From 073504444dfc46b5e5069b6b79fd8cee48e06d1c Mon Sep 17 00:00:00 2001 From: Light Chain Date: Tue, 8 Sep 2026 20:46:53 +0800 Subject: [PATCH] notes(#635/#660): identity write-up against the wrapping-type tables Verdict A at finite L (wrapping-form, Mertens-Ziff 2016), degenerate both-same sectors; L=3 Bernstein tripwire reproduced bit-for-bit this session. Axis L=5 / diamond L=4: A-continues, both-two mass 0. Jacobsen map gap stated (order of limits, one-operator vs two-lattices, novelty boundary). Does not fund #636. No STATUS.md edit; no ticket closed. --- notes/issue-635-identity-20260908.md | 243 +++++++++++++++++++++++++++ 1 file changed, 243 insertions(+) create mode 100644 notes/issue-635-identity-20260908.md diff --git a/notes/issue-635-identity-20260908.md b/notes/issue-635-identity-20260908.md new file mode 100644 index 00000000..86419dd9 --- /dev/null +++ b/notes/issue-635-identity-20260908.md @@ -0,0 +1,243 @@ +# #635 identity write-up — is M(p) the difference of two topological-sector amplitudes? + +**Ticket:** LightChainr/Matching-One#660 (write-up deliverable of #635). **Date:** 2026-09-08. +**Mode:** write-up against wrapping-type tables that already exist. No new census. No +`docs/STATUS.md` edit; no ticket closed or merged. + +This note consumes, and does not re-enumerate: + +- **PR #646** — probe635 sector map, verdict A, degenerate both-same sectors + (`notes/probe635-sector-map-20260907.md`, `results/probe635-sector-map/latest.json`). +- **PR #653** — #632 bond-lab repair; wrapping-type 4×4 tables at axis L=3,4 and diamond L=3 + (`notes/wrapping-type-census-l3l4-20260908.md`, `results/wrapping-type-census/`). +- **PR #657** — axis L=5 (`2^25`) and diamond L=4 (`2^32`) censuses; Bernstein tripwire vs + PR #649 pass; **#651 A-continues** (`both-two` mass 0 at every k) + (`notes/wrapping-type-census-axis-L5-diamond-L4-20260908.md`). +- **PR #654** — wrapping literature retrieval (preferred over #656 for this note) + (`notes/torus-wrapping-retrieval-20260908.md`). +- **PR #645** — Jacobsen retrieval (#637) + (`notes/jacobsen-sector-retrieval-20260907.md`). + +All exact claims below use integers / `Fraction` only. The tripwire integers were +re-run this session from `scripts/exact_matching_polynomial.py --geometry axis --L 3` +and reproduce bit-for-bit. + +--- + +## 1. Sector labels on finite-L configurations + +The #635 question is whether + +```text +D(C) = 1{black NN wraps} − 1{white NN+NNN wraps} +M(p) = Σ_k a_k p^k (1−p)^(N−k) +``` + +decomposes as a difference of two topological-sector amplitudes, **exact at finite L**. +To make that decidable at finite L, the sector labels must be defined combinatorially, +on configurations, not on asymptotic objects. + +**Five-name labels (PR #646).** For a configuration C on the L×L torus, compute the +displacement lattice of each occupied cluster (greedy spanning-forest lift into Z²): + +| label | definition | +|---|---| +| `none` | no cluster has nontrivial winding | +| `x` | some cluster winds exactly the x-axis (up to sign) | +| `y` | some cluster winds exactly the y-axis (up to sign) | +| `both-same` | a **single** cluster whose displacement lattice has rank 2 in H₁(T²;Q) | +| `both-two` | two distinct clusters, one winding each axis | + +Each configuration carries a black label (primal, NN) and a white label (matching, +NN+NNN). The candidate map of #646 is + +```text +M(p) = Σ_k [ #(both-same | black, |C|=k) − #(both-same | white, |C|=k) ] · p^k (1−p)^(N−k) +``` + +i.e. the two sector amplitudes are the black- and white-winding indicators restricted +to the `both-same` homology class. + +**Coarse 4×4 labels (PR #653).** The wrapping-type census classifier +(`scripts/wrapping_type_census.py`, `torus_homology.wrapping_channels`) uses +`neither / dir0 / dir1 / both`, where `both` is `dir0 ∧ dir1` and therefore **lumps +rank-2 cross-wrapping with rank-1 spirals**. The correspondence: + +| #646 5-name | coarse 4×4 | Mertens–Ziff type (Newman–Ziff (13)–(15) language) | +|---|---|---| +| `none` | `neither` | no wrap | +| `x` | `dir0` (or `dir1`, orientation convention) | R^{(1)} / R^{(h)} single-axis | +| `y` | `dir1` (or `dir0`) | R^{(1)} / R^{(h)} single-axis | +| `both-same` | `both` (cross **and** spiral component) | R^{(b)}: both-axes wrap | +| `both-two` | folded into `both` | two single-axis wraps by distinct clusters | + +The fold `both-same ∪ both-two → both` is exactly the distinction MZ do not make in +wrapping-type language; it is the one structural question the coarse 4×4 cannot see +and the 5-name labels can (#651's `both-two` mass). + +**Mertens–Ziff cross vs spiral.** The retrieval in PR #654 fixes the identification: +MZ (2016) prove, at every finite L on the torus, + +```text +M_L(p) := N_L(p) − N̂_L(1−p) − L² χ(p) = R^x_L(p) − R̂^x_L(1−p), x ∈ {c, b, e, h} +``` + +with the **only contribution to the right-hand side coming from the cross-wrapping +probabilities** — all other wrapping types cancel by the MZ pairing (no black wrap ⇒ +exactly one white cross; single-direction wraps pair; cross-wrapping is exclusive; +spiraling counts match). The repository observable (x = `either`) therefore equals the +cross difference by the published theorem. In 5-name language this predicts: D mass +only on cells where exactly one side is `both-same` (cross) and the other is `none`; +`x×x`, `y×y`, `both-same×both-same` (spiral pairings) carry D = 0. + +## 2. The candidate identity, written so it can fail + +The identity under test at finite L, in committed-integer form: + +```text +a_k = #(black ∈ both-same, |C|=k) − #(white ∈ both-same, |C|=k) +``` + +Equivalently, as a two-amplitude difference: + +```text +M(p) = A_black^{both-same}(p) − A_white^{both-same}(p) +A_black^{both-same}(p) = Σ_k #(black ∈ both-same, |C|=k) · p^k (1−p)^(N−k) +``` + +**Acceptance gate.** The committed Bernstein integers at axis L=3 (N=9), re-run this +session and reproduced bit-for-bit: + +```text +bernstein_counts: [-1, -9, -36, -78, -90, -36, 36, 36, 9, 1] +polynomial: -4p^9 + 18p^8 - 18p^7 + 6p^3 - 1 +physical root: 0.5865114551126756356545589766069017348243 +``` + +Any decomposition that does not reproduce these integers exactly (Fraction arithmetic, +no floats) is unused. The run also re-confirms the #628 §8 anchor by exact arithmetic: +M(1/2) = −21/64, so u_c = (1 + M(1/2))/2 = **43/128** at L=3. + +PR #646's script, run this session at axis L=3, gives the two amplitude rows and their +sum: + +```text +reference bernstein: [-1, -9, -36, -78, -90, -36, 36, 36, 9, 1] + A_white (m-both-same): [-1, -9, -36, -78, -90, -45, 0, 0, 0, 0] + A_black (p-both-same): [ 0, 0, 0, 0, 0, 9, 36, 36, 9, 1] + sum: [-1, -9, -36, -78, -90, -36, 36, 36, 9, 1] ← exact +``` + +with the independent wrap-label diagnostic path (union-find, not the +displacement-lattice rank path) agreeing position-by-position. **The identity holds +exactly at finite L on every size the enumerations reach — verdict A, in the +wrapping-form.** + +## 3. Verdict A / B / C + +**Verdict: A at finite L, with a degeneracy caveat that is now evidence-backed, not +speculative.** + +- **A (exact at finite L)** — attained in the wrapping-form on all tested sizes: + axis L=3 (2⁹), L=4 (2¹⁶), L=5 (2²⁵); diamond L=2 (2⁸), L=3, L=4 (2³²). The L=3 + integers are reproduced bit-for-bit (§2); the larger sizes pass the same tripwire + against the PR #649 committed Bernstein rungs (PR #657, both kernels on diamond L=4). +- **Not B** — no finite-L defect exists at any reachable size; there is nothing to + exhibit or scale. +- **Not C** — no obstruction: the MZ theorem (PR #654, their (20)) proves the + difference-of-wrapping-amplitudes form at every finite L on a torus, so the + wrapping-form identity cannot be false. + +**The degeneracy caveat, now closed through axis L=5 / diamond L=4.** At axis L=3, the +per-sector rows above show the D-carrying cells are exclusively +`none × both-same` (D=−1) and `both-same × none` (D=+1); `x`, `y` carry D=0 wherever +they occur, and `both-two` never occurs with D≠0. The structural worry (PR #646 §3) +was that this collapse into a single `both-same` label measured twice might be a +small-L accident. It is not, on the last independently checkable axis rung: + +- **Axis L=5** (2²⁵ configs, PR #657): `both-two` mass **0 at every k**; support is + exactly the five cells `none×both-same` (−1), `both-same×none` (+1), `x×x` (0), + `y×y` (0), `both-same×both-same` (0). Coarse 4×4 is still the five Mertens–Ziff + cells; no sixth cell. +- **Diamond L=4** (2³² configs, K1 and K2 kernels, identical 5×5 tables): same five + cells, `both-two` mass 0. **A-continues.** + +Axis L=3 per-k D detail (PR #653 §3, exact): the sign change k=5→6 is two-term +cancellation — k=5 has `neither×both`=45 (D=−1) and `both×neither`=9 (D=+1), net −36; +k=6 has `both×neither`=36 plus a 6-count `both×both` (D=0), net +36. In 5-name terms +this is the same statement: D lives only on the exclusive cross mismatch. + +## 4. What axis L=5 / diamond L=4 changed — and what they cannot change + +**Changed.** They upgraded the degeneracy caveat from "observation at L ≤ 4" to +"A-continues at the enumeration frontier": `both-two` has zero mass at every k on +2²⁵ and 2³² configurations, and the coarse 4×4 support is unchanged at five MZ cells +on both geometries. They also confirm `dir0 = dir1` at every k on both geometries. + +**They cannot change: the Jacobsen map (PR #645).** The published object +(arXiv:1507.03027) is a statement about the **leading eigenvalues of two topological +sectors of one periodic Temperley–Lieb transfer matrix**, with **m → ∞ taken first** +(semi-infinite cylinder), the criterion `P_B = 0 ⇔ Λ_open = Λ_closed` proved by +intermediate value theorem at finite n, and the n → ∞ extrapolation conjectural +(tower Δ_k = 2(k+1) marked as a conjecture in the source). Three specific gaps remain +between our finite-L verdict A and that map, none of which more enumeration can close: + +1. **Order of limits.** Our identity is exact at finite L. Jacobsen's criterion is + proved after taking the cylinder length to infinity. A finite-L amplitude identity + does not imply equality of leading eigenvalues of the two sector transfer matrices; + that is an asymptotic statement requiring its own limit. +2. **One operator, two sectors vs two lattices.** Jacobsen's sectors are open/closed + blocks of **one** pTL operator. Our two amplitudes are black-NN on the primal + lattice and white-NN+NNN on the **matching** lattice — two different edge sets, + and the white indicator is not a sector of the primal transfer matrix. Whether the + matching-lattice content is exactly the open/closed sector distinction is the crux + #635 named and is untouched by census evidence. (PR #645 Q2: no published + boundary-state theory for NN+NNN strip connectivity exists; the frontier relation + admits crossings, so Catalan counting should not be assumed to survive.) +3. **Novelty boundary.** The published nearest neighbor is Mertens–Ziff 2016 itself: + their identity **is** `M(p)` as a difference of wrapping amplitudes, exact at + finite L — the pairing is published (PR #654). What is **not** published (PR #645 + Q1) is the sector reading: two sectors of *one* pTL operator vs a signed difference + on *two* lattices. Any claim from this line must be phrased as the sector + decomposition, with citation obligations to both Mertens–Ziff 2016 and + Jacobsen 2015. + +So: L=5 / L=4 strengthen A as a finite statement; they are silent on the eigenvalue +identification, which is a limit statement about a different object. + +## 5. Boundary: this note does not fund #636 + +Nothing in this note promotes #636 (transfer-matrix machinery). The honest statement +of position: + +- The wrapping-form of verdict A is **published** (Mertens–Ziff 2016). Building + machinery to re-derive it buys nothing. +- The unpublished piece — the sector reading — is a *conceptual identification* + (matching-lattice content ↔ open/closed sectors of one pTL operator), and the two + census facts that would de-risk it (does `both-two` ever carry mass; does a sixth + 4×4 cell ever appear) stayed at zero through the enumeration frontier. Zero mass at + L=5/L=4 is evidence of degeneracy, not a theorem for all L, and degeneracy, if + exact, makes the "two sectors" degenerate into one label measured twice — which + would make #636's sector machinery carry **less** structure than its cost suggests, + not more. +- If machinery is ever worth building, the reason would be the **crossing frontier + state space** of the matching side (NN+NNN admits crossing connectivities; PR #645 + Q2 gap, #638 is the correct instrument to measure it) — a measurement question for + #638 first, a funding question for #636 after. That recommendation lives here, not + in `docs/STATUS.md`, and does not promote anything. + +**Tripwire standing rule:** any future claimed decomposition must reproduce +`[-1, -9, -36, -78, -90, -36, 36, 36, 9, 1]` at axis L=3 bit-for-bit, integers / +Fraction only, or it is unused. + +--- + +**Claim boundary.** Finite identity, not a threshold theorem, not a low-dimensional +TM claim. The 2015 pc interval is superseded in the literature (PR #645 §2: Yang–Zhou +2024, Jacobsen Reply 2024) — recorded, not acted on here. Do not cite #628's 118133 +bond `dual_fail` as physics (repaired compound artifact, PR #653/#646; the corrected +bond census has `dual_fail = 0` and exact `r_b + r_w = 2`). `M(p) + M(1−p) ≠ 0` stands +on the site side (complement-transpose of the 4×4 fails as soon as single-direction +wrap appears — the generating fact behind the non-self-complementing F). + +Related: #635, #637, #640, #642, #646, #651, #653, #654, #657, #636 (not funded).