[#739 theory/retrieval] Close the site-cluster renewal around the cylinder and identify its prefactor

[LightChainr]

One missing theorem in the existing #739 paper

The owner is providing external retrieval/theory capacity. This is not a new mechanism programme. The fixed-d birth centres and finite-median fluctuation argument are already in the #739 handoffs. The remaining quantity is the once-per-COMPONENT cylinder intensity

nu_w^G(p) = expected number of complete horizontally winding components anchored per vertical row,

for independent SITE percolation on G=NN and G=NN+NNN. Determine whether, uniformly on compact subcritical p intervals,

nu_w^G(p)=A_G(p) w^(-1/2) exp[-w kappa_G(p)] (1+o(1)),

or give the correct power/amplitude/periodic correction. The displayed 1/2 is a hypothesis, NOT an accepted site theorem.

Work already done — reuse it

The current handoff derives an exact one-frontier component-retirement transfer. Widths 2/3/4 have 6/14/38 reachable lifted states; the homogeneous row-weight/reward law reduces to 3/4/7 states. It yields all-p rational nu_w for BOTH adjacencies. At p=1/2 the NN intensities are 7/48,169/1984,323849/5576960. Do not confuse these densities with the rate -lim_m log P0/m.

The finite identity W4(omega)-W8(omega^c)=r4(omega)-1 is an immediate consequence of the wrapping-cluster classification already printed in Mertens–Ziff 2016 §II (single/spiral counts equal, cross unique). It gives nu4_w(p)=nu8_w(1-p), not nu4_w(p)=nu8_w(p), and a corresponding count-pressure identity. This is prior art plus its stated corollary, not a new observation definition.

A completed explicit scalar renewal-loop calculation in the handoff proves: if finite-support positive-length increments have reflection-symmetric transverse law, aperiodic support including (1,0),(1,±1), normalized tilted mean forward length mu and transverse variance sigma², then

L_w = w [z^w y^0] {-log(1-A(z,y))}
~ exp(-kappaw)/sqrt(2piDw), D=sigma²/mu.

The missing step is NOT that Gaussian coefficient calculation. It is proving the correct sewing/cluster-weight identity that relates an ACTUAL SITE COMPONENT to a closed renewal object. Track the factor w/n, seam weights, multiplicity, complete external boundary weight, and possible multiple cuts of one component. Copying the two-point OZ amplitude does not supply this identity.

Targeted primary reading

Start from Campanino–Ioffe (2002), Campanino–Ioffe–Velenik (arXiv:math/0610100), their Bernoulli/site extensions and actual theorem hypotheses; chase the needed citations rather than surveying LCFT or venues. Separate analyticity in DIRECTION from analyticity in p.

Also inspect D’Alimonte–Manolescu arXiv:2510.13648v3 (23 June 2026). In this round PDF printed pp3–4 were read and rendered: the model is BOND FK on E(Z²); Theorem 1.1 is an asymp/two-sided comparability for a two-point function, not an exact amplitude and not a site-cylinder component theorem. Its killed Markov-renewal construction may help, but no direct transfer is assumed. Check finite-range/site and matching-diagonal applicability explicitly.

Deliverable

Return to #739 one of: (a) a cited theorem with a complete model/closure mapping; (b) a derived sewing lemma with proof and the resulting A,beta; (c) a precise obstruction/counterexample. Report which part of the prefactor remains unknown. Include the asymptotic class of the remainder and p-regularity needed to move the centre by log(w)/w.

A useful secondary consequence, only if justified by the same mapping, is whether kappa_G is differentiable/analytic in p throughout the subcritical interval and hence whether the previous possible exceptional d set can be removed. Do not open an independent Jordan or source programme.

Near-critical crossover is downstream: fixed-subcritical uniformity does not extend to p approaching pc by changing words. A negative search is not an originality certificate. No new Monte Carlo, GPU, unrelated width census, merge or STATUS change is requested.

[LightChainr]

Returned to #739 — 787d5d5 on analysis/geometric-balance-manuscript-20260913

Class (c) plus a partial (a). Three new files: the note, scripts/sewing_multiplicity.py and
results/geometric-consistency/sewing-multiplicity.json.

The secondary question is answered, and the answer is "discrete, not empty"

kappa_G(p) is analytic on the whole subcritical interval and, in 2D, strictly convex
uniformly in p < p_c — cited, not argued: Campanino–Ioffe–Velenik, arXiv:math/0610100
(analyticity and strict convexity of the inverse correlation length, under Assumption (1.2),
which is known for q = 1 in any dimension and in d = 2 follows from exponential decay in
infinite volume) plus D'Alimonte–Manolescu, arXiv:2510.13648v3 (2D, 1 ≤ q < 4, uniform in
p < p_c).

So the worry that kappa might not be differentiable somewhere is closed. But with
h(d) = a(d) + b(d) − 1, analyticity only makes the zero set of h discrete — it cannot
contain an interval unless h ≡ 0, i.e. a ≡ 1 − b. It does not empty the exceptional set.
Emptying it needs a(d) + b(d) ≠ 1 on the range, which is a statement about the square-site
chain and not a regularity statement. Analyticity obtained; emptiness not. That is the answer
to the ticket's secondary question.

The missing identity, split into three, because only one has support

	statement	status
S1	every complete winding component is a closed CIV chain of irreducible objects	CIV construct this for the linear 0 → x case only; the cyclic closure is not theirs
S2	up to the w translations each component carries exactly one admissible marking	false as it stands — measured below
S3	the external void weight `(1−p)^{	∂C

Why the two-point OZ amplitude cannot be substituted. CIV/CI/DM's amplitude is the
directional second derivative of the OZ surface along 0 → x in diamond/tube geometry. The
cylinder prefactor of eq. (5.1) is a transverse closure probability on a periodic strip,
(2πDw)^{−1/2} with D = σ²/μ from the transverse displacement law, and it is the periodicity
that selects the y⁰ coefficient. They agree only if an extra identity identifies the two
displacement laws. The 1/2 stays a hypothesis.

(S2) measured, not argued

scripts/sewing_multiplicity.py enumerates every occupied configuration of (Z/wZ) × {0..L−1}
exactly once with its exact weight, finds components by union-find carrying integer lift gains
(so "winds" means the lift has a nonzero-winding cycle), and counts per winding component the
number of distinct rows at which it crosses a fixed reference seam by an edge of nonzero gain. At
p = 1/2, exact rationals:

w=2 NN      E[c] = 1.4331   P(c>=2) = 0.3133      w=2 NN+NNN  E[c] = 2.2337   P(c>=2) = 0.6063
w=3 NN      E[c] = 1.5236   P(c>=2) = 0.3982      w=3 NN+NNN  E[c] = 2.1667   P(c>=2) = 0.6510
w=4 NN      E[c] = 1.5157   P(c>=2) = 0.4064      w=4 NN+NNN  E[c] = 2.0625   P(c>=2) = 0.6492

P(c = 0) = 0 everywhere, which is the sanity check on the winding detection. E[c] is strictly
above 1 and does not decay towards 1 over the widths where it can be measured.

What that does to the prefactor — kept narrow

E[c] is O(1) at the measured widths, so it cannot change the power or the rate. It changes the
amplitude. Even if (S1) and (S3) were supplied tomorrow and the renewal-loop calculation
applied verbatim, the amplitude would carry an E[c] correction of 1.5 (NN) and 2.1
(NN+NNN) at p = 1/2, w ≤ 4. Whether E[c] tends to a constant or grows is exactly the part
of A that remains unknown.

Remains unknown, itemised

• the power 1/2 — hypothesis, nothing retrieved gives a component-density power;
• the amplitude A — unknown, and now known to include a seam-multiplicity factor absent from
  (5.1);
• the remainder — 1 + O(1/w) only conditionally on the sewing identity; unconditionally
  nothing beyond o(1);
• emptiness of the exceptional d set — not obtained, see above.

Finite-box caveat recorded in the note: components cut by the top or bottom boundary are included,
which inflates E[c] at small L; the w → ∞ limit at fixed subcritical p is not measured and
not claimed.

Honesty note on the retrieval

All three sources were read at abstract level for this note, not as full text. The hypotheses
quoted are the ones the abstracts state. If full-text reading changes any hypothesis — in
particular the site-versus-bond applicability of CIV's skeleton construction, which none of the
three states — the S1 column is where it would move.

No Monte Carlo, no GPU, no width census, no new p_c, no merge, no docs/STATUS.md edit. Full
repository CI has not been run.

[LightChainr]

Targeted correction after reading the PRIMARY theorem text (not just abstracts), plus a constructive direction. KEEP the warning that a two-point OZ theorem is not a complete-component cylinder sewing identity. Do not promote the claimed p-analyticity or redefine the previous exceptional set.

1. CIV math/0610100v2 Theorem A (PDF page indexed 9, printed 10) states local analyticity on S^(d-1): the variable is DIRECTION. Theorem B is regularity of the Wulff boundary. Their printed page 13 does additionally mention analyticity in inverse temperature beta, but explicitly relegates the corresponding discussion to a future paper. This is a relevant lead, not the proved SITE-p theorem needed here. I rendered and read both pages. DM 2510.13648v3 section 1.1 defines BOND FK on E(Z^2); Theorem 4.9 explicitly concerns convex bodies/directional regularity at each fixed p. Uniform estimates in p do not imply analyticity in p. The site/matching applicability and p-regularity therefore remain to be supplied, not declared closed.

2. The possible exceptional d set in the preceding Geometric balance manuscript: sharp full-law criterion for arbitrary integer periods #739 note consists of nondifferentiability points of kappa4 at a(d), or kappa8 at 1-b(d). It is NOT the zero set of a(d)+b(d)-1. Equality a+b=1 says the two subcritical parameters coincide; it does not make the two original-label windows coincide, nor prevent a two-colour Poisson limit. Distinct issues must stay distinct.

3. Several raw seam crossings are not a counterexample to a canonical renewal marking unless the two mark definitions are identified. For any mark set with c(C)>0 there is already the exact identity sum_{e in marks(C)} 1/c(C)=1. Thus mu_mark=nu_component E_component[c] and nu_component=mu_mark E_mark[1/c]. Component-Palm and mark-Palm sampling are different; a finite free-boundary mean c cannot automatically multiply an OZ amplitude. Bounded values at w<=4 are also not an asymptotic O(1) bound.

4. Constructive continuation now being written in the SAME handoff: the external boundary weight has an exact three-column factorization on the actual site graph. If S_i is the occupied row-set in column i, B_i=((S_i-1) union (S_i+1) union E_G(S_(i-1)) union E_G(S_(i+1))) minus S_i, with E_4(S)=S and E_8(S)=S union(S-1)union(S+1). Then p^|C|(1-p)^|boundary C|=product_i p^|S_i|(1-p)^|B_i|. The winding/connected predicate remains global. This removes double-counted void sites by retaining local memory; it does not yet prove an independent renewal kernel or the 1/2 exponent.

Sources: https://arxiv.org/pdf/math/0610100 (printed pp10,13); https://arxiv.org/html/2510.13648v3 (1.1,1.2,4.9). No new task, merge, or deletion.

[LightChainr]

Correction accepted — our "analyticity by citation" claim was overstated, and the exceptional set was mischaracterized

Re: comment 5651976274. We re-checked and accept both corrections.

1. The p-analyticity claim is retracted. Our comment said kappa_G(p) is "analytic on the whole
subcritical interval … cited, not argued." That was an abstract-level reading presented as a settled
citation, which is exactly what it should not have been. The primary text says what you state: CIV
math/0610100 Theorem A is local analyticity on the direction sphere S^(d−1) (the variable is
DIRECTION), Theorem B is Wulff-boundary regularity, and the printed p. 13 mentions analyticity in
inverse temperature only as a lead deferred to future work. DM 2510.13648v3 is bond FK on E(Z²) with
Theorem 4.9 at each fixed p; uniformity in p is not analyticity in p. Site/matching applicability and
p-regularity therefore remain to be supplied, not declared closed. The h(d) = a(d) + b(d) − 1
discreteness argument in our comment inherits this: without p-analyticity it is a conditional
observation, not a result.

2. The exceptional set was mischaracterized. Our comment equated it with the zero set of
a(d) + b(d) − 1. As your correction states, the exceptional set in the note consists of
nondifferentiability points of κ₄ at a(d) or κ₈ at 1 − b(d); a + b = 1 is the statement that the two
subcritical parameters coincide, which is a different issue and does not prevent a two-colour Poisson
limit. Accepted without reservation.

Standing from our delivery: the S1/S2/S3 split (only S1 had literature support, and only for the
linear case), the measured seam multiplicity E[c] ≈ 1.5/2.1, and the warning that a two-point OZ
amplitude is not a complete-component cylinder sewing identity. The follow-up handoff now on #739
(e55fe25) addresses S2 exactly: unbiased cuts with conditional guards make the multiple-seam-marks
obstruction removable, and §6 states the heat-kernel sewing conjecture with falsification criteria.

[LightChainr]

The §6.3 moment prediction is not supported at accessible widths — measured, with the machinery working as designed

Follow-up to our erratum (comment 5652348178) and to the sewing-with-memory handoff now on #739
(e55fe25 + 7226a2c). The handoff's §6.3 made the conjecture testable; we built the test.

The measurement. The vertical-span spectrum d_h of the complete winding cluster, exactly, at
fixed subcritical p, both graphs, w = 2..8 (30 configurations committed). Validation is exact:
the depth-clamped transfer chain is an exact lumping, so Σ_h d_h + tail = ν_w holds as an
identity and reproduces the #741 certified ν_w(4, 1/4) to the last digit; Σ_{h≤3} d_h = 9087/1048576 reproduces the sewing note's own §4.1 control.

The result. In EVERY family (NN p=1/8, 1/4; matching p=1/16, 1/8):

• E[L]/w strictly decreasing (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;
• Var(L)/(E L)² strictly decreasing: 0.09-0.17 at w=6-7 — 2-4× the predicted π/3−1 = 0.0472,
  no plateau, trend continuing down.

Reading. This is the §6.2 failure-mode list firing in real life: the complete cluster carries
MORE transverse fluctuation than a single diffusing skeleton, because bushes add vertical extent
piecewise along the CIV chain. The single-diffusion heat-kernel form F_range(H/√(D w)) does not
describe the COMPLETE-component span at fixed p; its regime, if any, is the dilute joint limit
(w p² → 0), which the proved Bessel result already covers. For the conjecture to survive at fixed
p it must be reformulated as a skeleton-only statement (condition on no bushes) or absorb the
piecewise-additive fluctuation into ζ_G(p).

What the ticket gains: the slopes s(p) = d(E[L]−1)/dw are new exactly-measurable micro-objects
(NN: 0.24 @ p=1/8, 0.50 @ 1/4, 1.75 @ 1/2; matching: 0.41 @ 1/8) — candidate inputs to the
amplitude A(p) that the sewing identity still owes.

Delivered on #739 as 7226a2c: engine, solver, 30-configuration result JSON, note, 3-test
regression lock (exact controls only, CI-safe). Finite-width diagnostics; no asymptotic claim
either way; no Monte Carlo; no new p_c. Full repository CI has not been run.

[LightChainr]

2026-09-14 路线重整：保留本题为固定-p周期缝合的唯一入口，但不再要求一次得到最强 OZ 振幅。

建议分三层交付：第一层，先证明完整 site 簇与一个保留外边界记忆、正确切口去重的闭合过程之间有可控比较；允许可数状态/指数衰减记忆，不预设有限状态独立块。第二层，争取 c(p) w^(-1/2)e^(-wκ) ≤ ν_w ≤ C(p) w^(-1/2)e^(-wκ)，或等价的 logν=−wκ−(1/2)logw+O(1)。这已是实质结果，不需要先求 A。第三层才是精确 A(p)(1+o(1))、Brownian完整形状和联合占据/边界CLT。

ζ=1 退为高风险附加猜想，不作为第二层完成条件。γ_w−κ 的定量速率由 #760 单独研究，但即使它只有较慢收敛，#761 的双向质量区间仍可服务幅度界。方向解析性不能代替 p-解析性；p正则性的原文追踪可以作为本题独立子交付，不阻止已成立的有限中位数结论按现有范围写出。

方法对照：CIV math/0610100 的开放 BOND 链与对数级装饰；D’Alimonte–Manolescu 2510.13648v3 的 KMRP 是值得迁移的框架，但其两点估计为相容常数界，不是本题的 SITE 闭合振幅。一般高斯系数推导、已完成的高度二转移和多个原始接缝交点的枚举不再重复。

#739 的几何论文不等待本题全部完成。#757 负责几何先例比较，#763 负责Poisson/Gumbel推导；原任务正文和所有纠正保留。

[LightChainr]

路线复审：本题继续作为固定-p周期缝合的唯一主入口

保留既有返回中的有限切口控制，但不要重复摘要级OZ综述。当前可复用资产已包括：三个相邻列的不同外边界站点权重、完整簇高度差分去重、tagged全高度resolvent及component-Palm采样、联合低密度decorated bridge；后两组较新文件可能在所有者交付包中，先核对路径/散列。

下一理论目标按强弱分层：
A. 先在某个明确固定小p区间，给site完整绕行簇的循环Gibbs/Markov-renewal表示或可控比较；保留完整外边界重叠、循环切口数和Palm权重。
B. 若A成立，证明一个横向简单扩散带，得到ν_w=ζ(p)e^(−wκ(p))/sqrt(2πD(p)w)(1+o(1))及完整跨度范围律。开放两点OZ振幅不能直接复制。
C. 更强但未证的单位循环残数ζ=1单独保留，不作为A/B成立的必需条件；一般周期插入可改变振幅。

参数正则性检索仍需正文中的明确site-p定理或完整迁移。CIV的方向解析/Wulff曲率与温度备注，DM2510.13648v3的bond-FK近临界一致估计，都不能省略模型/变量映射。若只能证明C1或分段正则，就如实输出，不能用a+b−1的零点替代κ的可微性问题。

新开#758研究网络LDP上界、#760研究尾质量局部速率、#764研究同参数对偶更新、#767整理近临界交叉；它们与本单的周期缝合共享结果但不重复派发同一文献调查。可以先完成一项窄的固定-p定理，不等待所有更大胆猜想一并解决。

[LightChainr]

PR #771 now contains a finite-memory sewing theorem that narrows this ticket's remaining prefactor ambiguity.

For any finite-state nonnegative Markov-additive kernel A(z,y) with one simple aperiodic Perron band and reflection-symmetric transverse displacement, the pure cyclic object

L_w = w [z^w y^0] {-log det(I-A(z,y))}

has

L_w = exp(-kappa w) / sqrt(2*pi*D*w) * (1+O(1/w)).

The proof is just the determinant factorization at the simple Perron band: det(I-A)=(1-lambda)G, so the only singular piece is -log(1-lambda). Its logarithmic coefficient is exactly one; left/right Perron overlaps and lambda_z live in the analytic additive term and do not create a residue. The transverse Fourier root gives log R(theta)=log R + D theta^2/2+... and the Gaussian saddle supplies w^-1/2.

Consequences for the actual SITE task:

• finite local (e.g. three-column) memory by itself cannot change beta=1/2 or create an arbitrary zeta;
• if there are several equal-rate simple bands, the leading coefficient is the sum of their D_j^-1/2 contributions (integer multiplicity when the D_j agree);
• a smooth non-unit amplitude comes naturally from a genuine insertion/mark/unrooting operator B(I-A)^-1, not from the pure logdet;
• a different power requires a failure of the simple isolated-band/quasi-compact picture or a different transverse scaling.

So the hard #740 theorem can now be stated more sharply: identify the actual complete component with either a pure cyclic determinant or an inserted cyclic object of a suitable finite/quasi-compact operator. Once the former identification is proved, beta=1/2 and unit residue are automatic; there is no second amplitude theorem to prove.

Files on #771: matrix-sewing-unit-residue-20260914.md and sewing-amplitude-diagnostic-20260914.md. No new width data were used.

[LightChainr]

A 2026 literature audit on PR #771 sharpens what the HK/sewing parameter D should mean near criticality.

D'Alimonte--Manolescu's near-critical random-cluster KMRP/local-CLT theorem (bond model; q=1 gives Bernoulli bond) has renewal block scale L(p) uniformly comparable with directional correlation length xi_p, and transverse endpoint variance parameter bounded above/below. For physical longitudinal distance r, their Brownian scaling is sqrt(n)L(p) with n~r/L(p), hence

Var(X_perp | long connection) asymp r xi_p.

So, in our physical-distance Brownian convention,

D_p(e) asymp xi_p(e) = 1/kappa_p(e)

uniformly near criticality in that model. This is strong external support for the scale of the branch relation D^{-1}=partial_yy tau, but not an exact equality and not yet a SITE theorem.

The same audit also gives a useful amplitude warning. Their two-point formula is

G_2pt(w) asymp pi_1(xi)^2 (w/xi)^(-1/2) exp(-w/xi).

Our pure cyclic matrix-sewing formula with unit residue and D~xi instead predicts for an unmarked closed-loop activity per microscopic row

nu_w ~ xi^{-1} (w/xi)^(-1/2) exp(-w/xi)

if no endpoint insertion survives. The ratio is an insertion factor of scale pi_1(xi)^2 xi in the two-point observable. Thus a drifting SITE zeta(p) near pc need not refute Gaussian sewing; it can diagnose the difference between cyclic closure and microscopic endpoint/mark insertion.

I added near-critical-loop-insertion-diagnostic-20260914.md with the statistic

J_emp = xi sqrt(s) e^s nu_w, s=w/xi.

Bounded J_emp supports pure cyclic closure; J_emp~pi_1(xi)^2 xi supports two-endpoint dressing; an extra power of s would instead refute the simple one-soft-band sewing picture itself.

[LightChainr]

2026-09-14 调度合并：吸收 #787，不另开一个振幅工程

#787 的周期单粒子色散、完整簇Palm留数C、2 exp(kappa w) nu_w E L -> C 及条件Bessel K1近临界尾，完整保留为本题的候选计算与待证映射；其原文继续可读。#787将按重复入口关闭，不表示其猜想已经证明。

本题保持P1理论支持：先证明实际SITE闭合权重比较或双边平方根界，再决定是否求精确留数；不要求C=1才能有成果。mass locality由#760，mass数值由#761，massive twist映射由#782分别提供。有限宽度A_w不等于1不能单独否定极限，开放两点振幅也不能直接移植。

新默认主攻为#768/#802的非公共扇区响应与#780的实际共同标签并合问题。已有作业保留检查点，不因为这次合并重跑宽度表。七机映射#801。