[P2][THEORY/ANALYSIS] Extract the matching-odd RG tangent by thermal orthogonalization in the exact self-matching family

[LightChainr]

2026-08-31 上下文恢复（原提案保留在下）

• 精确 N10 odd tangent、wrapping rank1 和 local nonparallel controls 已完成；更后面的 injective R8 general-period thermal-null pilot 也已实际运行（c679c32，branch_only）。
• N260/N340 各20k的 R8 lambda z=.218/−.088，条件数约189/384；R4的可见信号主要 thermal-parallel。该冻结 readout 尚未分辨独立 odd irrelevant 方向，不等于所有微观 odd tangent 都不存在。
• 后续需改变能分辨 thermal 与新方向的具体 source/readout/几何或提出对应数值预测；不再把同一 exact tangent、N130/N170首测或 N260/N340 R8 pilot 当作未完成的接口准备。
• 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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New opportunity from PR #148

The C4 self-matching checkerboard family now supplies an explicit two-parameter microscopic odd tangent:

p_even = 1/2 + t + lambda
p_odd  = 1/2 + t - lambda

with exact complement action

(t,lambda) -> (-t,-lambda).

Thus the linear matching action on this microscopic tangent plane is exactly J=-I. This is much closer to the object sought in Issue #61 than an abstract OPE-parity argument: we now have a concrete two-dimensional matching-odd UV tangent space that can be followed toward the fixed point.

The existing PR #148 focuses mainly on searching an improved matching-even H4 amplitude. A second, largely zero-extra-compute program should instead ask:

How does the two-dimensional odd tangent (t,lambda) decompose into the relevant thermal eigenvector plus the first matching-odd irrelevant direction under RG?

Score-function derivatives at the exact center

At t=lambda=0, all sites are independent Bernoulli(1/2). Let

K_e = occupied count on even sublattice
K_o = occupied count on odd sublattice
N_e, N_o = corresponding site counts.

For any configuration observable O, the exact likelihood-score identities are

d_t <O> = < O * S_t >
d_lambda <O> = < O * S_lambda >

with

S_t      = 4[(K_e-N_e/2) + (K_o-N_o/2)]
S_lambda = 4[(K_e-N_e/2) - (K_o-N_o/2)].

So one ordinary p=1/2 configuration stream can measure both tangent responses with no finite-difference bias and no lambda grid.

PR #148 supplies an exact N=10 oracle for the central odd wrapping response:

d_t R_minus      = 15/8
d_lambda R_minus = 5/4
ratio             = 2/3.

The first implementation should reproduce these exact rational derivatives from the score estimator before any larger simulation.

RG decomposition

The uniform t direction contains the thermal relevant field, with

y_t = 3/4.

The staggered/local-basis lambda direction is also matching odd, but it need not be an RG eigenvector. Write schematically

O_lambda = c_t O_thermal + sum_a c_a O_a^(odd,irrelevant).

For a dimensionless critical torus observable, define the finite-size tangent ratio

c_N = (d_lambda <O>)/(d_t <O>).

If c_N -> c_*, construct the thermal-orthogonal response

R_perp(N) = d_lambda <O> - c_* d_t <O>.

Operationally, do not estimate c_* on the same target block. Use a source-size rule, for example:

1. estimate/freeze c_* or a simple asymptotic model from smaller compatible tori;
2. predict the residual on larger held-out tori;
3. fit an effective RG exponent only after a fixed-power/model challenge.

A nonzero residual with a stable negative RG exponent would be direct evidence that the exact matching-odd UV tangent contains an irrelevant continuum direction in addition to the thermal field.

Stronger two-observable version

One scalar topological observable only measures one projection of the tangent plane. When feasible, add a second independent odd readout, for example:

• another topological/homology channel after the canonical [P2 exact/protocol] Canonical wrapping-channel algebra and type-safe scorer semantics #146 algebra proves independence;
• a local energy/sublattice observable with known transformation;
• an orientation-resolved odd H4 projector on compatible N=130/170 tori.

Then form the 2x2 response matrix

R_N = d(observable vector)/d(t,lambda)

and study its singular/eigen-directions across N. The goal is not a free matrix fit but identification of one growing thermal direction and one decaying odd direction.

Why this matters for the x=21/4 program

This experiment does not assume that the orthogonal odd direction is the measured spin-4 x=21/4 field. Instead it attacks the missing structural step in #61:

exact microscopic matching involution
        -> RG tangent grading
        -> continuum odd/even blocks.

Possible outcomes are highly informative:

A. lambda is asymptotically parallel to thermal

c_N stabilizes rapidly and the orthogonal residual is tiny.

Then this particular local tangent gives no accessible independent odd irrelevant field; the square-site H4 sector must arise from a different microscopic perturbation/readout.

B. a clean irrelevant residual appears

Then the self-matching model provides an explicit matching-odd irrelevant RG direction. Its spin should be measured separately with orientation projection before comparing its dimension to x=21/4.

C. more than one residual power is required

Then the odd tangent block is already visibly multi-dimensional at accessible scales, supporting the matrix/generalized-eigenvector picture rather than a scalar parity assignment.

Cheap execution plan

1. Extend the PR Construct exact self-matching tangent family #148 exact N=10 enumerator/report to output the score-function estimators and verify 15/8, 5/4 exactly.
2. On compatible N=130 and N=170 self-matching tori, run p=1/2 only; record K_e,K_o sufficient statistics together with the declared topological/orientation observables.
3. Use aligned batches to retain covariance of d_t and d_lambda.
4. Report scaling of both raw responses and the source-frozen thermal-orthogonal residual.
5. Only then decide whether N=290 or another size is worth adding.

This should be substantially cheaper than scanning a two-dimensional (t,lambda) grid.

Relation to improved-action search

The matching-even H4 zero search in PR #148 and this odd-tangent spectroscopy answer different questions and can share samples/sufficient statistics. Do not let one overwrite the other.

Claim boundary

An observed odd irrelevant tangent would establish an empirical RG-direction decomposition for this self-matching lattice. It would not by itself prove a full matching/OPE automorphism or identify the square-site x=21/4 field.

Related: #44, #61, #100, #118, #121, #125, PR #148.

[LightChainr]

There is an exact normalization/orthogonality baseline worth building into this RG-tangent program.

At t=lambda=0, with equal even/odd sublattice sizes, the Bernoulli score vectors are

S_t=4 sum_i (X_i-1/2),
S_lambda=4 sum_i sigma_i (X_i-1/2).

Their Fisher matrix is exactly

I_tt=I_ll=4N,
I_tl=4 sum_i sigma_i=0.

So the uniform and staggered UV parameter directions are already orthogonal and equally normalized in the natural product-measure information metric. The observed N=10 response ratio d_lambda R-/d_t R-=2/3 is therefore an observable coupling ratio, not evidence that the microscopic parameter coordinates themselves are non-orthogonal.

I would use the Fisher-normalized score basis as the fixed UV basis for the 2x2 response-matrix version. Then any rotation/mixing of the two response directions with N has a cleaner interpretation as RG/observable mixing rather than a coordinate artifact. This also interfaces with PR #162: higher lambda derivatives are signed Walsh/Banzhaf interaction moments, so the same p=1/2 stream can supply both the first-order Fisher-normalized tangent and higher interaction tomography.

A useful diagnostic would be to report the singular vectors of the response matrix after whitening by the exact score Fisher matrix, with the whitening fixed analytically rather than estimated from target observables.

[LightChainr]

Repository organization sync: the exact finite microscopic tangent requested as the starting point is now canonical via #175. For p_even=1/2+t+lambda, p_odd=1/2+t-lambda, complement acts exactly as (t,lambda)->(-t,-lambda) and the N=10 response matrix is [[0,0],[15/8,5/4]]; independent Bernoulli score-function derivatives reproduce those rational responses. Leaving #155 open for its real large-N task: freeze a source-size thermal orthogonalization rule and determine whether a separate matching-odd irrelevant RG direction survives.

[LightChainr]

Stronger RG extraction: use a response-matrix pencil, not a pre-estimated thermal subtraction

The 2x2 response matrix idea in this issue has a stronger asymptotic invariant that can remove both the unknown UV mixing of (t,lambda) and the unknown readout mixing.

Choose two fixed odd readouts O_1,O_2 that remain linearly independent in the scaling limit and define

R_N[a,mu] = d_<g_mu> <O_a>_N,
mu in {t,lambda}.

The derivatives are measured by the exact score estimators already described in this issue. A useful candidate readout pair is one scalar/topological odd response plus one independent orientation/H4-sensitive odd response; the exact choice must be frozen before looking at the large-N eigenvalues and must pass a conditioning gate.

If the microscopic odd tangent is dominated by two continuum RG eigenoperators with length exponents y1,y2, then asymptotically

R_N = H * diag(L^y1, L^y2) * C + subleading,
L ~ sqrt(N),

where:

• C is the nonuniversal UV mixing of (t,lambda) into the continuum eigenbasis;
• H is the nonuniversal overlap of the two continuum eigenoperators with the chosen readouts.

For two sizes N1,N2, form the generalized eigenvalue problem

det(R_N2 - rho R_N1) = 0.

At leading order,

R_N2 R_N1^-1
 = H * diag((N2/N1)^(y1/2), (N2/N1)^(y2/2)) * H^-1,

so the generalized eigenvalues are independent of both H and C:

rho_i = (N2/N1)^(y_i/2).

Equivalently,

y_i = 2 log|rho_i| / log(N2/N1).

This is a direct finite-size RG-spectrum estimator. It does not require first estimating a thermal coefficient c_*, and it is invariant under any nonsingular change of microscopic tangent basis or readout basis.

Concrete frozen adversaries

The largest eigenvalue should reproduce the thermal direction

y_t = +3/4
rho_t = (N2/N1)^(3/8).

For the second eigenvalue do not assume the square-site Q4 result. Score fixed alternatives only after the readout symmetry is understood. A useful declared comparison set is:

thermal-Q4-like odd irrelevant: y = 2 - 21/4 = -13/4
x=4 odd scalar candidate:       y = -2
x=33/4 V_<1,4>-like scalar:     y = -25/4
free second eigenvalue:         diagnostic only, after fixed scores

If the second tangent carries a unit-cell/staggered quantum number that forbids overlap with the square-site Q4 sector, remove Q4 from the physical interpretation but retain the matrix-pencil extraction itself.

Statistical implementation

The matrix entries are correlated score-function estimates, so do not diagonalize only the central matrix and attach naive errors afterward. Recompute the entire generalized eigensystem inside each aligned delete-one batch replicate (or bootstrap block) and report:

- eigenvalue/effective-y covariance;
- matrix condition numbers and determinant significance;
- eigenvector stability only as a secondary diagnostic;
- sensitivity to the frozen readout pair.

Hard gate: if R_N is statistically rank-1 or ill-conditioned, report that only one RG direction is resolved; do not regularize a second eigenvalue into existence.

Three-size closure

With three sizes N1<N2<N3, the same eigenoperator interpretation predicts composition. After matching eigenvalue branches,

rho_i(N1->N3)
 ~= rho_i(N1->N2) * rho_i(N2->N3).

This gives a held-out RG-semigroup closure test that is structurally analogous to the Gaussian-isogeny program but occurs in coupling/tangent space rather than geometry space.

If this works, the project would have two independent representation-level probes:

Gaussian covers -> eigenvalues in finite-size geometry space
self-matching tangent -> eigenvalues in microscopic coupling space

Agreement on a common exponent/parity sector would be considerably stronger than another single-channel power fit.

This should be treated as an upgrade of #155, not a separate production campaign: use the same score sufficient statistics and only add a second readout if it creates a well-conditioned response matrix.

[LightChainr]

PR #208 completes the first large-N wrapping-readout stage.

Frozen results:

• N130 five-channel response family: no pair passed condition <= 50 and |det z| >= 3; the rows are statistically rank one.
• Equal-area H4 orientation rescue: primary cross determinant z = -0.0983, angular condition = 69.94; no diagnostic channel exceeded |z| = 0.74.
• Prospectively frozen scalar holdout on unseen N170: orthogonal residual z = -0.02445.
• Fixed thermal scaling control d_t proportional to N^(3/8): z = -0.28472.
• c_130 = 0.6740950243 and c_170 = 0.6740655896.

Interpretation: for the tested topological projections, the microscopic lambda tangent remains parallel to the thermal response from N130 to N170. More samples of these same rows are low value. The issue should remain open only for a genuinely local/sublattice-odd readout, which is the next credible way to expose an independent irrelevant direction.

PR: #208

[LightChainr]

PR #219 implements the local/sublattice-sensitive readout requested after PR #208's rank-one result:

#219

Freeze the complement-odd observable

O_local = [H4_pivotal(black) - H4_pivotal(complement)] / 2

on the C4 self-matching checkerboard triangulation. The full N=10 (3,1), R=1 enumeration gives the exact score-response matrix

rows = (global_cross_half_difference, local_pivotal_H4_half_difference)
cols = (t, lambda)

R10 = [[ 15/8,   5/4  ],
       [ -3/64, 11/64 ]]

det(R10) = 195/512 != 0.

The local twice-observable occurs with counts -1:88, 0:848, +1:88; complement oddness, exact C4 rotation, and the pi/4 landing-registry sign flip have zero violations.

This crosses the exact microscopic observability gate: unlike the topological rows tested in PR #208, the new row is not parallel to the thermal/global row. It does not yet claim that the second direction survives at large N. The next bounded experiment is the aligned-batch N130/N170 2x2 score-response matrix, with the second singular value/determinant recomputed in every delete-one replicate; add a third size only if that conditioning gate passes.

[LightChainr]

The large-N conditioning gate is complete in stacked PR #224 (depends on #219):

#224

N130: det z=-0.505, singular values=(5.68616,0.003371), condition=1686.75
N170: det z= 1.553, singular values=(6.29215,0.010341), condition= 608.45

Both matrices fail the frozen |det z|>=3 && condition<=50 gate. The analyzer therefore reports only one resolved dimension and deliberately does not form the generalized eigensystem/effective y.

Important distinction: all local-H4 response entries are themselves clearly nonzero. The negative result is that this local complement-odd pivotal readout remains nearly parallel to the global thermal wrapping response at N130/N170. More samples of the same rows are unlikely to be the highest-value next move; exposing a second odd RG direction now requires a genuinely different microscopic perturbation or readout, not merely a higher-statistics repeat.