Prove the digital Alexander rank bridge

[LightChainr]

Scope

This is the claimed Phase B slice of Issue 269. It proves the rank bridge for honest periodic square-cell tori and leaves degenerate short-period quotient geometry outside the theorem statement.

What changes

• proves that complementary compact subsurfaces of a closed oriented surface have ambient first-homology images that are orthogonal complements under the intersection pairing;
• connects the black nearest-neighbor graph and white matching graph to complementary regular neighborhoods;
• adds an executable certificate over all 16 unit-square color patterns;
• prunes a white diagonal unless its endpoints are exactly the two white corners, replacing every removed diagonal by an in-face nearest-neighbor path;
• records the resulting torus identities
  r_black + r_white = 2 and 2q = r_black - r_white.

The local certificate retains diagonals only for masks 5 and 10, verifies six redundant-diagonal path replacements, preserves matching-graph connectivity in all 16 patterns, and never retains crossing diagonals.

Proof chain

The surface step identifies the annihilator of im H1(U) with the kernel of restriction to H1(U); the long exact sequence of the pair, excision to (V, boundary V), Poincare-Lefschetz duality, and naturality identify that kernel with im H1(V).

Validation

• python -m unittest tests.test_digital_alexander_local_bridge: 6/6
• combined Phase A + Phase B focused suite: 12/12
• generated JSON and Markdown exactly reproduce the checked-in results
• Python bytecode compilation passes

Evidence boundary

The proof assumes each unit square embeds with four distinct corners. Degenerate short-period quotients remain finite-oracle evidence, not part of this theorem. This PR deliberately does not change configuration Euler/Betti work or close the parent issue.