From cf2d49e936727f7161eaae7dcf3ba2d1ca354b61 Mon Sep 17 00:00:00 2001 From: Light Chain Date: Tue, 8 Sep 2026 19:22:20 +0800 Subject: [PATCH 1/2] Cite-or-gap retrieval for torus wrapping vs the matching function (#642). Q1-Q5 against Pinson (via Newman-Ziff / Pruessner-Moloney), Mertens-Ziff 2016, and the L=3,4 census. The finite matching identity is published; the projected shape exponent is not. Does not close #642 or edit STATUS. --- notes/torus-wrapping-retrieval-20260908.md | 216 +++++++++++++++++++++ references.bib | 67 +++++++ 2 files changed, 283 insertions(+) create mode 100644 notes/torus-wrapping-retrieval-20260908.md diff --git a/notes/torus-wrapping-retrieval-20260908.md b/notes/torus-wrapping-retrieval-20260908.md new file mode 100644 index 00000000..dcf0516f --- /dev/null +++ b/notes/torus-wrapping-retrieval-20260908.md @@ -0,0 +1,216 @@ +# #642 torus wrapping retrieval (cite-or-gap) + +Theory input for [issue #642](https://github.com/LightChainr/Matching-One/issues/642). Distinct from #637 (transfer matrices / critical polynomials). **Does not enter** `docs/STATUS.md`. Does not close #642. No claimed exact threshold is ingested through `scripts/threshold_claim_intake.py` because none is claimed. + +Read this session, at first hand unless marked: + +- Mertens–Ziff, *Percolation in finite matching lattices*, PRE **94**, 062152 (2016), [arXiv:1603.07289v2](https://arxiv.org/abs/1603.07289) +- Newman–Ziff, *A fast Monte Carlo algorithm for site or bond percolation*, PRE **64**, 016706 (2001), [cond-mat/0101295](https://arxiv.org/abs/cond-mat/0101295) (PDF) +- Pruessner–Moloney, *Winding clusters in percolation on the torus and the Möbius strip*, [cond-mat/0310361](https://arxiv.org/abs/cond-mat/0310361) +- Repository notes already on published wrapping ground: `notes/literature-officer-20260905-issue576-wrapping.md`, `notes/pinson-arguin-primitive-baseline.md` +- This session’s axis/diamond wrapping-type census: `results/wrapping-type-census/`, `notes/wrapping-type-census-l3l4-20260908.md` + +Pinson, *Critical percolation on the torus*, J. Stat. Phys. **75**, 1167 (1994) is **not** opened as a PDF this session. Formulae and ten-figure values below are quoted from Newman–Ziff and Pruessner–Moloney, who attribute them to Pinson. That is a read-through, not a primary-page check. + +--- + +## Q1 — exact wrapping results, by type + +**Universality class.** Two-dimensional percolation (Fortuin–Kasteleyn / Q=1 Potts, c=0). Pinson’s torus wrapping probabilities are **universal at criticality for a given aspect ratio / modular parameter**, not lattice-specific. Newman–Ziff evaluate them for the **square torus** (aspect 1) and use them as the L→∞ targets of **square-site** wrapping. Pruessner–Moloney check the same formulae numerically for **both site and bond** percolation on very large tori and report no excess site-versus-bond deviation from the formula. + +**They do transfer to square site.** Newman–Ziff §III.A treat square **site** percolation as the working example and quote Pinson’s evaluations as the exact R_∞(p_c) for that geometry. Akhunzhanov–Eserkepov–Tarasevich (2022), already scored in the #576 wrapping note, give exact **square-site** Bernstein polynomials for wrapping along one specified direction through L=12 (L=10 ancillary file corrupt). Independent enumeration at L=3,4 matched those polynomials bit-for-bit (`notes/literature-officer-20260905-issue576-wrapping.md`). + +**They are for one lattice copy, not for the matching lattice.** Pinson / Newman–Ziff / Akhunzhanov give wrapping of occupied clusters on one graph. Matching-lattice wrapping (vacant sites, NN+NNN) is **not** a second table in those papers. The matching half enters only through Mertens–Ziff (Q2–Q3). + +**Type resolution, Newman–Ziff definitions** (cond-mat/0101295, p. 10). On an L×L torus: + +- R^{(h)}: wraps the specified axis, and **may** also wrap the other +- R^{(e)}: wraps either axis, or both +- R^{(b)}: wraps **both** axes (cross **or** spiral; their Fig. 8) +- R^{(1)}: wraps one specified axis **and not** the other + +Identities (their (11)–(12)): + +```text +R^{(e)} = 2 R^{(h)} − R^{(b)} +R^{(1)} = R^{(h)} − R^{(b)} = (R^{(e)} − R^{(b)}) / 2 +``` + +**Pinson evaluations at p_c, aspect 1**, quoted by Newman–Ziff as (13)–(15): + +```text +R^{(e)}_∞(p_c) = 1 − [ϑ3(e^{−3π/8}) ϑ3(e^{−8π/3}) − ϑ3(e^{−3π/2}) ϑ3(e^{−2π/3})] + / (2 [η(e^{−2π})]^2) + +R^{(1)}_∞(p_c) = [3 ϑ3(e^{−6π}) + ϑ3(e^{−2π/3}) − 4 ϑ3(e^{−8π/3})] + / (√6 [η(e^{−2π})]^2) +``` + +and, to ten figures, + +```text +R^{(h)}_∞(p_c) = 0.521058290 +R^{(e)}_∞(p_c) = 0.690473725 +R^{(b)}_∞(p_c) = 0.351642855 +R^{(1)}_∞(p_c) = 0.169415435 +``` + +Duality remark, Newman–Ziff p. 10–11, verbatim in substance: R^{(e)}_∞(p_c) = 1 − π(Z×Z), because if there is no wrapping around either axis then there is a cross configuration on the **dual** lattice. That duality is **bond / dual**, not site / matching. Do not import it as a site identity. + +**Neither-wrap at aspect 1.** 1 − R^{(e)} = 0.309526275. Pinson’s surprising identity, restated by Pruessner–Moloney: the probability of a **cross** topology equals the probability that **all** clusters are homotopic to a point, π(X, r) = π(0, r). At r=1 this is the same number as 1 − R^{(e)}. + +**Winding-number formula** (Pruessner–Moloney eq. (1), attributed to Pinson): + +```text +P̂((a,b), ≥1, r) + = Σ_ℓ Z_{a 3ℓ, b 3ℓ}(2/3; r) + − ½ Σ_ℓ Z_{a(3ℓ+1), b(3ℓ+1)} + − ½ Σ_ℓ Z_{a(3ℓ+2), b(3ℓ+2)} + − Σ_ℓ Z_{a 2ℓ, b 2ℓ} + + Σ_ℓ Z_{a(2ℓ+1), b(2ℓ+1)} +``` + +with Z_{m,n}(g; r) as in that paper (g=2/3 in **this** normalization). This is Arguin’s π({a,b}), **not** Newman–Ziff R^{(h)}. At r=1, π({1,0}) = 0.169415435… = R^{(1)}, already frozen in `notes/pinson-arguin-primitive-baseline.md`. At r=2,4 they diverge: see Q5. + +**Gap inside Q1.** No published exact table for **Sq8 / NN+NNN site** wrapping polynomials. Akhunzhanov is square NN only. + +--- + +## Q2 — joint law of primal vs matching wrapping + +**Cite, combinatorial, all finite L.** Mertens–Ziff §II, not a CFT formula. On a torus, Euler’s formula plus the matching construction give, configuration-wise (their (9)–(11)): + +```text +N_black − N_white − (V − E + F0) + = +1 if black is cross-wrapping + −1 if white is cross-wrapping + 0 otherwise +``` + +and the pairing + +- no black wrap ⇒ exactly one white **cross**-wrapping cluster +- k single-wrapping black clusters ⇔ k single-wrapping white clusters +- black cross-wraps ⇔ white has **no** wrapping +- spiraling counts match: R^{s}(p) = R̂^{s}(1−p) +- one-direction only: R^{1}(p) = R̂^{1}(1−p) (their (19)) + +**Not independent copies of Pinson.** The two wrapping events live on complementary colourings of the **same** configuration. At p=p_c both sides are critical (p_c(matching)=1−p_c), so each **marginal** tends to the Pinson numbers, but the **joint** is supported only on the pairing above. + +**Exact identity for the difference**, Mertens–Ziff (20), the main theorem: + +```text +M_L(p) := N_L(p) − N̂_L(1−p) − L² χ(p) + = R^x_L(p) − R̂^x_L(1−p) + for x ∈ {c, b, e, h} +``` + +with χ_□(p) = p − 2p² + p⁴ on square site. They state explicitly: **the only contribution to the right-hand side is the cross-wrapping probabilities**; the other wrapping types cancel by the pairing. + +This **is** the repository observable. `scripts/exact_matching_polynomial.py`: + +```text +D(C) = 1{black NN wraps} − 1{white NN+NNN wraps} +M(p) = E_p[D] = Σ_k a_k p^k (1−p)^{N−k} +``` + +“Wraps” here is Newman–Ziff **either** (any nontrivial homology). By MZ (20) that equals the **cross** difference. `scripts/matched_torus_reference.py` already names the equality as the Mertens–Ziff finite matching relation. + +**What is not published.** A closed modular/CFT formula for the **joint** law (primal type, matching type) at p_c, beyond the combinatorial support and the one-dimensional difference M. Pinson does not treat the matching colouring. That is a real gap for a continuum joint, and it is **not** a gap for the finite-L pairing. + +**#640 is therefore a verification of a published pairing, not a rediscovery of the pairing**, if the wrapping classifier implements MZ types. The L=3,4 census (`notes/wrapping-type-census-l3l4-20260908.md`) finds **exactly five nonempty cells** and **zero MZ-forbidden cells**. D mass lives only on `neither×both` and `both×neither`. Complement-transpose of the 4×4 **fails** (NN ≠ NN+NNN), which is the finite-L source of `M(p)+M(1−p)≠0`. + +**Do not cite #628’s 118133 bond dual_fail as physics.** That count is an implementation artifact, repaired in PR #653. Site `M(p)+M(1−p)≠0` still stands, and is the expected square-site (non-self-matching) statement. + +**Scullard–Jacobsen connection**, MZ after (21): the criterion R^c_L(p) − R^0_L(p) = 0 is identical to M_L(p)=0, because R^0(p)=R̂^c(1−p). That sentence belongs to #637 as well; it is recorded here because it is how wrapping types become a threshold polynomial. + +--- + +## Q3 — the matching function D(C) = 1{black wraps} − 1{white wraps} + +**Where introduced.** Mertens–Ziff 2016. Two faces of the same object: + +1. Cluster-count form (15): M_L(p) = N_L(p) − N̂_L(1−p) − L² χ(p) +2. Wrapping form (20): M_L(p) = R^x_L(p) − R̂^x_L(1−p), x ∈ {c,b,e,h} + +The repository uses (2) with x = either. They prove (1)=(2) at **every finite L** on a torus. + +**What is proved about the root.** + +- M_L is strictly increasing, range in [−1,1] (because it equals a difference of probabilities). +- Unique root p*_L ∈ (0,1). +- lim_{L→∞} M_L(p) = −1 for pp_c (their (31)). Hence p*_L → p_c. This is the same qualitative convergence the repository already has from subcritical decay + matching duality (`notes/homological-balance-root-ledger-20260906.md` §2; `docs/astra/ANSWER-610-20260907.md`). +- On **self-matching** lattices (triangular site, square bond, …), M_L(p_c)=0 for **every** L, so p*_L = p_c with zero displacement (MZ (22), (24)). The L^{−4} displacement is a square-site matching-odd residual. The repository already isolated that as a theorem in the homological-balance note. + +**Rate.** **Not a theorem.** MZ: empirically p*_L − p_c ∼ L^{−w} with w≈4, citing Jacobsen 2014/2015; their own fit from M_L(p_c)∼L^{2−x} gives w=2−x−1/ν≈4.17, and a direct plot of p*_L−p_c gives slope −4.07. They say larger L is needed. Jacobsen’s transfer-matrix critical polynomials are the high-precision engine behind the “~L^{−4}” lore; that engine is #637, not a wrapping-rate theorem. + +#618’s report that Q_N(u)→p_c gives **no polynomial rate from H1–H3** is compatible: MZ also do not prove a rate. The empirical L^{−4} is an extra input, not a corollary of the finite identity. + +**Do not promote L^{−4} through threshold_claim_intake.** It is not an exact threshold. + +--- + +## Q4 — finite-size corrections: wrapping vs the difference + +**Wrapping probabilities themselves.** Newman–Ziff Fig. 10 and eq. (18): they **conjecture** R_L(p_c) − R_∞ ∼ L^{−2} (fits −1.95(17) site, −2.003(5) bond for R^{(1)}). Combined with the critical-window slope L^{1/ν}, the estimator defined by R_L(p)=R_∞(p_c) converges as + +```text +p_L − p_c ∼ L^{−2 − 1/ν} = L^{−11/4} +``` + +This is the wrapping-only rate. It is **not** the matching-function rate. + +**The difference M.** MZ scaling (36)–(39): in the scaling limit M_L(p)=f(z)−f(−z), z∝(p−p_c)L^{1/ν}. Even powers cancel, so M is analytic in z even at criticality. Corrections: M_L(p_c)∼L^{2−x} with a numerical 2−x≈−3.42, hence the root shifts as L^{2−x−1/ν}≈L^{−4.17}. If one assumes w=4 exactly, 2−x=13/4, which is the L^{−13/4} the repository already writes for M_L(p_c). + +**Shape after location/scale.** The repository’s ω≈0.970±0.031 (`results/p612-n725-score/latest.json`, spin0 exponent fit; unity not excluded) is the residual of the **threshold-law shape** after projecting out location and scale. MZ do **not** quote a correction exponent for that projected shape. + +A literature comparison, not a new fit: Newman–Ziff’s R_L(p_c)−R_∞ ∼ L^{−2} is N^{−1} in site count N=L². A leftover N^{−1} after location/scale would be ω=1. The measured 0.97±0.03 does not exclude unity. That is a **possible** identification, not a published theorem that the matching-function shape correction is the wrapping θ=2. + +**Gap.** No paper found that analyses the affine-invariant shape of M_L, or of Q_N, or quotes ω≈0.97. Q4 is cite for the **unprojected** wrapping and matching-root corrections, gap for the projected shape. + +--- + +## Q5 — aspect ratio vs the N=580 ladder + +Pinson’s wrapping probabilities depend on the modular parameter. The repository already evaluated π({1,0})(ir) at r=1,2,4 (`notes/literature-officer-20260905-issue576-wrapping.md`): + +| r | π({1,0})(ir) | ratio to r=1 | +|--:|--:|--:| +| 1 | 0.169415435321 | 1 | +| 2 | 0.503035897695 | **2.969244784222** | +| 4 | 0.855969321054 | **5.052487215408** | + +Newman–Ziff R^{(h)}(i)=0.521058290 is a **different** observable (specified-direction wrap, including simultaneous wrap). Do not score the ladder against 0.521 when the competitor is π({1,0}). + +**Does this predict the aspect-ladder amplitude ratio?** Only if the measured object **is** a Pinson wrapping. The #576 wrapping note already recorded a **non-claim**: matching-odd slope is not identified with π({1,0}). N=290’s measured 1.880±0.177 is ~6σ from 2.969, so Pinson does not explain that run. It remains a named competitor (`pinson_pi10_ratio`) that must sit on the next freeze **before** the data, alongside: + +| competitor | r=2 | r=4 | +|---|--:|--:| +| weight-4 Ê4(ri)/Ê4(i) | 2.75 | 10.99 | +| Pinson π({1,0}) ratio | 2.969 | 5.052 | +| bare aspect r | 2 | 4 | +| area r² | 4 | 16 | + +The three 11/4’s that must stay apart: modular weight-4 ratio 11/4; Newman–Ziff estimator L^{−11/4}; Pinson r=2 ratio 2.969. + +**Score.** Theory number exists for **wrapping homology class {1,0}**. It does **not**, on present evidence, predict the matching-odd aspect ladder. Converting the underpowered three-hypothesis result into a Pinson test would first require identifying the ladder’s observable with π({1,0}) — which #576 explicitly refused. That identification is still refused here. + +If a later readout **is** wrapping-flavoured, score it in the Cardy/Pinson function space, not against E4, and use the table above. + +--- + +## What this does to the live tickets + +- **#640 / #651.** The finite-L 4×4 is the MZ pairing, resolved by wrapping type. Axis L=5 is a **classifier tripwire** (does a sixth cell appear?) more than an open topological question. both-two mass is the one structural caveat MZ do not name in wrapping-type language; it is the rank-2-versus-two-component distinction inside `both`. +- **#635.** MZ prove, at every finite L, that M is exactly the difference of two **cross-wrapping** amplitudes. Verdict A in the wrapping-form is published. Whether that is the same map as Jacobsen’s two transfer-matrix sectors is #637/#635, not a wrapping-probability gap. #646’s “degenerate both-same” labels need to be matched onto MZ cross vs spiral; the L=3,4 census is compatible with that degeneracy and does not prove it. +- **#618 / #622.** Location Q_N(u)→p_c is proved (qualitative). Polynomial rate of the root is empirical L^{−4}, not a wrapping theorem. Projected shape ω is a gap relative to this literature. +- **#577 / aspect ladder.** Pinson supplies a number; it is not, without a new identification, the ladder’s theory value. + +--- + +## Boundaries + +- No STATUS edit, no ticket closed, no threshold ingested. +- #637 still owns: connectivity transfer matrices, critical polynomials, Jacobsen eigenvalue identities, Scullard–Jacobsen as a computational method. +- #620 still owns: inverse-CDF windows, DKS, F1 connecting lemma. +- #601 still owns: symmetry quotients and selection rules. diff --git a/references.bib b/references.bib index 9d04315d..99bf67eb 100644 --- a/references.bib +++ b/references.bib @@ -1,3 +1,70 @@ +@article{Pinson1994Torus, + author = {Pinson, Haru T.}, + title = {Critical percolation on the torus}, + journal = {Journal of Statistical Physics}, + volume = {75}, + number = {5-6}, + pages = {1167--1177}, + year = {1994}, + doi = {10.1007/BF02186762}, + note = {Primary JSP text not opened this session. Wrapping-type formulae and numerical values used in notes/torus-wrapping-retrieval-20260908.md are quoted from Newman--Ziff (2001) eqs. (13)--(15) and Pruessner--Moloney (cond-mat/0310361), which attribute them to this paper.} +} + +@article{NewmanZiff2001FastMC, + author = {Newman, M. E. J. and Ziff, R. M.}, + title = {A fast {Monte Carlo} algorithm for site or bond percolation}, + journal = {Physical Review E}, + volume = {64}, + number = {1}, + pages = {016706}, + year = {2001}, + doi = {10.1103/PhysRevE.64.016706}, + eprint = {cond-mat/0101295}, + archivePrefix = {arXiv}, + note = {Primary arXiv PDF read 2026-09-08. Wrapping definitions R^{(h,e,b,1)}, Pinson evaluations (13)--(15), and L^{-2} / L^{-11/4} FSS are quoted in notes/torus-wrapping-retrieval-20260908.md.} +} + +@article{MertensZiff2016Matching, + author = {Mertens, Stephan and Ziff, Robert M.}, + title = {Percolation in finite matching lattices}, + journal = {Physical Review E}, + volume = {94}, + number = {6}, + pages = {062152}, + year = {2016}, + doi = {10.1103/PhysRevE.94.062152}, + eprint = {1603.07289}, + archivePrefix = {arXiv}, + note = {Primary arXiv HTML v2 read 2026-09-08. Finite matching identity (20) is the repository matching function. Quotes: notes/torus-wrapping-retrieval-20260908.md.} +} + +@article{PruessnerMoloney2004Winding, + author = {Pruessner, Gunnar and Moloney, Nicholas R.}, + title = {Winding clusters in percolation on the torus and the {M\"obius} strip}, + journal = {Journal of Statistical Physics}, + volume = {115}, + pages = {839--853}, + year = {2004}, + doi = {10.1023/B:JOSS.0000022369.14889.4b}, + eprint = {cond-mat/0310361}, + archivePrefix = {arXiv}, + note = {Primary arXiv HTML read 2026-09-08 for Pinson's winding-number formula and the identity P(X)=P(0).} +} + +@article{Akhunzhanov2022ExactWrapping, + author = {Akhunzhanov, R. K. and Eserkepov, A. V. and Tarasevich, Y. Y.}, + title = {Exact percolation probabilities for a square lattice: site percolation on a plane, cylinder, and torus}, + journal = {Journal of Physics A: Mathematical and Theoretical}, + volume = {55}, + number = {20}, + pages = {204004}, + year = {2022}, + doi = {10.1088/1751-8121/ac5ce8}, + eprint = {2204.01517}, + archivePrefix = {arXiv}, + note = {Square-site torus wrapping along one specified direction, polynomials through L=12 (L=10 ancillary corrupt). Independent L=3,4 match in notes/literature-officer-20260905-issue576-wrapping.md.} +} + @article{Jacobsen2015EigenvalueIdentity, author = {Jacobsen, Jesper Lykke}, title = {Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley--Lieb algebras}, From bdc16eb6f6ec84159fe6faa8716001e5b32125ba Mon Sep 17 00:00:00 2001 From: Light Chain Date: Tue, 8 Sep 2026 19:39:03 +0800 Subject: [PATCH 2/2] Sharpen #642 Q2: hat R(1-p) is equal-p vacant matching wrap. Fold bibliographic corrections from PR #656 (Pinson 1167-1177, no arXiv) and four extra entries. Axis L=5 / diamond L=4 K2 five-cell support is now recorded as verification of the published pairing, not a gap. --- notes/torus-wrapping-retrieval-20260908.md | 8 ++-- references.bib | 55 +++++++++++++++++++++- 2 files changed, 58 insertions(+), 5 deletions(-) diff --git a/notes/torus-wrapping-retrieval-20260908.md b/notes/torus-wrapping-retrieval-20260908.md index dcf0516f..1c3a9898 100644 --- a/notes/torus-wrapping-retrieval-20260908.md +++ b/notes/torus-wrapping-retrieval-20260908.md @@ -10,7 +10,7 @@ Read this session, at first hand unless marked: - Repository notes already on published wrapping ground: `notes/literature-officer-20260905-issue576-wrapping.md`, `notes/pinson-arguin-primitive-baseline.md` - This session’s axis/diamond wrapping-type census: `results/wrapping-type-census/`, `notes/wrapping-type-census-l3l4-20260908.md` -Pinson, *Critical percolation on the torus*, J. Stat. Phys. **75**, 1167 (1994) is **not** opened as a PDF this session. Formulae and ten-figure values below are quoted from Newman–Ziff and Pruessner–Moloney, who attribute them to Pinson. That is a read-through, not a primary-page check. +Pinson, *Critical percolation on the torus*, J. Stat. Phys. **75**, 1167–**1177** (1994) is **not** opened as a PDF this session (Short Communication; no arXiv version; `hep-th/9309029` is a different paper). Formulae and ten-figure values below are quoted from Newman–Ziff and Pruessner–Moloney, who attribute them to Pinson. That is a read-through, not a primary-page check. --- @@ -78,7 +78,7 @@ with Z_{m,n}(g; r) as in that paper (g=2/3 in **this** normalization). This is A ## Q2 — joint law of primal vs matching wrapping -**Cite, combinatorial, all finite L.** Mertens–Ziff §II, not a CFT formula. On a torus, Euler’s formula plus the matching construction give, configuration-wise (their (9)–(11)): +**Cite, combinatorial, all finite L.** Mertens–Ziff §II, not a CFT formula. Notation trap: \(\hat R(1-p)\) is wrapping of the **vacant** matching colouring at black density \(p\), i.e. it is an equal-configuration (equal-\(p\)) statement. It is not a second independent copy of the primal lattice occupied at \(1-p\). On a torus, Euler’s formula plus the matching construction give, configuration-wise (their (9)–(11)): ```text N_black − N_white − (V − E + F0) @@ -116,9 +116,9 @@ M(p) = E_p[D] = Σ_k a_k p^k (1−p)^{N−k} “Wraps” here is Newman–Ziff **either** (any nontrivial homology). By MZ (20) that equals the **cross** difference. `scripts/matched_torus_reference.py` already names the equality as the Mertens–Ziff finite matching relation. -**What is not published.** A closed modular/CFT formula for the **joint** law (primal type, matching type) at p_c, beyond the combinatorial support and the one-dimensional difference M. Pinson does not treat the matching colouring. That is a real gap for a continuum joint, and it is **not** a gap for the finite-L pairing. +**What is not published.** A closed modular/CFT formula for the **joint** law (primal type, matching type) at p_c, beyond the combinatorial support and the one-dimensional difference M. Pinson does not treat the matching colouring. The equal-occupancy difference R(p)−R̂(p) (matching occupied at p, not vacant at 1−p) is a different observable and is also unpublished. Those are real gaps for a continuum joint. They are **not** a gap for the finite-L pairing. -**#640 is therefore a verification of a published pairing, not a rediscovery of the pairing**, if the wrapping classifier implements MZ types. The L=3,4 census (`notes/wrapping-type-census-l3l4-20260908.md`) finds **exactly five nonempty cells** and **zero MZ-forbidden cells**. D mass lives only on `neither×both` and `both×neither`. Complement-transpose of the 4×4 **fails** (NN ≠ NN+NNN), which is the finite-L source of `M(p)+M(1−p)≠0`. +**#640 is a type-resolved census of a published pairing**, not a rediscovery of an unknown identity. If the wrapping classifier implements MZ types, the 4×4 can only be supported on the five cells the pairing allows. The L=3,4 census (`notes/wrapping-type-census-l3l4-20260908.md`) finds exactly those cells and zero MZ-forbidden cells. Axis L=5 (2^25, tripwire vs PR #649 Bernstein pass) and diamond L=4 K2 (2^32, tripwire pass) repeat the same five cells; `both-two` mass is zero (#651 A-continues). D mass lives only on `neither×both` / `none×both-same` and `both×neither` / `both-same×none`. Complement-transpose of the 4×4 **fails** (NN ≠ NN+NNN), which is the finite-L source of `M(p)+M(1−p)≠0`. **Do not cite #628’s 118133 bond dual_fail as physics.** That count is an implementation artifact, repaired in PR #653. Site `M(p)+M(1−p)≠0` still stands, and is the expected square-site (non-self-matching) statement. diff --git a/references.bib b/references.bib index 99bf67eb..47cb91ca 100644 --- a/references.bib +++ b/references.bib @@ -7,7 +7,7 @@ @article{Pinson1994Torus pages = {1167--1177}, year = {1994}, doi = {10.1007/BF02186762}, - note = {Primary JSP text not opened this session. Wrapping-type formulae and numerical values used in notes/torus-wrapping-retrieval-20260908.md are quoted from Newman--Ziff (2001) eqs. (13)--(15) and Pruessner--Moloney (cond-mat/0310361), which attribute them to this paper.} + note = {Short Communication. Primary JSP text not opened this session; no arXiv version (hep-th/9309029 is a different paper). Wrapping-type formulae and numerical values used in notes/torus-wrapping-retrieval-20260908.md are quoted from Newman--Ziff (2001) eqs. (13)--(15) and Pruessner--Moloney (cond-mat/0310361), which attribute them to this paper.} } @article{NewmanZiff2001FastMC, @@ -51,6 +51,59 @@ @article{PruessnerMoloney2004Winding note = {Primary arXiv HTML read 2026-09-08 for Pinson's winding-number formula and the identity P(X)=P(0).} } +@article{ZiffLorenzKleban1999Shape, + author = {Ziff, Robert M. and Lorenz, Christian D. and Kleban, Peter}, + title = {Shape-dependent universality in percolation}, + journal = {Physica A}, + volume = {266}, + number = {1-4}, + pages = {17--26}, + year = {1999}, + doi = {10.1016/S0378-4371(98)00609-2}, + eprint = {cond-mat/9811122}, + archivePrefix = {arXiv}, + note = {Folded from PR #656. Torus wrapping depends on aspect and twist through modular functions.} +} + +@article{MorinDuchesneSaintAubin2009Homology, + author = {Morin-Duchesne, Alexi and Saint-Aubin, Yvan}, + title = {Critical exponents for the homology of {Fortuin}--{Kasteleyn} clusters on a torus}, + journal = {Physical Review E}, + volume = {80}, + number = {2}, + pages = {021130}, + year = {2009}, + doi = {10.1103/PhysRevE.80.021130}, + eprint = {0812.2925}, + archivePrefix = {arXiv} +} + +@article{Arguin2002Homology, + author = {Arguin, Louis-Fran\c{c}ois}, + title = {Homology of {Fortuin}--{Kasteleyn} clusters of {Potts} models on the torus}, + journal = {Journal of Statistical Physics}, + volume = {109}, + number = {1-2}, + pages = {301--310}, + year = {2002}, + doi = {10.1023/A:1019979326380}, + eprint = {hep-th/0111193}, + archivePrefix = {arXiv} +} + +@article{ScullardJacobsen2012Transfer, + author = {Scullard, Christian R. and Jacobsen, Jesper Lykke}, + title = {Transfer matrix computation of generalized critical polynomials in percolation}, + journal = {Journal of Physics A: Mathematical and Theoretical}, + volume = {45}, + number = {49}, + pages = {494004}, + year = {2012}, + doi = {10.1088/1751-8113/45/49/494004}, + eprint = {1209.1451}, + archivePrefix = {arXiv} +} + @article{Akhunzhanov2022ExactWrapping, author = {Akhunzhanov, R. K. and Eserkepov, A. V. and Tarasevich, Y. Y.}, title = {Exact percolation probabilities for a square lattice: site percolation on a plane, cylinder, and torus},