[P2] C4 self-matching site triangulation parity control

[LightChainr]

Goal

Build the closest exact control for the proposed matching-parity projector: a site-percolation, C4-symmetric, self-matching periodic triangulation.

See notes/c4-self-matching-triangulation-control.md.

Lattice

Take the square lattice and add one diagonal to every square, alternating / and \ by checkerboard face parity (equivalently connect the even-parity corners). Every face is triangular, so the graph is its own site matching graph.

The degree pattern is 8/4 on the two vertex parities and the diagonal pattern is C4 invariant up to translation/rotation.

For periodic amenable site percolation the self-matching relation gives exact p_c=1/2.

Why this control is decisive

At the finite-size critical center, with the same quotient and wrapping convention,

M(1/2)=R_G(1/2)-R_G(1/2)=0

should be an algebraic/runtime invariant.

But the individual wrapping probability can still carry a C4 spin-4 anisotropy. Therefore this model separates:

• matching-even square anisotropy (x=4 candidate);
• matching-odd central residual (x=21/4 candidate), which must vanish in the self-matching control.

This is more directly comparable to the square-site target than square-bond duality, while avoiding the C6 spin-4 suppression of the ordinary triangular lattice.

Compatible Gaussian sizes

Checkerboard parity descends when both period components are odd. Start with

• N=130: (11,3) vs (9,7);
• N=170: (13,1) vs (11,7).

If needed, confirm with N=290: (17,1) vs (13,11).

The existing cyclic label j=ax+by mod N satisfies j mod 2 = x+y mod 2 for odd (a,b), so the checkerboard sublattice is cheap to implement in the current same-N engine.

Exact correctness gate

Before stochastic runs:

1. verify all quotient faces are triangles;
2. verify the site matching generator adds no connectivity beyond the graph itself;
3. exhaustively verify central M=0 on the smallest compatible quotients;
4. verify C4 invariance up to quotient automorphism;
5. verify degree/local-face multisets across the two orientation representations.

Production observables

At p=1/2, retain individual either/cross wrapping probabilities for both orientations. Form

P4[R]=(R(theta1)-R(theta2))/DeltaCos4.

The matching difference should be asserted zero rather than estimated as a noisy target.

Compare, in this order:

• fixed P4[R] = A N^-1;
• zero anisotropy;
• only after fixed scoring, free exponent/log alternatives.

The control amplitude need not equal the target square/matching-pair amplitude.

Acceptance

The two-spin4 parity model is strongly supported if this C4 self-matching site lattice has a reproducible ordinary N^-1 cos4theta anisotropy while the central matching-odd sector remains exactly zero.

A failure of the finite self-matching identity is an implementation error until proven otherwise; a failure of the N^-1 ordinary anisotropy is a scientific negative result.

[LightChainr]

Exact cyclic geometry/checker is now committed:

scripts/c4_self_matching_cyclic_geometry.py (commit 7b570ce3384e05a16f4bf42bec4b1cc8d1dde775).

For primitive odd (a,b), it uses j=a*x+b*y mod N and adds checkerboard diagonals only from even j with steps a+b and a-b. It constructs all 2N triangular faces explicitly and validates:

• V=N, E=3N, F=2N;
• every graph edge is incident to exactly two triangular faces;
• even/odd vertex degrees are exactly 8/4;
• matching completion adds zero edges (self-matching);
• exact C4 automorphism j -> t*j, t=b*a^(-1) mod N, with t^2=-1 mod N and face/edge invariance.

Recommended correctness commands before Monte Carlo:

python scripts/c4_self_matching_cyclic_geometry.py 11 3
python scripts/c4_self_matching_cyclic_geometry.py 9 7
python scripts/c4_self_matching_cyclic_geometry.py 13 1
python scripts/c4_self_matching_cyclic_geometry.py 11 7
python scripts/c4_self_matching_cyclic_geometry.py 17 1
python scripts/c4_self_matching_cyclic_geometry.py 13 11

I independently checked these six designs plus (3,1) against the same incidence algebra: all give E=3N, F=2N, edge incidence 2, degree sets {8}/{4}, and exact C4 edge/face automorphism. The server should still run the committed checker from its clean source tree before using this as a scientific control.

[LightChainr]

Exact tiny-quotient gate completed (PR #82)

PR #82 adds the missing exhaustive gate for the smallest nondegenerate compatible quotient (a,b)=(3,1), N=10. The lifted graph agrees edge-for-edge with the previously committed cyclic geometry and has identical primal/matching edge sets.

All 2^10=1024 masks were enumerated. For each of cross, both, either, direction_0, and direction_1, the exact black-minus-complement Bernstein coefficients by occupation count k=0,...,10 are

[-1, -10, -45, -100, -100, 0, 100, 100, 45, 10, 1]

They satisfy M_k=-M_(N-k) and the independently retained identity M_k=R_k-R_(N-k), so the finite-quotient center is exactly

M(1/2)=0

for all five wrapping channels. The C4-invariant cross/both/either indicators also have zero failures under the exact quarter-turn automorphism. Focused tests: 3/3; general integer-period topology regressions: 8/8. Exact JSON, commands, checksums, and report are committed.

Evidence boundary

This is an exact finite-quotient implementation result, not the stochastic spin-4 amplitude result. It also must not be read as M(p) being identically zero: on the same quotient every recorded channel gives the exact off-center value M(1/3)=-47/81. The N=130/170 fresh-seed orientation production remains open.

PR: #82
Commit: 96c2551