Construct exact self-matching tangent family

[LightChainr]

Result

Constructs an explicit local one-parameter tangent on the C4 self-matching checkerboard triangulation:

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

The legal domain is |t+lambda|<=1/2, |t-lambda|<=1/2, and occupation complement realizes the exact local map (t,lambda)->(-t,-lambda) with J=-I.

Exhaustive N=10 enumeration gives, for every recorded wrapping channel, the exact response matrix (rows Rplus,Rminus; columns t,lambda)

[[0,    0],
 [15/8, 5/4]].

The exact center-slice odd polynomial is

Rminus(lambda) = (5/4) lambda - 4 lambda^5.

Its only legal root on [-1/2,1/2] is lambda*=0; the other real-root magnitude is (5/16)^(1/4)=0.747674.... This is a sharp finite-quotient negative result for a nonzero odd improved point.

The note therefore redirects the nontrivial improved-action search to the matching-even H4 amplitude

A_T4_plus(N,lambda) = N * DeltaRplus / DeltaCos4.

It freezes the first compatible orientation design, N=130 (11,3) versus (9,7), and the nonnegative grid 0,1/8,1/4,3/8: fit the first three in z=lambda^2, retain 3/8 as a no-refit check, then transport any accepted root to N=170 without refitting.

Artifacts

• exact rational bivariate enumerator and JSON
• derivation/claim-boundary note
• four focused regression tests
• local report, command log, checksum

Verification

python3 -m unittest discover -s tests -p 'test_*.py' -v
Ran 164 tests in 20.047s — OK

Relates to #44 and #61.

[LightChainr]

A second use of the exact (t,lambda)->(-t,-lambda) family is now tracked in #155. At the center, likelihood-score identities let one measure d_t and d_lambda responses from the same p=1/2 configurations via sublattice occupation counts, with the N=10 exact 15/8, 5/4 response as an oracle. The proposed goal is to subtract the thermal projection and expose any independent matching-odd irrelevant RG tangent, directly advancing #61 rather than only searching an improved even-H4 point.

[LightChainr]

PR #162 adds a complementary exact interpretation of the lambda tangent. For a Boolean wrapping event under p_i=1/2+sigma_i lambda, the unbiased Walsh expansion gives

E_lambda[f] = sum_S fhat(S) (2 lambda)^|S| product_(i in S) sigma_i.

Thus the coefficient of lambda^r is exactly 2^r times a signed Fourier-level sum (equivalently a higher Banzhaf interaction moment); the linear term is a signed pivotal/Birnbaum importance sum. Your exact Rminus(lambda)=(5/4)lambda-4lambda^5 therefore corresponds, in the declared sublattice sign convention, to signed level sums A1=5/8, A3=0, A5=-1/8.

This gives a second finite object besides the isotropic reliability signature: the N=10 Beta(3,3) law fixes only layer counts, while this tangent resolves how the Boolean event is distributed over variables/sublattices. I would preserve both. In particular, if the N=26 exact engine is extended to a (k_even,k_odd) table, it yields the entire odd interaction spectrum with no finite-difference noise. I would not freeze the N=10 absence of the r=3 term as a prediction; whether it persists is exactly the interesting result.

[LightChainr]

Superseded by merged PR #175. #175 preserved this PR's exact N=10 tangent note and result archive byte-for-byte, and combined them with #167's stronger score-function oracle on the current organized mainline. Closing this older base PR without deleting provenance.