diff --git a/notes/digital-alexander-rank-sum-scope-20260908.md b/notes/digital-alexander-rank-sum-scope-20260908.md new file mode 100644 index 00000000..d9dd8c7b --- /dev/null +++ b/notes/digital-alexander-rank-sum-scope-20260908.md @@ -0,0 +1,217 @@ +# #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.