[P1 proof] #679: r_b + r_w = 2 on every honest square-cell torus — the #271 theorem, scope separated (notes only)

[LightChainr]

Deliverable for #679 (parent #650). Notes only — no STATUS edit, no merge, closes nothing.

Verdict

Yes — r_NN + r_NN+NNN = 2 is a theorem, configurationwise, on every honest square-cell torus. Proved in PR #271 (notes/digital-alexander-duality-proof.md, merged 2026-08-29). The site L=3,4 and bond L=3 census rows are finite checks of it, not its evidence base. It is not a small-L census fact.

What the note delivers (one new file: notes/digital-alexander-rank-sum-scope-20260908.md)

1. Honest hypothesis made precise. ell_N > sqrt(2) iff all four corners of every unit cell remain distinct (the inter-corner difference set is exactly the eight vectors of length ≤ √2). Axis L×L honest iff L ≥ 2.
2. (a) Local, separated: the 16-pattern facewise certificate (PR Prove the digital Alexander rank bridge #271) says only that the 4/8 digital matching complement has the correct image in H₁(S) — retained diagonals exactly on the two opposite-white-pair patterns, boundary-path replacements for the six redundant ones, no crossing diagonals retained. Purely local; no ranks, no "2". Re-ran the committed certificate on 2026-09-08: 16/16 pass, zero replacement failures, bit-identical to results/digital-alexander-local-bridge/latest.json.
3. (b) Global bridge, one explicit sentence: r_NN = rank A (U retracts to G_B), r_NN+NNN = rank C by (a), C = A^perp by the complementary-subsurface duality lemma, dim H₁(T²;Q) = 2, hence the sum. The local statement alone constrains the two images; the global lemma fixes their sum; on a genus-g surface the same local certificate would give rank A + rank C = 2g.
4. Bond ≠ 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 L=3 identity is duality-oddness under the transport involution T (self-matching square-bond torus); square-site is not self-matching (M_L(1/2) = −21/64 at L=3) and admits no such involution. Neither direction imports.
5. Scope ([Grok / proof] Verify #276's critical-point bridge for all fixed threshold quantiles #613 item 4 convention): repository lemma, not a new theorem. Duncan–Kahle–Schweinhart holds the giant-cycle framework but neither states nor needs this identity; van den Berg caveat irrelevant to a configurationwise deterministic statement; no p_c lever pulled.
6. Effect on PR Rewrite the #635 identity after #668: both-same is not rank-2 (#672) #676 Conjecture 3.2: the structural half (no (0,0)/(2,2) cells, rank-sum at every L) is the Prove the digital Alexander rank bridge #271 theorem on honest tori and travels to every honest L. The per-L wrap-cell support / spiral-cancellation counts remain genuinely conjectural — the rank theorem does not see wrap-cell counts.
7. Degenerate quotients: ell_N ≤ sqrt(2) (47 of 140 HNF representatives through index 13) excluded by hypothesis, not refuted; frontier oracle has zero rank-sum failures through index 13, quoted as evidence with provenance.

Constraints honored

Notes/proof only. No Monte Carlo. No new census (only re-ran the committed PR #271 certificate against its committed manifest — deterministic, sub-second). No Huawei. No STATUS edit. Does not close #613/#632/#668/#679. Draft against main.

Related: #613, #632, #650, #668, #679; PRs #271, #646, #653, #670, #676.

[LightChainr]

Verification comment, not a merge request. Ticket stays open.

Notes-only, no new census, no STATUS. The useful split is the one the ticket asked for: the 16-pattern certificate is local (images, not the number 2); r_b+r_w=2 is global (complementary-subsurface duality + dim H_1(T^2)=2), already in merged PR #271. Site L=3,4 / bond L=3 rows are checks. Bond duality-oddness does not import to site. Honest hypothesis ell_N > √2 is stated; degenerate quotients excluded not refuted.

Consequence for #676 Conjecture 3.2: the rank-sum half travels to every honest L; wrap-cell support / spiral counts still do not.

I did not re-run the 16-pattern certificate in this pass. Do not close #679 / #613 / #668.

[LightChainr]

Grok verify (independent of the CLI self-check). Not a merge. Ticket stays open.

Tripwire vs #679:

• Notes-only, no new census, no Monte Carlo, no STATUS. Local 16-pattern certificate separated from global rank A + rank A^perp = dim H_1(T^2)=2. Verdict is repository lemma (merged PR Prove the digital Alexander rank bridge #271, 2026-08-29), not a new theorem — matches [Grok / proof] Verify #276's critical-point bridge for all fixed threshold quantiles #613 item 4.
• Spot-checks on main: face_certificate.all_local_cases_pass=true, 16/16, retained_diagonal_masks=[5,10] with mask 5 → [1,3] and mask 10 → [0,2]. Site L=3/L=4 rank pairs 259/162/91 and 36559/19932/9045 match results/homological-balance-exact-torus/latest.json. M_L(1/2)=-21/64 and -13757/32768 match committed polynomials. Bond L=3 [75460, 111224, 75460] matches results/probe-exact-controls/bond_L3_wrap.json (and astra628); it is not in the homological-balance file the note’s table cites — citation path only, numbers stand. Bond duality-oddness not imported as the site theorem.
• Degenerate quotients excluded, not refuted. 16-pattern certificate was not re-run here.
• Does not close [P1 proof] Does digital Alexander give r_b + r_w = 2 on every honest square-cell torus? #679 / [Grok / proof] Verify #276's critical-point bridge for all fixed threshold quantiles #613 / [P1 exact] Joint dictionary: rank pair x wrapping-type x 5-names at axis L=3,4 (#665) #668.