[P2 analysis] Test emerging Jordan geometry via response-vector coalescence

[LightChainr]

2026-08-31 上下文恢复（原提案保留在下）

• 真实 production 的 coalescence 分析已完成。 acd38cf（branch_only，analysis/p218-production-coalescence-elimination-20260830）使用 P154/277 四代档案的冻结 H/U 坐标与协方差，不是只有 synthetic Exact: add a rational Jordan geometry oracle #382。
• source/held-out determinant 仅 .645/1.458sigma；N520/N680 上 normal-diagonalizable .3839/4、Jordan1.059/4、generic .2614/4均未分辨。它是完成了的低信息结果，不是已证明或已排除 physical Jordan。
• 三个有限模板不等于互斥物理全集：generic2×2 也可对角化；PR Exact identifiability controls: Jordan closure and a C4-protected five-state quotient #385 的近碰撞普通族还揭示有限噪声下的非分离边界。后续需有真正不同的结构/观测预测，不能靠更精细重复同一特征值拟合宣称唯一机制。
• 当前总览与下一步见 Draft docs: recover scientific frontier and score production mechanisms #267 的上下文恢复与注意力顺序；原提案全文保留在下。

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Motivation

The post-P57/post-N290 state makes a rank-2 Jordan interpretation plausible but still only at the level of finite-size functional fits. A stronger diagnostic is available from the Potts/loop literature.

Liu, Jacobsen and Saleur, “Emerging Jordan blocks in the two-dimensional Potts and loop models at generic Q”, arXiv:2403.19830, emphasize that the lattice transfer matrix can remain diagonalizable at every finite size while the continuum limit develops a Jordan block. The mechanism is not merely “a logarithm fits”: two finite-size scaling states become degenerate and their eigenvectors coalesce as size grows.

That suggests a sharper test for #180/#200:

if the live S' structure is genuinely an emerging logarithmic block, the smallest covariance-aware finite-size response operator should show eigenvalue collision together with response-vector/eigenvector coalescence, not only an affine log N scalar law.

This is an analysis diagnostic, not a claim that the empirical curve-transfer operator is literally the microscopic Potts transfer matrix.

Existing data first

Use the structured Krawtchouk/Hermite state coordinates from #182 and the already archived norm-2, P57 norm-5, and N290 full-curve blocks. Do not choose a new PCA basis on the target.

Start with the smallest source-stable two-dimensional block containing the leading thermal H4 direction and the live S' correction direction after the already-declared thermal metric/width coordinates are removed.

For each admissible finite-size transfer estimate T, report all of the following jointly:

1. eigenvalue gap

   g = |lambda_1-lambda_2| / max(|lambda_1|,|lambda_2|)
   

2. eigenvector coalescence / principal angle in the covariance-whitened metric;

3. eigenbasis condition number (or an equivalent non-normality diagnostic);

4. Jordan minimal-polynomial residual after optimizing only the common eigenvalue on the source block,

   J2 = ||(T-lambda I)^2||_W
   J1 = ||T-lambda I||_W
   

   together with J2/J1;

5. predictive score of three fixed classes:

   two distinct diagonalizable eigenmodes
   one rank-2 Jordan block
   one parameter-capped generic 2x2 mixing block
   

The purpose is not to maximize reconstruction accuracy. The question is whether increasing size produces the joint signature gap -> 0 and angle -> 0 while the nilpotent/Jordan residual improves.

Chronology / leakage boundary

N290 and P57 are already revealed, so the first pass is explicitly post-reveal mechanism analysis. Do not relabel it as new evidence.

Use source-block deletion and lineage deletion only as stability diagnostics. Then freeze the coalescence statistic and its expected direction before any unrevealed norm-4 N260/N340 target is opened.

Prospective norm-4 implication

Norm 4 is unusually useful because

T4 = T2^2

is an exact semigroup composition target and the proposed conjugation-odd sine component is at a phase node.

If a rank-2 block has

T_Q = Q^-alpha (I + log(Q) K),   K^2=0,

then the nilpotent part must satisfy the composition law exactly:

K contribution at Q=4 = 2 * K contribution at Q=2

in the same frozen response basis. Score this matrix identity together with the existing scalar R_J / R_q2 views, not as an independent vote from the same histograms.

Decision rule

Supports an emerging-Jordan interpretation

• the suspect eigenvalues approach each other;
• the corresponding response directions coalesce rather than remain stably separated;
• the eigenbasis becomes increasingly ill-conditioned in the expected two-dimensional block;
• a rank-2 minimal-polynomial/Jordan residual closes across lineages and later under T4=T2^2;
• these features survive the cyclic/noncyclic quotient control.

Weakens the Jordan interpretation

• A+B log N looks acceptable but the response directions remain stably diagonalizable and separated;
• the apparent coalescence depends strongly on basis truncation or one raw block;
• norm-4 composition fails while a regular diagonalizable mixing model remains stable;
• the residual direction follows Smith/quotient class rather than scale.

Relation to current work

• [Speculative synthesis] Learn a finite-size RG transfer operator from the full threshold curves #180 supplies the low-rank/full-curve transfer framework.
• [Speculative analysis] Diagonalize threshold-rank response in log-odds / Krawtchouk score modes #182 supplies the exact score-mode coordinates.
• [P2 analysis/protocol] Two-generator thermal transfer after norm-5 and N290 #200 supplies the minimal two-generator restriction and norm-4 gate.
• [P2 control/analysis] Same-parent conjugate norm-5 coalescence tomography #205 supplies the strongest same-N cyclic/noncyclic arithmetic control.

This issue adds a Jordan-specific geometric diagnostic to those programs. It should reduce the risk of calling a descriptive logarithmic correction a logarithmic representation without the characteristic state coalescence.

Literature anchors

• Liu, Jacobsen, Saleur, arXiv:2403.19830 — emerging Jordan blocks at finite size.
• Grans-Samuelsson et al., arXiv:2007.11539 — Virasoro indecomposable/Jordan structure in Potts and loop models.
• He, arXiv:2411.18696 — rank-2 and higher-rank logarithmic multiplets at c=0.

Deliverable

A no-new-compute first implementation should produce a machine-readable table of eigenvalue gaps, covariance-metric principal angles, eigenbasis conditioning, minimal-polynomial residuals, and predictive model scores. Only after that table is frozen should norm-4 be used as a prospective composition test.