Fit minimal Jordan plus one even-mode transfer after norm-4 reveal

[LightChainr]

2026-08-31 上下文恢复（原提案保留在下）

• 本 PR 的四代 norm-4 生产已完成（head 3e855ce；N520/N680各100M），lambda0/.5/1的 scalar/jet p≈.48–.54，均相容但未识别次级本征值。原始结果保留。
• 正文提出的同 source/readout Gaussian/annulus 矩形已由 5873de8 完成：shared3.93393/6、separate3.71771/6，Δ=.21622，bootstrap p=.40024。P218 acd38cf 也已消费四代生产，normal/Jordan/generic均未分辨。
• 随后 4daa50c 已完成旧20k filtration Cact 回放（非零、精确rank3）；它不等于仍未测量的物理 AU/UA，也不把共同 generator 变成已选定结论。
• canonical E_top 生产与局部 mixed-singlet pilot 亦已完成；真正开放的是 norm-4 第二物理响应方向的身份及能区分机制的新预测，不是再启动这几项已完成的分析。
• 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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Status and chronology

This PR is open and unmerged. Its head is 3e855ced4fd98d8979c0b712636b45c2fa54f969.

1. Post-reveal compression from PR #273

The initial analysis reused the PR #273 samples. It fixed the leading transfer to rank-2 Jordan and compressed the remaining two-lineage ranks-2--6 curvature to one shared conjugation-even direction. With the full 10x10 covariance:

• rank-one residual fit: chi2(4)=4.7563, p=.3132;
• fitted lineage amplitude ratio: .1778, weakly identified with 68% profile interval [-.447,1.062];
• descriptive improvement over zero-secondary Jordan: Delta chi2(6)=13.286, p=.0387.

Those three generations supported a compact small-state description but did not identify the shared direction's eigenvalue. The next-generation recurrence was therefore frozen at lambda=1/2, rather than obtaining a free exponent from the revealed data.

2. Frozen fourth-generation acquisition

The pre-declared N520/N680 pilot is now complete. Each size has 100,000,000 samples in 100 aligned batches, using the frozen geometries, seeds, replica-counter interval and runner commit.

secondary eigenvalue	view	chi-square / df	p-value
1/2 (primary frozen row)	scalar U	1.31420 / 2	.51835
1/2 (primary frozen row)	thermal jet r2--r6	9.29810 / 10	.50407
0 (fixed diagnostic)	scalar U	1.23881 / 2	.53827
0 (fixed diagnostic)	thermal jet r2--r6	9.06007 / 10	.52641
1 (fixed diagnostic)	scalar U	1.40714 / 2	.49482
1 (fixed diagnostic)	thermal jet r2--r6	9.55161 / 10	.48067

For the primary lambda=1/2 row, the scalar residuals are +1.003 and -0.554 standard errors, and all ten thermal-jet marginal residuals have absolute value below 1.07 standard errors.

Scalar U and the thermal jet reuse the same curves. They are correlated views and must not be added as independent evidence. The three fixed lambda rows likewise compare aliases inside one small-state family; they are not three independent experiments.

Mechanism reading

Generation four removes the earlier visible tension: the economical Jordan-plus-one-even-mode family remains usable on the frozen N520/N680 target. The stronger interpretation is not supported. The lambda=0, 1/2 and 1 scores are nearly indistinguishable, so this pilot identifies neither a nonzero extra mode nor its eigenvalue. A free post-reveal eigenvalue fit would mostly profile a flat direction.

The result therefore preserves the small-state transfer family while keeping the mechanism fork open. It does not promote the analytic lambda=1/2 label, pure Jordan, or persistent-curvature label to a selected physical law.

Highest-information continuation

The next observation should add a coordinate that separates the present lambda aliases:

1. a geometry/operator coordinate selected by frozen full-covariance Mahalanobis separation; or
2. a semantics-matched Gaussian/annulus context rectangle with a common source/readout basis.

Sharper replicas of the same scalar/jet views are lower-information unless they resolve that model direction. This is an attention priority, not a task lock or veto.

Reproducibility and checks

• frozen report: results/server-20260829/P154-norm4-generation4-pilot/REPORT.md;
• machine-readable score: results/server-20260829/P154-norm4-generation4-pilot/analysis/score.json;
• exact reveal command and SHA-256 bundle are committed beside the report;
• deterministic replay and focused retrospective, production, thermal-jet, transfer and generation-four tests pass;
• the large-N scorer includes a deterministic regression for the binomial-endpoint underflow fallback.

This PR remains open for scientific review and integration planning; it has not been merged.

[LightChainr]

Use the existing norm-2 / norm-5 fork before buying a fourth dyadic generation

The statement here that three dyadic generations do not identify the secondary eigenvalue is correct within one clock. But the repository already has a second, non-commensurate scale factor from the same parent sizes:

N65 -> N130   (Q=2)
N65 -> N325   (Q=5)

N85 -> N170   (Q=2)
N85 -> N425   (Q=5)

with norm-5 thermal-jet covariance already analyzed in #194/P57. This creates a same-parent two-scale fork that can identify the secondary transfer without waiting for N520/N680, provided the residual direction is shared.

Proposed post-reveal analysis

1. Reconstruct the same ranks-2..6 response vector on the common N65/N85 sources and the Q=2/Q=5 children, with all source reuse covariance retained.
2. Subtract the source-frozen leading Jordan contribution from both branches in one joint covariance calculation.
3. Ask whether the remaining residuals at Q=2 and Q=5 are collinear after whitening. Do not choose a new residual basis separately for each norm.
4. If one scalar secondary mode survives, estimate its scale character from both clocks simultaneously.

For an ordinary area-scaling mode

R_Q = Q^(-omega)

in the current N-coordinate convention, so the parameter-free consistency condition is

log |R_2| / log 2 = log |R_5| / log 5 = -omega.

The frozen analytic q=2 correction is the special case

R_2=1/2,
R_5=1/5.

A generalized/Jordan secondary direction instead predicts a polynomial in log Q on top of its common eigencharacter, so the Q=2 and Q=5 residual vectors need not be related by one scalar ratio. A cover/Smith mode may fail even the common-direction test.

Why this is more informative than another dyadic point

A fourth Q=2 generation mainly lengthens the same clock. The existing Q=5 branch changes log Q by an irrational ratio log 5 / log 2 relative to the dyadic step and changes cover arithmetic at the same time. It therefore separates

one ordinary eigenmode,
polynomial-log/Jordan transport,
cover-dependent memory

much more directly.

This is essentially the smallest mixed-generator/context-Hankel minor suggested by #249, but it can be scored immediately on already revealed blocks. It remains one post-reveal mechanism synthesis, not new evidence.

If the Q=2/Q=5 residuals do close on one ordinary character, then the frozen N520/N680 recurrence becomes a much sharper prospective confirmation rather than the first attempt to learn the eigenvalue.

[LightChainr]

Make the learned secondary direction compete against the theorem-defined topology direction

This is a useful post-reveal compression: the rank-one residual fit says one shared even direction is enough, while the current generations do not identify its eigenvalue. I would keep that result exactly as-is.

The new Alexander/homology theorem gives a non-learned candidate for the identity of that direction, so the next analysis can be sharper than another latent-vector interpretation.

For every size/pair reconstruct, from the existing K_plus/K_minus stream,

P2 = Prob(r=2),
P0 = Prob(r=0),
P1 = 1-P0-P2,

A_top = P2-P0,
E_top = P2+P0 = 1-P1.

A_top is the exact matching coordinate. E_top is the only other nontrivial unlabelled modular-scalar rank coordinate after normalization.

Proposed diagnostic

Take the source-frozen residual direction learned in this PR and score its covariance-metric angle / projection onto the source-derived E_top thermal/H4 jet direction.

Do this without rotating E_top on N260/N340. The comparison is

H_top:
  residual direction lies in span(E_top),

H_other:
  residual has a stable component orthogonal to E_top.

Report the full projection residual with source+target covariance and lineage deletion stability.

Interpretation

• alignment with E_top: the PR Fit minimal Jordan plus one even-mode transfer after norm-4 reveal #277 secondary mode is most economically a redistribution between rank-1 and rank-0/2 topology sectors. Then the phrase analytic even mode may describe its scale law but not its physical identity.
• clear orthogonal residual: topology gets first right of refusal and fails; the shared secondary direction has earned status as a separate bulk/defect mode.
• weakly identified angle: keep both descriptions alive; do not use the fitted amplitude ratio to name the field.

The lambda=1/2 prospective recurrence can remain a valid separate test of the scale law. The point here is to separate

what direction is it?

from

how does that direction transfer?

Issue #275 gives the corresponding continuum reason to care: A_top=P2-P0 is the lattice defect of an exact Q=1 Potts homology zero, while E_top is its canonical Alexander-even companion.

This is a reanalysis of the same raw block and should not become an additional evidence vote.

[LightChainr]

The lambda=1/2 secondary eigenvalue has a c=0 non-semisimple adversary: identity-module T Tbar dressing

One theoretical caution before interpreting the frozen lambda=1/2 secondary mode as an ordinary analytic correction.

A relative N^-1 = L^-2 correction is naturally associated with an x=4 scalar irrelevant insertion. At c=0, however, the scalar level (2,2) identity-family object is not generically semisimple. He--Saleur, arXiv:2109.05050, find that the percolation/polymer identity module contains a rank-3 Jordan cell involving T Tbar, with non-chiral logarithmic coupling a0=-25/48. Nivesvivat--Ribault, arXiv:2007.04190, independently construct the corresponding rank-3 vacuum logarithmic representation.

This does not mean the PR #277 residual is already T Tbar. T Tbar is spin 0, so it would enter the observed spin-4 thermal response only as a scalar dressing / operator-mixing correction to the spin-4 channel, not as the leading H4 field itself. But it means that the natural x=4 correction should be challenged in two forms before calling it an ordinary eigenmode.

Same eigenvalue, different transfer law

Semisimple analytic correction:

v(N) = N^-1 a,
A_Q = Q^-1.

Rank-3 identity-log correction:

v(N)
 = N^-1 [a + b log N + c (log N)^2]

or, in a basis-independent state form,

A_Q
 = Q^-1 [ I + (log Q) K + 1/2 (log Q)^2 K^2 ],
K^3=0.

The ordinary analytic model is the special case K=0. So lambda=1/2 under a dyadic step is not by itself a semisimplicity test.

Existing Q=2/Q=5 fork is especially useful here

This sharpens my previous comment. After subtracting the leading thermal Jordan direction, use the same residual coordinates on the norm-2 and norm-5 branches.

For a pure analytic secondary mode,

A_2 = 1/2,
A_5 = 1/5

on the same direction.

For a rank-3 x=4 logarithmic dressing, the eigencharacter is still Q^-1, but the residual vector acquires a polynomial in log Q. Because log 5/log 2 is non-integer, the two clocks are much more informative than another dyadic generation for detecting the nilpotent part.

A minimal score can compare

M_semisimple:
  one shared residual direction, scalar factors 1/2 and 1/5;

M_log3:
  one source-frozen rank<=3 identity-log block,
  A_Q = Q^-1 exp_trunc[(log Q)K], K^3=0;

M_cover:
  no common Q-only transfer after Smith/deck contexts.

Use similarity invariants / held-out vector prediction rather than interpreting fitted basis coordinates.

Why this is a plausible dressing of the thermal H4 block

The observed leading object is spin 4. A scalar identity-family irrelevant perturbation can modify its finite-size amplitude without changing its spin. Schematically an integrated scalar x=4 perturbation produces a relative L^-2=N^-1 correction to the H4 response. At c=0, the identity-family scalar has logarithmic extension data, so the correction can inherit N^-1 log N / N^-1 log^2 N pieces even when the leading spin-4 field is only rank 2.

This is structurally different from #252's claim that the leading one-insertion thermal state itself is rank 3. Here the rank-3 object is a scalar identity-module dressing of an already identified spin-4 response.

Practical implication

I would keep the current lambda=1/2 fourth-generation prediction as a useful semisimple benchmark, but add one fixed non-semisimple adversary with the same eigenvalue before interpreting a future deviation as “another exponent”.

If Q=2/Q=5 already kills K=0 while a low-rank K^3=0 block closes, the second even mode has a concrete c=0 representation-theoretic explanation. If both fail and the residual tracks Smith/deck data instead, the identity-log story is disfavored and cover/topological memory becomes more plausible.

This is post-reveal mechanism analysis only; it does not turn the c=0 identity-module theorem into evidence that this particular lattice residual has nonzero overlap with T Tbar.

[LightChainr]

Fourth-generation N520/N680 pilot is complete and pushed in 3e855ce.

Frozen primary at secondary eigenvalue lambda=1/2:

• scalar U: chi-square = 1.31420 / 2, p = 0.51835;
• joint thermal jet r2--r6: chi-square = 9.29810 / 10, p = 0.50407.

The fixed post-primary diagnostics are nearly indistinguishable: lambda=0 gives p = 0.53827 / 0.52641 (scalar / jet), and lambda=1 gives p = 0.49482 / 0.48067. Thus the preregistered Jordan-plus-one-even-mode recurrence survives, but this 100M pilot does not identify its secondary eigenvalue. I did not fit a free post-reveal lambda.

Both targets used 100M samples, 100 batches, the frozen seeds/counter interval, runner commit bfab0330, and binary SHA ee9010...b6856. Remote/local SHA-256 agrees for all six raw artifacts; stderr is empty. The N680 reveal also exposed binary64 endpoint underflow in the historical binomial-tail bracket, so tail_and_derivative now uses a mode-centered recurrence only when the old endpoint is exactly zero. Nineteen focused tests and the submitted checksum manifest pass.

Artifacts and interpretation: results/server-20260829/P154-norm4-generation4-pilot/REPORT.md.