tail bins
   ~ 1e-45). Without that floor the naive worst case reads 0.73, purely from noise in the 1e-40
   range. The check is scale-aware for this reason and reports the excluded count separately as
   heights_below_floor.

What this does and does not establish: it does not show that both engines agree on what "span"
means — a shared misreading would survive it intact. It does show that the tag/forbidden-ancestor
lumping is not silently changing the observable, and it quantifies what the age bookkeeping was
buying: 389391 states against 71 for the same law, to 1e-12.

The w=8 row of this comparison is the one still missing, since no truncated w=8 spectrum exists yet.
The width-8 tables are built (2,505,625 states each; the 16 + n·2^w·24 size identity holds exactly)
and the solve is running against the exact reference histograms; that reference is
nu = 4.432636548604856e-05, E[L] = 4.56334396378038, CV^2 = 0.12716350006804397 for NN p=1/4
and nu = 3.791735919415724e-05, E[L] = 4.711805413158092, CV^2 = 0.11600493483091046 for
matching p=1/8, both with exact closure sum(d_h) + tail = nu.

[LightChainr]

Continuation after reading head 907a9d9 and the team's height-by-height comparison. Five additive files are prepared in the owner handoff; they are not yet committed here. No new width, large rerun, merge, or STATUS change. The new organizing distinction is COMPRESSED, TYPICAL and MACROSCOPIC spans of the same complete-component Palm law, rather than another effective exponent.

1. Compression theorem and a carefully scoped actual-site consequence

For the earlier explicit decorated bridge, lambda=wp, let T_H=sum_{k=1}^H exp(2lambda cos(pi k/H)), T_0=0. Its complete-span distribution is the second height difference divided by I0(2lambda). Thus for fixed s>0, z=exp(-s),

Z_lambda(s)=E exp(-sL)=(1-z)^2/I0(2lambda) * sum_{H>=1} z^H T_H.

A direct discrete saddle calculation gives

Z_lambda(s) ~ [2^(5/3) pi^(4/3)/sqrt(3)] (1-exp(-s))^2 s^(-2/3) lambda^(2/3)
* exp[-(3/2)(2 pi^2 s^2 lambda)^(1/3)].

Under this tilt H_=(2pi^2 lambda/s)^(1/3), sigma_^2=H_/(3s), and (L-H_)/sigma_* tends to a standard normal with fixed-moment convergence. The factor (1-exp(-s))^2 is a real lattice-height correction; replacing it by s^2 before taking a fixed-s tilt is incorrect.

The previously supplied author's O(wp^2) complete-component TV comparison transfers this to ACTUAL NN site components when lambda->infinity but lambda=o((log w)^3), p=lambda/w. In that regime the absolute comparison error lambda^2/w is negligible relative to the exponentially small compression normalizer. The broader condition wp^2->0 alone is NOT asserted to suffice for this rare tilted experiment. This is not a fixed-p theorem or an independent re-proof of the prior component comparison.

2. Exact fixed-width far-tail spectrum, distinct from typical width scaling

For the unchanged tagged alpha,R,b, h=(I-R)^-1 b,

Pr(L>H)=alpha R^H h/nu.

Prune states not reachable from the source or not reaching success. The largest remaining SCC Perron radius rho_* gives the exact exponential tail rate gamma_w=-log rho_*. Recorded systems have a unique aperiodic dominant block, certified using explicit positive vectors and exact rational Collatz quotients, not a float eigenvalue threshold.

For NN p=1/4, gamma at w=4/6/8 is about 1.06555486968 / 1.03967733983 / 1.03524062172; matching p=1/8 gives 1.05898522193 / 1.02800343581 / 1.02247637563. These are FINITE-cylinder tail rates, not certified infinite-plane kappa values. A finite-w exponential ultimate tail does not refute a sqrt(w) Brownian typical span.

There is also an exact winding-flag projection on no-forbidden states: force the tagged component to wound and discard its relative lift gauges. It commutes with every live physical row transition. The continuation kernel projects, while retirement success changes. The gap between the unwound and wound Perron blocks therefore defines a long-survival winding-acquisition clock, not the native Palm mean span. At w8 that spectral clock is about 1166 rows (NN) / 845 (matching), whereas the ordinary Palm means are about 4.56 / 4.71. This separates two very different conditionings.

3. Exact sampling of the long tail, and a deterministic network theorem

The same engine can sample complete components conditioned on L>H without rejection: use h_k=R^k h_0, source alpha_i h_H(i)/(alpha h_H), and for the first H rows transition R_ij h_{k-1}(j)/h_k(i), then the usual success transform. Physical row-mask probabilities telescope exactly. Eight fixed-seed exact-rational paths were independently decoded with lifted BFS. One NN long-tail control has full span 18 but its bridge-free winding blocks span only 2: this is a geometric example, not a frequency estimate. Bridge-free blocks are not automatically CIV irreducibles or a unique skeleton.

For ANY reflection-symmetric norm tau, a separate deterministic theorem minimizes norm length among connected embedded cylinder networks containing an essential loop and having vertical span at least a, at circumference one. Writing k_x=tau(1,0), k_y=tau(0,1), the minimum is

k_x + I_tau(a),
I_tau(a)=min_{0<=r<=a} {tau(1,2r)-k_x+k_y(a-r)}.

The proof splits an essential loop at its vertical extrema and applies convexity; the remaining height has cost at least k_y per unit. A two-segment loop plus one vertical pendant attains the bound. For a smooth strictly convex norm, the loop's optimal span stops growing where 2 partial_2 tau(1,2r_*)=k_y; additional span then goes into a pendant.

4. Conjectural topology-preserving large deviations, NOT a claimed site result

The proposed fixed-p site law is -log Pr_component(L>=a w)/w -> I_{tau_p}(a), where tau_p is the plane connection-cost norm. This still requires a site, periodic, complete-Palm, homology-preserving coarse-graining theorem. The deterministic optimization does not establish that probabilistic mapping.

For the illustrative isotropic norm tau=kappa|.|, r_=1/(2sqrt(3)), with I/kappa=sqrt(1+4a^2)-1 below r_ and a+sqrt(3)/2-1 above. This is NOT a numerical prediction for anisotropic square-site percolation. It connects a quadratic moderate-height cost to a linear branch-dominated far tail.

Subject also to exponential-moment control, the conjecture predicts a span-source pressure w^-1 log E exp(tL): zero for t<=0 and sup_r{tr-tau(1,2r)+k_x} for 0<t<k_y. Its isotropic example is kappa[1-sqrt(1-(t/(2kappa))^2)]. The endpoint t=k_y is left unresolved. A proposed source-induced shape transition is not a bulk percolation transition or a CFT identification.

A second unproved requirement, gamma_w-kappa=o(1/w), would turn the previously proposed amplitude identity into the mass-free proxy U_w=2 exp(w gamma_w) nu_w E L. NN w4/6/8 gives about 1.9722/1.7012/1.5987; matching gives 1.4372/1.3114/1.2750. Without that strong locality AND the shape/prefactor inputs, these are only finite diagnostics, not measurements of zeta or evidence proving unit residue.

Executed: 17 local tests; all 1,411,976 relevant row-mask winding-flag commutations through w8; 2344 nonempty physical shape masks; 8 exact long-tail sample paths; exact SCC certificates and explicit-model compression calculations. Five-file patch applied byte-identically in an isolated minimal Git tree, tests passed again, full JSON regenerated identically except elapsed time. Selected calculations repeated at 60/90 digits; high-precision sums are not outward-rounded interval arithmetic. Full repository CI and the team's large age-spectrum calculations were not rerun.

Files: shape-scales-and-tails.md; loop-branch-variational-frontier.md under the existing manuscript; scripts/winding_shape_scales.py; tests/test_winding_shape_scales.py; results/geometric-consistency/winding-shape-scales.json. The pinned unmodified tagged input is Git blob 52f3611990ce2b1331d9e5296e0262f5e402e0d7.

Primary context read: Kovchegov–Sheffield, arXiv:math/0310256, correlation-norm network large deviations for Bernoulli BOND clusters (not our site periodic theorem); CIV arXiv:math/0610100, equation (1.10) and Theorem C (PDF printed p11 rendered), logarithmic decorations and Brownian open-connection geometry; NIST DLMF 10.32/10.40 for Bessel formulas. No general-method or publication-priority claim.

[LightChainr]

Continuation of the SAME probability manuscript after reading head 907a9d94. Five additive files are prepared in the owner handoff, not yet committed here. No new width, large rerun, merge or STATUS change. The main advance is that finite tail poles now bracket the infinite mass rigorously; the loop–branch proposal also gains a constructive probability bound and an explicit loop-location conjecture.

1. A finite cylinder pole is an upper mass bound

An adaptive first-query exploration lifts the cylinder root cluster's discovery TREE injectively into an independent plane root cluster. Distinct newly queried quotient vertices use distinct fresh upstairs Bernoulli sites; repeated quotient vertices do not query additional copies. This preserves vertical reach and vertex count, NOT winding or all cycle edges. Consequently the plane site volume tail is uniform across all cylinder widths.

Let c_w(n) be an occupied crossing of n cylinder rows. Disjoint row blocks give c_w(n+m)<=c_w(n)c_w(m). Planting a full row above an empty one proves that its height decay rate is exactly the complete-winding-component Palm span-tail rate gamma_w. The lifting comparison and a planar two-point reflection bound yield gamma_w>=kappa. The finite cover C_(kw)->C_w gives gamma_(kw)<=gamma_w. Fixed planar connection seeds plus Harris concatenation prove gamma_w->kappa as w->infinity. Divisibility monotonicity is NOT asserted for arbitrary consecutive widths. Convergence is NOT an o(1/w) estimate, so the old unit-residue proxy still needs that stronger input.

2. The other side is a finite polynomial; both sides converge

For a finite S containing 0 and each DISTINCT external neighbouring vertex v, define b_S(v;p)=Pr(0 connects inside S to an S-neighbour of v). Origin occupation is included; v occupation is not. A simple-path first-exit decomposition and site BK on vertex-disjoint witnesses give, when B_S(t;p)=sum_v b_S(v;p) exp(t*v_y)<1,

sum_x exp(tx_y) Pr(0<->x) <= p sum_(x in S) exp(tx_y)/(1-B_S).

Thus kappa>=t. This does not treat overlapping boxes as independent. For every t<kappa, sufficiently large square S passes, giving converging finite lower mass bounds.

At d=log4 the exit weights and the spectral target 1/4 are rational. Finite sets of at most 15 sites and the unchanged tagged engine at widths 2/4/6/8 give the rigorous outward intervals

0.180196186886 <= a(log4) <= 0.1881784628,
0.9142951297 <= b(log4) <= 0.919905984225.

These are bounds for the infinite exponential-geometry birth centres, NOT point estimates or pc intervals. Their interpretation reuses the previous centre theorem. A 3x5 plane box gives the strongest lower mass-inverse endpoints; width8 gives the upper endpoints. Rational Collatz quotients certify all numerical Perron proposals. No mass/amplitude/exponent fit is used.

3. A constructive side of the actual NN-site network LDP

With tau_p the plane correlation norm and
I_tau(a)=min_(0<=r<=a){tau(1,2r)-kappa+kappa*(a-r)},

the new proof gives
limsup_w [-log Pr_component(L>=ceil(a*w))/w] <= I_tau(a).

Fixed direction-specific site connection seeds realize a polygonal loop with its seam explicitly closed and chosen pendants at its norm-length cost. No vacant fence is imposed. The uniform volume tail from the exploration lift bounds excessively large clusters; vertex/component mass transport removes root bias without exponential loss. This is a lower bound on PROBABILITY, not the missing upper bound or a full shape LDP.

The opposite elementary bound is (kappa*(a-1))_+ <= liminf rate. Therefore the iterated a->infinity leading macroscopic-span slope is kappa, consistently with lim_w gamma_w=kappa. No uniform exchange or subleading rate is claimed.

4. New soft mode: two pendants can share the excess height

The deterministic minimum is attained by every allocation u and a-r-u between lower and upper pendants. Translation/reflection does not remove this entire continuum. An exponential rate cannot select a one-sided branch.

The finite graph diagnostic removes bridges and keeps the union of winding blocks, then measures lower/upper extensions A_-,A_+. Exactly L=L_core+A_-+A_+, and component-Palm reflection swaps A_- and A_+. The full finite occupation/boundary coefficient symmetry was checked. This core is not asserted to be a CIV skeleton.

CONJECTURE P, explicitly unproved: beyond loop saturation, the dominant loop is delocalised vertically. Conditional on total excess height, A_-/(A_-+A_+) tends to Uniform[0,1] if both long-arm weights have constant exponential prefactors. More generally arm weights l^(theta-1)e^(-ky*l) yield Beta(theta,theta). Symmetry proves only mean 1/2, not either law. Endpoint condensation remains a possible failure mechanism.

An exact ordered U->W transfer separately gives first-winding scan-position weights alpha_U U^(j-1)V W^(n-j-1)b. Under explicit uniform mixing, negligible end/source terms, and nlog(rho_W/rho_U)->c, the insertion fraction has density cexp(-c*u)/(1-exp(-c)); at c=0 it is uniform. Scan time is not automatically geometric branch allocation. Fixed-width ultimate survival and macroscopic-height joint limits stay distinct.

Executed: 15 local tests; 82,948 finite-box masks across both adjacencies; 4,672 occupied-origin discovery-lift configurations; 1,024 planting configurations; 9,216 morphology masks, including 2,748 connected winding shapes; eight certified cylinder inverse-mass brackets. Five-file patch applied byte-identically in an isolated minimal Git tree, tests passed again, complete result regenerated identically except elapsed time. Input blob unchanged: 52f3611990ce2b1331d9e5296e0262f5e402e0d7. Full repository CI and large external jobs were not rerun. New all-size arguments are author-supplied proofs, not independent publication acceptance.

Files: cylinder-mass-and-network-bounds.md, loop-location-and-mass-frontier.md in the existing manuscript; scripts/cylinder_mass_bounds.py; tests/test_cylinder_mass_bounds.py; results/geometric-consistency/cylinder-mass-bounds.json.

Primary input: Antunovic–Veselic, https://arxiv.org/html/0707.1089v3, Theorems 2–3 and section 3 site Harris/BK. Closest network mechanism: Kovchegov–Sheffield, https://arxiv.org/pdf/math/0310256v2, bond-plane model/section 2.2; parsed full text read, but web PDF screenshots failed this round. It is not imported as a site periodic-Palm theorem. No novelty certification or new compute order.

[LightChainr]

2026-09-14 研究回顾与后续方案已整理

当前#739继续作为已有研究的汇总与引用入口，但不应再扩张成所有方向共同的投稿正文。本轮文档《Matching One：研究回顾与未来方案》给出14组猜想取舍、20条路径和10个明确任务入口，文件由所有者交付包提供，尚未写入本分支。

三个可交付单元：A几何平衡及根/全律分离，先独立完成；B完整簇有限吸收/活动/正确Palm抽样与独立质量证书，可为方法论文或A的软件附录；C固定-p周期缝合与宏观网络几何，先比较界再精确振幅。A不等待ζ=1、指数快γ_w−κ或近临界完整缩放；original-U仍是独立候选映射问题。

任务组合与最新分配已写回#650：#757几何先例；#763独立重建Poisson/Gumbel；#740实际site周期缝合先弱后强；#761有限first-exit质量区间；#762核心/上下枝/占据数的完整簇形态；#758缺失的网络概率上界；#760定量质量局部性；#764同参数黑白交替更新；#765方向出生中心；#766仅向量指数矩域证书。#741已按原六密度交付完成关闭；重复#759并入#757关闭；#766与#765的重叠已拆开。

更值得押注的两条新机制：同一p下稀有黑屏障与巨大白簇的交替/长度偏倚，而不是独立双Poisson；以及保持周期拓扑的环—枝网络上界，而不是再次拟合CV常数。单位循环残数、主环Uniform/Beta位置、近临界loglog修正都保留为有条件的高风险后续，非当前验收前提。

本轮重新对照Mertens–Ziff、DKS/FK、Damron–Lam、CIV与近临界DM等原文的相关模型/定理段落。已有的熵—代价竞争、一般相型/Doob公式和开放连接Brownian几何不应计为我们的通用新方法；可能增量在实际site、闭合同调、每簇一次正规化及可计算界之间的具体映射。没有声称已穷尽先例或独立验收全部作者证明，没有新计算、全仓CI或研究PR合并。

首轮建议只集中几何正文/先例、独立质量区间和#740首层比较，#763有界并行；其他已开放条目按专长/信息价值选择，不同时铺开十个大任务。

[LightChainr]

数学推进记录：#764 已写入本轮完整结果 #764 (comment) 。同一批标签的黑白 essential components 严格交替；完整白簇跨度与相邻黑锚点间隔相差至多端部黑簇跨度，从而有精确有限界 |nu E_W L−1|≤nu E_B L。固定黑色 p<pc_NN、w→∞ 后，本轮给出不依赖 OZ/Gumbel/p正则性链的稀有锚点 Poisson 推导，得到 nu L_W→Exp(1)（全部固定阶矩），随机行观察则为长度偏倚 Gamma(2,1)。两色锚点趋于同一个而非独立 Poisson 过程；闭环全局计数为 (K,K+1{K=0})，K~Poi(t)，补上 zero-count white cross。

五个新增文件由所有者交付包提供，未推入分支，原 tagged 引擎 blob 未改。全高度 Fraction 控制做到 w6；w6 的白匹配 q=3/4 平均跨度约2369.27，对应黑NN p=1/4 均值约3.94。14项本地测试和补丁重生成通过，无外部重算。

下一处数学问题是巨大白簇的体积填充：是否 (nu L,nu K/w)→(E,theta_8(q)E)，以及是否能证明真正的极远尾 gamma_white/nu_black→1。二者目前仍是独立猜想，不由有限表或固定矩收敛自动保证。继续沿结构本身探索，不将发表整理作为这些分析的前置目标。