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

notes/digital-alexander-rank-sum-scope-20260908.md

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+# #679 — Does digital Alexander give r_b + r_w = 2 on every honest square-cell torus?
+
+2026-09-08. WorkBuddy CLI (local machine). Notes/proof only — no Monte Carlo, no new
+census, no Huawei, no STATUS edit, does not close #613/#632/#668. Parent #650.
+
+**Verdict in one line: yes — `r_NN + r_NN+NNN = 2` is a theorem, configurationwise,
+on every honest square-cell torus; it is proved in PR #271
+(`notes/digital-alexander-duality-proof.md`) and the site L=3,4 / bond L=3 census
+rows are finite *checks* of it, not its evidence base. The 16-pattern certificate
+is only the local half; the "`= 2`" is the global half and needs one further
+lemma, which we make explicit below.**
+
+---
+
+## 0. Honest tori: `ell_N > sqrt(2)` iff four distinct corners per cell
+
+Let `T^2 = R^2/Lambda` with `Lambda <= Z^2` an index-`N` sublattice, so the
+standard unit-square grid descends to a periodic square-cell decomposition with
+`N` cells and `N` vertices. The four corners of the cell `[0,1]^2` are the
+cosets of `(0,0), (1,0), (0,1), (1,1)`, and the differences between two corners
+of (possibly different) cells range exactly over
+
+```text
+(+/-1, 0), (0, +/-1), (+/-1, +/-1),
+```
+
+which are precisely the eight vectors of `Z^2` with Euclidean length `<= sqrt(2)`.
+Hence two corners are identified in the quotient iff `Lambda` contains a nonzero
+vector of length `<= sqrt(2)`, i.e. iff `ell_N <= sqrt(2)`:
+
+```text
+four distinct corners per embedded cell   <=>   ell_N > sqrt(2).
+```
+
+Call such quotients **honest**. Axis `L x L` tori are honest iff `L >= 2`.
+Degenerate (self-identifying) quotients are *excluded* by the theorem, not
+refuted; see section 6.
+
+## 1. What the census establishes, and what it cannot
+
+The committed exact censuses confirm, with zero failures:
+
+| source | scope | rank pairs `(0,2)/(1,1)/(2,0)` |
+|---|---|---|
+| site L=3 (512 configs) | honest axis | 259 / 162 / 91 |
+| site L=4 (65 536 configs) | honest axis | 36 559 / 19 932 / 9 045 |
+| bond L=3 (262 144 configs, after the #653 repair) | honest axis | 75 460 / 111 224 / 75 460 |
+
+(`results/homological-balance-exact-torus/latest.json`, independently rerun in
+`notes/literature-officer-20260906-homological-balance-verify.md`; bond row from
+PR #653.) A null census through any finite `L` cannot distinguish "structural"
+from "accidental" — that is exactly the worry PR #676 raised about its own
+Conjecture 3.2. The question is therefore whether a proof exists in scope.
+It does, and it predates the census-based uses.
+
+## 2. (a) The local statement: complementary neighbourhoods, face by face
+
+Fix any configuration on an honest torus. Let `G_B` be the black
+nearest-neighbour graph (embedded, because cells are honest) and let `U` be a
+closed regular neighbourhood of `G_B`; put `V = closure(S \ U)`. Let `W` be the
+white matching graph: white NN edges plus both diagonals of every square.
+
+The **local** claim is facewise, over the 16 black/white corner patterns:
+
+> In each face, every active matching diagonal either (i) has its endpoints
+> exactly the two white corners of the face — retained (this happens for
+> exactly the two opposite-white-pair patterns, masks 5 and 10); or (ii) has
+> its endpoints joined by a white NN path along the boundary of that same
+> face — replaced by that path; and no pattern retains crossing diagonals.
+
+Replacing each redundant diagonal by its boundary path changes every cycle by a
+chain contained in a single contractible face, hence preserves classes in
+`H_1(S)`. Call the result `G_W`. Then `G_W` is embedded, `G_W subset V`, and
+cell by cell `G_W` is a spine of `V` (each face's part of `V` is a disk or an
+annulus-adjacent piece that deformation-retracts onto its `G_W` part; glued
+over faces, `V` retracts to `G_W` up to disks). Consequently
+
+```text
+im[H_1(W)   -> H_1(S)] = im[H_1(G_W) -> H_1(S)] = im[H_1(V) -> H_1(S)].   (*)
+```
+
+This is the precise role of the 4/8 complementary-adjacency convention, and it
+is *purely local*: nothing about ranks, and nothing about the number 2, has
+been used. The machine certificate
+(`scripts/digital_alexander_local_bridge.py` against
+`analysis/digital_alexander_local_bridge_manifest.json`, merged in PR #271)
+checks all 16 patterns; re-run on 2026-09-08 it reports 16/16 pass, zero
+replacement failures, retained diagonals exactly `{mask 5: (1,3), mask 10:
+(0,2)}`, `all_local_cases_pass = true`, bit-identical to the committed
+`results/digital-alexander-local-bridge/latest.json`.
+
+## 3. (b) The global statement: the one-sentence bridge, made explicit
+
+The global input is the complementary-subsurface duality lemma, quoted from
+`notes/digital-alexander-duality-proof.md` section 2 (PR #271):
+
+> Let `U, V` be complementary compact subsurfaces of a closed oriented surface
+> `S`, meeting on their common boundary. Over `Q`, put
+> `A = im[H_1(U) -> H_1(S)]`, `C = im[H_1(V) -> H_1(S)]`.
+> Then `C = A^perp` for the nondegenerate intersection pairing on `H_1(S)`.
+
+(Proof chain: Poincare duality identifies `A^perp` with
+`ker[H^1(S) -> H^1(U)]`; the long exact sequence of `(S,U)` identifies that
+kernel with the image of `H^1(S,U)`; excision gives `H^1(V, boundary V)`;
+Poincare-Lefschetz gives `H_1(V)`; naturality matches the maps.)
+
+The bridge from (a) to (b) — the sentence the ticket demands — is exactly:
+
+> `r_NN = rank A` because `U` deformation-retracts to `G_B`;
+> `r_NN+NNN = rank C` by the local bridge (*);
+> `A` and `C` are orthogonal complements by the lemma;
+> `dim H_1(T^2; Q) = 2`;
+> therefore `r_NN + r_NN+NNN = rank A + rank A^perp = 2`, for every
+> configuration on every honest square-cell torus. `[]`
+
+The only torus-specific input is the dimension of `H_1`. On a closed oriented
+surface of genus `g` the *identical* local certificate would give
+`rank A + rank C = 2g`: the local statement constrains the two images, and the
+global statement fixes their sum. Neither half alone yields the identity.
+
+Allowed rank pairs, exhaustively: `(0,2), (1,1), (2,0)`. In particular no
+configuration has `(0,0)` or `(2,2)`, which is the structural half that PR
+#676's Conjecture 3.2 leans on (see section 5).
+
+## 4. Bond is not site (#646/#653 vs #271)
+
+The bond-side identity at L=3 rests on a **different mechanism**: geometric
+dual transport `T` (`notes/square-bond-transport-parity-theorem.md`,
+`notes/square-bond-duality-tiny-torus.md`) is a bijection on bond
+configurations that swaps primal and dual wrapping, so the odd combination `D`
+satisfies `E[D] = 0` at the self-dual point `p = 1/2` — duality-oddness under a
+measure-preserving involution. It is structurally tied to the square-bond torus
+being self-matching (the dual grid *is* the primal grid).
+
+The square-site pair (NN vs NN+NNN) is not self-matching — exactly,
+`M_L(1/2) = -21/64` at L=3 and `-13757/32768` at L=4 — admits no such transport
+involution, and its rank-sum is proved by the digital-Alexander route of
+sections 2-3, not by oddness. Conversely, the #271 theorem says nothing about
+bond observables. Neither direction imports. Both identities are
+configurationwise and deterministic; the probability statements (such as
+`E[D] = 0` at `p = 1/2` for bond) are additional and separate.
+
+## 5. Scope against the literature (#613 item 4 convention)
+
+- **Duncan-Kahle-Schweinhart (arXiv:2011.11903, AIHP 2025).** Supplies the
+  ambient-`H_1` "giant-cycle" observable framework and the sharp-threshold
+  mechanism for the square-bond torus. It neither states nor needs the site 4/8
+  matching rank-sum; the proof obligations do not overlap. The rank identity
+  here is a finite-`L`, configurationwise input; DKS-style asymptotics sit on
+  top of it and are out of scope.
+- **Classical topology.** The subsurface duality lemma is classical
+  Poincare-Alexander duality for complementary regular-neighbourhood
+  subsurfaces. The repository claims no novelty for it. The repository-specific
+  obligation — that the 4/8 digital matching complement has the correct image
+  in `H_1(S)` despite crossing diagonals and boundary-redundant diagonals — is
+  discharged by the 16-pattern certificate (PR #271).
+- **Cote-Uzcategui-Aylwin (arXiv:2503.17861).** Modern digital-connectivity
+  treatment of 4/8 adjacency; cited as consistent context, not as a dependency
+  of the proof.
+- **van den Berg caveat (recorded in #670 item 3).** Concerns
+  matching-lattice critical-point *relations* under dependence or
+  generalization. Irrelevant to the configurationwise deterministic identity
+  proved here — but equally, the identity must not be levered into a `p_c`
+  statement without the #613/#614 probability inputs (exponential decay for
+  site on both graphs; the amenable matching relation via Grimmett-Li). No
+  such lever is pulled in this note.
+- **Verdict per #613 item 4: repository lemma, not a new theorem.** Proved in
+  PR #271 (merged 2026-08-29); the census rows are checks.
+- **Effect on PR #676.** Conjecture 3.2's structural half — "`r_b + r_w = 2`
+  at every `L`, hence no rank-2 x rank-2 and no rank-0 x rank-0 cell" — is
+  exactly the #271 theorem on honest tori and travels to every honest `L`. The
+  per-`L` wrap-cell support claim (exclusive crosses at each `L`; spiral
+  cancellation `a_k = Delta #(both-same)`) remains genuinely conjectural: it is
+  a statement about wrap-cell *counts*, which the rank theorem does not see.
+  A-prime and the proposed `F = (1+M)/2` identity may use
+  `r in {0,1,2}` with complementary ranks at every honest `L`; any remaining
+  gap in those proposals lives in the count-level claims (and, per #668, in
+  sector rank-purity), not in the rank-sum.
+
+## 6. Degenerate quotients: excluded, not refuted
+
+The hypothesis "honest" excludes short-period/self-identifying quotients
+(`ell_N <= sqrt(2)`; 47 of the 140 HNF representatives through index 13). For
+these the cellwise proof does not apply, because some face fails to embed with
+four distinct corners and the regular-neighbourhood/spine construction breaks
+down. The frontier oracle
+(`notes/digital-alexander-short-period-frontier.md`,
+`results/digital-alexander-quotient-frontier/latest.json`) exhausts all 140
+representatives through index 13 — 101,140,028,118 complete filtrations — with
+**zero** `rank_sum` failures across honest and self-identifying geometries
+alike. That is evidence, not proof: a degenerate-quotient counterexample, if
+one exists, would delimit the honest-cell theorem rather than contradict it.
+
+## 7. Non-claims
+
+No Monte Carlo; no new census; no threshold value, rate, or scaling claim; no
+CFT field identification, no `V_(2,2)` selection rule; no STATUS edit; no
+ticket closed. The identity is deterministic and probability-free; the #276/#613
+qualitative-convergence targets remain exactly where they were.
+
+## References
+
+- PR #271 (merged): `notes/digital-alexander-duality-proof.md`,
+  `scripts/digital_alexander_local_bridge.py`,
+  `analysis/digital_alexander_local_bridge_manifest.json`,
+  `results/digital-alexander-local-bridge/latest.json`.
+- #613 (item 4) and PR #670: repository-lemma vs new-theorem convention.
+- #676: Conjecture 3.2 and the structural/per-`L` split.
+- #653: repaired bond L=3 census; #646: bond rank defect fix.
+- `notes/digital-alexander-rank-oracle.md` (weak vs strong identity, finite
+  oracles); `notes/digital-alexander-short-period-frontier.md` (degenerate
+  frontier); `notes/square-bond-duality-tiny-torus.md` and
+  `notes/square-bond-transport-parity-theorem.md` (bond transport parity);
+  `notes/literature-officer-20260906-homological-balance-verify.md`
+  (independent L=3,4 rerun).
+- Duncan-Kahle-Schweinhart, arXiv:2011.11903 (AIHP 2025); Cote and
+  Uzcategui-Aylwin, arXiv:2503.17861; Mertens and Ziff, arXiv:1603.07289.