Add exact digital Alexander rank oracle

[LightChainr]

Purpose

Complete the bounded finite Phase-A test of the proposed digital Alexander rank identity without modifying the files in the open configuration-Betti PR.

The parent theory task intentionally remains open for a general periodic digital-topology proof.

Exact algebra

For r_b,r_w in {0,1,2}, exhaustive enumeration of the nine abstract rank pairs proves

[1(r_b>0)-1(r_w>0)] = [1(r_b=2)-1(r_w=2)]
    => 2q = r_b-r_w.

The only pairs satisfying the premise are (0,0),(0,2),(1,1),(2,0),(2,2).

Finite result

Every configuration was enumerated on:

• axis L=2 and L=3;
• Gaussian (2,1);
• diamond L=2;
• C4 self-matching N=10.

Only (r_b,r_w)=(0,2),(1,1),(2,0) occurred. Consequently both

2q = r_black-r_white
r_black+r_white = 2

have zero failures on all 1,840 exhaustive configurations.

The older raw equality q=r_black-r_white occurs exactly for q=0, i.e. the (1,1) configurations.

Counterexample search

A fixed-seed scan evaluated 85,152 configurations on 160 nonsingular integer-period tori of orders 5 through 31. It found zero common-channel, weak-rank, or strong-rank-sum counterexamples.

This scan is a deterministic search, not statistical evidence or a proof.

Files

• standalone stdlib oracle reusing the canonical general-period homology backend;
• frozen machine-readable manifest;
• six focused regression tests;
• generated JSON/Markdown results;
• derivation note with explicit proof boundary.

Validation

python3 -m unittest tests.test_digital_alexander_rank_oracle -v
Ran 6 tests in 6.892s — OK

python3 -m py_compile scripts/digital_alexander_rank_oracle.py tests/test_digital_alexander_rank_oracle.py

Scientific boundary

A general theorem still needs a 4/8-adjacency periodic digital Alexander/relative-homology argument. No CFT field, universal amplitude, or closed form for the square-site threshold is inferred.