analysis: span spectrum shows a coalescing effective slow doublet

[LightChainr]

Scope

Small stacked diagnostic PR on top of #771. It uses only the already committed span-spectrum-20260913.json; no transfer matrix is rebuilt and no new width production is run.

Main finding

The complete-component span response does not rapidly relax to a single visible Perron pole as circumference grows.

A frozen projected two-pole/Prony extraction on the late d_h sequence finds, for NN p=1/4, two positive slow visible poles that rapidly coalesce while their scalar source/readout residues are nearly equal and opposite.

Representative relaxation lengths xi_rel=1/log(rho1/rho2):

w=3   12.0
w=4   33.4
w=5   84.1
w=6  203.7
w=7  629.4

so xi_rel/w grows roughly 4,8,17,34,90. The same near-opposite slow doublet is present in stored NN p=1/8 and matching p=1/16 controls. Direct tail hazards also show that h=w,2w are not yet at the fixed-w ultimate pole mass.

Scientific consequence

This does not falsify either the #758 simultaneous h=A w loop/branch LDP candidate or the #760 ultimate cylinder-mass locality conjecture. It falsifies a shortcut between them: do not first take h->infinity at fixed w and assume visible one-Perron relaxation o(w).

A simultaneous-limit description with a coalescing slow subspace is more appropriate.

Two-ended branch mechanism

span-double-pole-branch-convolution-20260914.md gives a more specific mechanism than generic metastability. In the saturated loop/branch regime, write

L=L_core+A_-+A_+.

If the two far one-sided branch excursions have simple near-equal poles, the span generating function contains their product. Its coefficient is a difference of two exponentials with opposite residues; in the equal-pole limit it becomes (a+b h)rho^h, i.e. an Erlang/double-pole prefactor.

This produces a direct #762 prediction: if the two excess branches become independent equal-rate exponentials, then conditional on total excess span

U=A_-/(A_-+A_+) -> Uniform(0,1).

Thus the pole diagnostic, #758's saturated linear branch, and #762's planned U morphology test become one mechanism-level hypothesis. Endpoint condensation would falsify this two-ended interpretation.

Tentative splitting conjecture

The finite-width pole splitting itself appears exponentially/super-polynomially small. For NN p=1/4 it is numerically comparable to w nu_w, but the ratio is not universal across graph/parameter controls. No universal splitting amplitude is claimed. Direct tagged-operator eigenvectors/source overlaps are the next decisive object.

Files

Adds 4 files:

• docs/manuscripts/geometric-balance/span-visible-pole-splitting-20260914.md
• docs/manuscripts/geometric-balance/span-double-pole-branch-convolution-20260914.md
• scripts/span_visible_pole_splitting.py
• results/geometric-consistency/span-visible-pole-splitting-20260914.json

No existing files modified; no new transfer build, Monte Carlo/GPU, STATUS or original-U change.

[LightChainr]

possible operator-level interpretation of the slow doublet

A cross-branch synthesis is now on #771: rare-topology-fugacity-and-mass-clock-20260914.md (cd33d5837c4b91c34c935fdff92cb81ed53638f4).

The concrete conjecture for #800 is stronger than “the poles coalesce exponentially”:

lim_w -w^-1 log Delta gamma_w = kappa(p),

with the optional stronger refinement

Delta gamma_w/[w nu_w] -> c_split(p) in (0,infinity)

if the same one-winding renewal event supplies the off-diagonal tunnelling matrix element. The current NN p=1/4 ratios 1.7--1.9 are consistent with the stronger form, but I would not fit the constant yet.

A minimal effective block is

R_w = rho_bar,w [I - epsilon_w L_w + o(epsilon_w)],
epsilon_w=exp[-w kappa(p)] * poly(w),

with a source/readout approximately antisymmetric between the two slow states. This automatically produces the nearly opposite residues. The high-value next object is therefore not another Prony width: directly identify the two tagged eigenvectors and ask whether the matrix element coupling them is supported on states/transitions carrying one additional essential/wrap witness.

This also predicts the doublet should stop being parametrically slow when w kappa=O(1) in the near-critical crossover, the same place where #780's pure-fragmentation no-merger argument loses its small parameter. That shared breakdown scale is more informative than a universal finite-p splitting ratio.

[LightChainr]

修正我上一条 lim -log Delta gamma/w = kappa 的表述

上一评论把 slow doublet 直接放在 one-extra-witness kappa action上，证据不足。#800 的 Delta gamma 是 projected spectral splitting；Delta gamma/(w nu)=O(1) 在 p=1/4 很 suggestive，但没有识别实际 tunnelling transition，更没有概率 BK 下界/上界配对。

新 #771 topological-action-spectrum-beyond-witness-count-20260914.md (60d70ab1) 改用

sigma_split = liminf -w^-1 log Delta gamma_w

并与 sigma_wrap,sigma_merge 分开。当前只把 sigma_split≈kappa 保留为候选。

因此下一步仍是原建议中真正高信息的部分：直接取 tagged operator 的两个 slow eigenvectors / left-right overlaps，定位 off-diagonal coupling来自哪些 frontier states/transitions。若能把 coupling 支撑映射到一个明确 constrained winding/network defect，再用 Wulff action预测 sigma_split；否则不要从 Prony scalar sequence把它先叫 one-witness tunnelling。

近反号 residues仍强烈支持“source/readout近 antisymmetric slow subspace”的现象学，但不决定其 topological action。