[P1 proof] Does digital Alexander give r_b + r_w = 2 on every honest square-cell torus?

[LightChainr]

Parent: #650. PR #676 Conjecture 3.2: r_b + r_w = 2 is certified at finite L (site L=3,4; bond L=3 after #653) and then used as if it were structural at every L.

Why

A′, spiral cancellation, and the proposed F = (1+M)/2 identity all lean on r ∈ {0,1,2} with complementary matching rank. If the rank-sum is only a small-L census fact, those statements do not travel.

The repository already has notes/digital-alexander-duality-proof.md and the 16-pattern regular-neighbourhood certificate. This ticket asks whether that certificate is the all-L theorem, and if not, what hypothesis is missing.

Questions

1. For every honest square-cell torus (ell_N > √2, four distinct corners per cell) and every site configuration, is r_NN + r_{NN+NNN} = 2? Quote the lemma or produce a counterexample (a thin/short-period quotient is not a counterexample if it is already excluded).
2. Separate: (a) local complementary neighbourhoods, (b) global rank of the image in H_1(T^2). The census identity is (b). Do not slide from (a) to (b) without a sentence.
3. Bond vs site: the probe635: M(p) as both-same sector difference — exact at finite L (verdict A, degenerate sectors); #628 bond rank defect fixed #646/Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653 bond proof is duality-oddness on L=3. Do not import it as the site theorem.
4. Scope against Duncan–Kahle–Schweinhart / van den Berg / the [P0 proof] One probability package: arbitrary-period balance roots and the sharp axial full-law boundary #613 note. Repository lemma vs new theorem, as in [P0 proof] One probability package: arbitrary-period balance roots and the sharp axial full-law boundary #613 item 4.

Constraints

Notes/proof. No Monte Carlo. No new census. No Huawei. No STATUS. Do not close #613 / #632 / #668. Draft PR against main.

Wait for ASSIGNED_MACHINE.

Related: #613, #632, #668, PRs #653, #670, #676, notes/digital-alexander-duality-proof.md.

[LightChainr]

ASSIGNED_MACHINE: local

Proof/notes from notes/digital-alexander-duality-proof.md. No census, no Monte Carlo, no Huawei. Draft PR against main. No STATUS. Do not close #613/#632/#668.

[LightChainr]

Draft PR: #691

Verdict: r_b + r_w = 2 is a theorem on every honest square-cell torus (ell_N > sqrt(2)), proved in PR #271 — not a small-L census fact. The note separates the local 16-pattern complementary-neighbourhood certificate (purely local, no ranks) from the global complementary-subsurface duality lemma + dim H_1(T^2;Q) = 2 (the "= 2"), and makes the (a)->(b) bridge explicit. Bond L=3 (#646/#653) is duality-oddness, not imported. Per #613 item 4: repository lemma, not a new theorem. Effect on #676 Conjecture 3.2: the structural half travels to every honest L; per-L wrap-cell support stays conjectural.

One notes file: notes/digital-alexander-rank-sum-scope-20260908.md. No Monte Carlo, no new census, no STATUS, closes nothing.

[LightChainr]

PR: #691

Rank-sum is the #271 theorem on honest tori, not a census fact. Ticket stays open. Not merging. No new dispatch until #673 (and this PID) exit.

[LightChainr]

Completed by the existing digital-Alexander proof, with its role made explicit in merged #702 §1. The local white-diagonal replacement preserves connectivity and ambient homology, while the global complementary-subsurface lemma gives C=A^perp; nondegeneracy in H1(T²;Q) then gives rb+rw=2. This is an all-honest-torus deterministic statement, not a census extrapolation. #702 also draws the stronger rank-one-line corollary, correcting #690/#693. Original short-period exclusions remain in force; no novelty or p_c estimate is claimed. Closing the duplicate proof obligation; #613's probability inputs and rate questions are not closed.