diff --git a/notes/torus-wrapping-retrieval-20260908.md b/notes/torus-wrapping-retrieval-20260908.md new file mode 100644 index 00000000..b458db08 --- /dev/null +++ b/notes/torus-wrapping-retrieval-20260908.md @@ -0,0 +1,172 @@ +# Torus wrapping probabilities: the exact results, sector structure, and primal-versus-matching — cite-or-gap retrieval + +**Ticket:** [LightChainr/Matching-One#642](https://github.com/LightChainr/Matching-One/issues/642). **Role:** theory input, retrieval only. **Does not enter** `docs/STATUS.md`. **No claim-ledger rows.** Distinct from #637 (transfer matrices / critical polynomials), #620 (location-to-rate), #601 (symmetry quotients). + +Companion to existing notes: `notes/pinson-arguin-primitive-baseline.md` (frozen continuum values), `notes/literature-officer-20260905-mertens-matching.md` (first pass on 1603.07289), `notes/literature-officer-20260905-issue576-wrapping.md` (Pruessner–Moloney ratios, Akhunzhanov polynomials), `notes/square-bond-duality-tiny-torus.md` (finite-L transport identity). + +Primary sources were read at first hand this pass (arXiv full texts, Springer DOI metadata). Verified vs inferred is marked per item. Two bibliographic corrections to earlier passes are recorded in §0. + +--- + +## 0. Bibliographic corrections (verified this pass) + +1. **Pinson 1994 is a Short Communication: J. Stat. Phys. 75(5–6), 1167–1177** (received 15 Oct 1993, accepted 9 Feb 1994; DOI 10.1007/BF02186762). The page range 1167–1190 sometimes quoted is wrong. The paper appears to have **no arXiv version**; `hep-th/9309029` is a different paper (Deguchi, colored-link invariants — do not cite it for Pinson). +2. **Mertens–Ziff arXiv:1603.07289 is "Percolation in finite matching lattices", Phys. Rev. E 94, 062152 (2016)** — not J. Phys. A 49, 065002 as one earlier note implied. +3. The Scullard–Jacobsen critical-polynomial paper cited by MZ as their Eq.-21 provenance is the **2012** series (Scullard, *The percolation critical polynomial as a graph invariant*, PRE 86, 041131 (2012), arXiv:1111.1061; Scullard–Jacobsen, *Transfer matrix computation of generalized critical polynomials in percolation*, J. Phys. A 45, 494004 (2012), arXiv:1209.1451). No 2008 Scullard–Jacobsen J. Math. Phys. paper could be verified; MZ cite them by criterion, not by that reference. (The 2008 items are Scullard–Ziff PRL 100, 185701.) + +--- + +## Q1 — Pinson 1994 and later checks: what exactly is known + +**Cite.** H. T. Pinson, *Critical percolation on the torus*, J. Stat. Phys. **75**, 1167 (1994), DOI 10.1007/BF02186762. + +**What is computed (verified via abstract and three independent secondary readings).** Verbatim abstract: *"We compute the various crossing probabilities defined by R. Langlands, P. Pouliot, and Y. Saint-Aubin for the critical percolation on the torus."* + +- The sectors are **homology-resolved**, not Newman–Ziff coarse sectors: the subgroup of H₁(T²)≅ℤ×ℤ generated by the cluster configuration — {0} (no wrap), primitive {a,b} with gcd(a,b)=1 (wrap along one primitive cycle), and ℤ×ℤ (cross, rank 2). Morin-Duchesne–Saint-Aubin (arXiv:0812.2925, PRE 80, 021130 (2009)): *"Pinson [17] obtained analytic expressions for the probability of these various subgroups as functions of the quotient τ of the fundamental periods."* +- Coarse Newman–Ziff sectors (neither / horizontal / vertical / both) are **not** Pinson's primary output. They are recoverable by summing homology sectors (the relation `R_h = π({1,0}) + π(ℤ×ℤ) + (diagonal covers)`; see `notes/literature-officer-20260905-issue576-wrapping.md` for the non-identification of `R_h` with `π({1,0})`). The cross probability π₊ (simultaneous winding in both directions, lattice-vs-dual resolved) is given separately: Ziff–Lorenz–Kleban (cond-mat/9811122, Physica A 266, 17 (1999)) quote Pinson: *"Pinson [11] has shown π₊(r,t) = ½[Z_c(8/3) − Z_c(2/3)]"*, algebraic at rational r; π₊(1,0) = 0.309 526 275…. +- **Modular/aspect dependence (verified).** All expressions are theta/Dedekind-eta functions of the modulus τ. ZLK: *"These transformations generate the modular group, and functions invariant under them are called modular. Thus, b(r, t) must necessarily be a modular function."* The exact modular laws in the paper convention (frozen in `notes/pinson-arguin-primitive-baseline.md`): π_τ({a,b}) = π_{τ+1}({a+b,b}), π_τ({a,b}) = π_{−1/τ}({−b,a}). + +**Universality class and lattice (verified).** Continuum critical percolation via the Q→1 Potts/FK Coulomb-gas route (di Francesco–Saleur–Zuber FK→six-vertex→SOS→compactified boson, with Nienhuis RG for the continuum limit). The exact statement is for the **continuum Q=1 FK theory**; transfer to a specific lattice is by universality. ZLK: the quantities are *"independent of the 'microscopic' features of the system (lattice type or continuum model used, whether site or bond percolation, etc.)"* — so the formulas hold for square **site** percolation at p_c exactly as for square bond, up to the universality hypothesis itself. + +**Square-site transfer (verified boundary statement).** Universal amplitude in the continuum; the lattice supplies only the metric (modulus). `notes/pinson-arguin-primitive-baseline.md` already records the operational caveat: the physical isotropic modulus, not a bare combinatorial aspect ratio, enters when the lattice metric is anisotropic. Pruessner–Moloney (cond-mat/0310361, J. Stat. Phys. 115, 839 (2004)) verified numerically for **both site and bond** square-lattice percolation that the winding-cluster numbers follow the Pinson formulas — that is the published square-site confirmation. + +**Matching lattice (the item the ticket most needs).** Pinson's own text does not, as far as could be verified, publish sector-by-sector **matching-lattice** wrapping probabilities. What exists in the same circle of ideas is the lattice-vs-dual resolution of the **cross** event via Pinson's oriented-boundary/hull indicator — ZLK: *"When I=1, there is a cross-configuration on the lattice, when I=−1 there is a cross configuration on the dual-lattice (these two events are clearly mutually exclusive), and when I=0 there is neither"* — and the FK level relation Z_Q(ℤ×ℤ) = Q·Z_Q({0}) (Arguin, hep-th/0111193), which is a duality statement about the two extreme sectors at Q=1. A full matching-side table of π({a,b}) is **not in the published record**; it would have to be built from the same formulas with the matching lattice's own modulus/homology labeling, which no paper we found does. **Mark: gap within Q1**, feeding Q2. + +**Rigor status (verified quote).** Morin-Duchesne–Saint-Aubin: *"His computation is mathematically rigorous, except for the step taking the limit as the mesh goes to zero; for this, he used Nienhuis' renormalization group argument… A more rigorous treatment of this step remains open."* So: exact prediction, one non-rigorous continuum-limit step. Do not inherit it as a theorem. + +**Later checks (verified).** +- Arguin, *Homology of FK clusters of Potts models on the torus*, J. Stat. Phys. 109, 301 (2002), arXiv:hep-th/0111193: extends to Q=1..4, numerics agree for Q=1,2,3. (Q=4 disagreement is expected — different universality class.) +- Pruessner–Moloney, J. Stat. Phys. 115, 839 (2004): Monte Carlo on site and bond square tori *"fully consistent with predictions from conformal field theory"*; also gives the clean Z_{m,n} form of the formulas (their eqs. (1)–(2), (9), (15)–(16)), reproduced in `notes/literature-officer-20260905-issue576-wrapping.md`. +- Ziff–Lorenz–Kleban 1999: Monte Carlo vs Pinson as a function of **twist** t at fixed r (Figs. 2–3), and the excess-cluster-number shape values b(1)=0.8836, b(2)=0.9918, b(4)=1.5163. They also state the twist dependence: *"Besides the aspect ratio of the torus, the universality class depends upon the twist in the periodic boundary conditions…"*. +- Morin-Duchesne–Saint-Aubin 2009: critical exponents for the homology sectors (asymptotics of Pinson's formulas). This is #637-adjacent machinery, not the observable; see the boundary note at the end. +- Akhunzhanov–Eserkepov–Tarasevich, J. Phys. A 55, 204004 (2022), arXiv:2204.01517: exact finite-L wrapping **polynomials** for square site, one specified direction, L≤12 (L=10 ancillary corrupt — already flagged in `notes/literature-officer-20260905-issue576-wrapping.md`). Primal only, no matching side, no continuum formula. + +**What 0.52105829 is and is not (verified attribution).** The universal torus value for the Newman–Ziff "wrap in one specified direction" event is R_∞^(h)(p_c) = 0.521058290…, with R^(e) = 0.690473725, R^(b) = 0.351642855, R^(1) = 0.169415435 (quoted by Newman–Ziff, cond-mat/0005264, PRE 64, 016706 (2001), crediting Pinson). Note R^(1) = R^(h) − R^(b) equals the primitive-sector value π({1,0})(i) = 0.16941543532… in the frozen baseline — consistent, two labelings of the same sector content. These numbers are **continuum CFT values**; their application to square site at p_c (NZ's numerics) is by universality. + +--- + +## Q2 — the joint law of primal and matching wrapping on a torus + +**Verdict: gap. No published joint law, no published exact expression for the difference of wrapping probabilities at equal p, and no exact complementarity-defect value — beyond the Mertens–Ziff difference identity, which is a statement at (p, 1−p), not at equal p.** (Verified by full-text inspection of arXiv:1603.07289 this pass, plus targeted searches 2016–2026; see also the first-pass note `notes/literature-officer-20260905-mertens-matching.md`, which found zero repo hits for 1603.07289 before that pass.) + +What IS published (all verified from the MZ full text): + +1. **The finite-size Sykes–Essam identity** — MZ's main result, eqs. (4)/(20): for any x ∈ {cross, both, either, horizontal}, + ```text + N_L(p) − N̂_L(1−p) − L² χ(p) = R^x_L(p) − R̂^x_L(1−p) + ``` + exactly, for every L and every p, with χ the matching polynomial (square site: χ_□(p) = p − 2p² + p⁴, their eq. (2)). The right-hand side is the difference of wrapping probabilities on the (square, Sq8) pair — at matching-side occupancy **1−p**, by construction of white = vacant. +2. **Per-configuration complementarity, stated as deterministic iff's, not a joint law** (MZ, duality section): *"a cross-wrapping black cluster on the primal lattice exists if and only if there is no wrapping white cluster on the matching lattice, and vice versa"*, giving R^0_L(p) = R̂^c_L(1−p); and the single/spiral equalities R^1_L(p) = R̂^1_L(1−p) (their eq. (19)). "Both wrap or neither" is not forbidden — those events' contributions cancel in M_L. The complementarity defect is exactly the {+1, 0, −1} of their eq. (11); P(both wrap) never appears as an independent quantity. +3. **Exact zero only for self-matching/self-dual pairs** (their eqs. (22)/(24)): M_L(p_c) = 0 for all L when the lattice is its own matching (square bond at p=1/2, triangular-honeycomb bond, martini) — consistent with the repo's exact tiny-torus result `notes/square-bond-duality-tiny-torus.md` (E[D(1/2)] = 0 for every L, every channel). +4. **Convergence of the root** (their assertion + numerics): M_L is monotone with a unique root p*_L ∈ (0,1), *"which converges to the critical density p_c as L→∞"* — asserted via the limit M_L → ∓1 off criticality, resting on cluster-number asymptotics, **not a rigorous proof**. The scaling ansatz gives M_L(p_c) ~ L^{2−x} with fitted 2−x = −3.42 (exact data L=3..11, MC to L=128); x−2 = 3.25 exactly is **Jacobsen's conjecture** (w = 2−x−1/ν = 4), an ansatz-consistency argument, not a theorem. + +**Consequences for the repo:** + +- **#640 is the frontier, not a rediscovery.** The joint distribution P(primal wraps, matching wraps), P(both), P(neither) on a torus is unpublished; the exact equal-p difference R_primal(p) − R̂_matching(p) is unpublished; M_L(p_c) for square site has no exact value in the literature, only the −3.42 fit and the −3.25 conjecture. +- The **bond** `dual_fail = 118133` datum belongs to the implementation artifact category (PR #653 repairs the bond laboratory; the geometric transport T, not bit-complement, is the correct duality involution — `notes/square-bond-duality-tiny-torus.md` already said bit-complement fails on L=2,3 by 138 and 147560). **Do not cite 118133 as physics.** What remains standing from #628 is the **site** statement M(p) + M(1−p) ≠ 0 exactly at finite L — which is the discrete shadow of the Q2 gap: MZ's identity evaluates matching-side quantities at 1−p, and no published identity transports the complement to equal p on the torus. On the plane the Sykes–Essam complementarity is exact; on the torus it fails, and that failure is exactly what #640 measures and what the literature does not yet contain. +- **Rigorous probability: also a gap** (searched, not exhaustive): no CLE₆/near-critical result on joint primal–dual noncontractible loop events on a torus was found. On the plane primal and dual crossings are exclusive (cardioid/hexagon dichotomy); on the torus both can wrap or neither, and the continuum theory of that defect appears to be open territory. + +**Boundary note:** the exact FK-level relation Z_Q(ℤ×ℤ) = Q·Z_Q({0}) (Arguin) is the closest published cousin of a duality statement among sectors, but it is a partition-function weight relation inside one model, not a primal/matching joint law. At Q=1 it is part of why trivial and cross weights are equal in the frozen baseline. + +--- + +## Q3 — the Mertens–Ziff matching function D(C): provenance and what is proved + +**Cite.** Mertens–Ziff, PRE 94, 062152 (2016), arXiv:1603.07289 (v2 20 Nov 2016). **Cite-or-gap verdict: this is the introduction of the per-configuration matching function D(C) ∈ {+1,0,−1} with M_L = E[D] as a threshold observable.** + +- **What it is there (verified, eq. (11) and eq. (20)):** the per-configuration difference of wrapping indicators between primal and matching side takes values {+1, −1, 0}; averaging gives the matching function M_L(p) = R^x_L(p) − R̂^x_L(1−p) (their eq. (20)) for each wrapping class x. The repo's `D(C) = 1{black NN wraps} − 1{white NN+NNN wraps}` is this object at equal p — note again that MZ's identity evaluates the white side at 1−p; the equal-p version is exactly the unpublished gap of Q2. +- **Precedence (verified):** the *criticality criterion* "all equals none", R^c_L(p) − R^0_L(p) = 0, is explicitly labeled by MZ as the Scullard–Jacobsen criterion (Scullard 2012 arXiv:1111.1061 / PRE 86, 041131; Scullard–Jacobsen 2012 arXiv:1209.1451 / J. Phys. A 45, 494004). In that literature the torus wrapping criterion is a **conjectural estimator**: for exactly-solved lattices it is an identity, otherwise *"the root of Eq. (6) provides an estimate"*, with the conjecture *"essentially equivalent with the universality conjecture for crossing probabilities"*. Sykes–Essam 1964 is the plane-level ancestor (matching relation, χ polynomial), but the finite-torus wrapping difference is MZ's. +- **What is proved (verified):** M_L(p) ∈ [−1,1], monotone, unique root in (0,1); M_L(p_c) = 0 for all L for self-matching/self-dual pairs; Sykes–Essam recovered as lim_{L→∞} L^{−2} M_L = 0. The convergence of p*_L to p_c is asserted via the off-critical limit (cluster-number asymptotics), not rigorously proved. **Mark: the p_c-convergence itself is an assertion in MZ, not a theorem.** +- **Rate (verified quotes):** *"Empirically, the rate of convergence is p*_L − p_c ~ L^{−w} with w ≈ 4."* And: *"This is significantly faster than the convergence of estimators derived from wrapping probabilities in the primary lattice alone, which converge like L^{−2.75}."* Their scaling argument (not rigorous): p*_L − p_c ~ L^{2−x−1/ν} (eq. (39)); measured 2−x = −3.42 with slope M′ ≈ 1/ν = 3/4 gives w ≈ 4.17; fitting with p_c fixed gives w = 4.07. So **w ≈ 4 is empirical; w = 4 exactly is a conjecture (attributed to Jacobsen)**. +- **Unblocking value for #618/#622/#613/#276:** the literature does supply a *named, faster* estimator with an empirical rate and a conjectural mechanism (2−x−1/ν), but **no proved polynomial rate**. So #618's finding ("no polynomial rate from H1–H3") is confirmed as the state of the art: if a proof of rate ≥ polynomial for M(p_N) = 0 does not exist in MZ, it does not exist yet anywhere we could find. + +--- + +## Q4 — finite-size corrections to torus wrapping probabilities and to their difference + +**Cite.** Newman–Ziff (cond-mat/0005264, PRE 64, 016706 (2001)); Ziff (arXiv:1103.3243, PRE 83, 020107 (2011)); MZ 2016 (above); Ziff–Lorenz–Kleban 1999; Pruessner–Moloney 2004. + +**The structure (verified quotes).** + +- **R_L(p_c) itself:** NZ, verbatim: *"For each of the definitions of R_L we find numerically that the difference R_L(p_c) − R_∞(p_c) scales approximately as L⁻²."* So the leading correction to the wrapping probability is **L⁻²** — empirical ("approximately"), consistent with an analytic-background correction; the derivation via the stretched-exponential tail is inferred, not quoted. Subleading terms beyond L⁻² are not resolved in NZ. +- **Derivative-based estimators:** the p_c estimator built from R has shift (correction)/(dR/dp) with dR/dp ~ L^{1/ν}, giving **L^{−2−1/ν} = L^{−11/4}**: *"Since the width of the critical region scales as L⁻¹/ν, this implies that our estimates of p_c in finite systems should have a leading order finite-size correction which goes as L⁻²⁻¹/ν = L⁻¹¹/⁴."* Ziff 2011 confirms for wrapping estimators. **The 11/4 is 2 + 1/ν, not 1/ν + θ with an irrelevant exponent** — the θ = 3/4 coincidence with the old Gaunt–Sykes guess is a numerical accident; the modern leading-irrelevant story (Ω = 72/91 ≈ 0.791, from Aharony–Asikainen on den Nijs, and Cardy's annulus result per Ziff 2011) gives different numbers. The repo already keeps the three 11/4s apart (`notes/literature-officer-20260905-issue576-wrapping.md`); this retrieval confirms the reading and adds: **none of them is a leading-irrelevant exponent ω of percolation.** +- **Correction to the difference:** no published treatment was found of FSS corrections to a *difference* of two wrapping probabilities (primal vs matching, or any two models). For self-dual bond at p=1/2 the difference vanishes identically for all L (MZ eq. (24); repo tiny-torus identity). For square site, the difference observable is M_L(p_c) itself, whose scale is L^{2−x} with fitted −3.42 / conjectured −3.25 (Q3) — i.e., the published record has no statement of the form "corrections to the difference are smaller than to either factor". **Mark: gap**, with the one relevant structure note: the leading L⁻² correction of R is dominated by the analytic background and is *the same* for primal and matching up to amplitudes (both are the same universality class at their own critical points); a difference could in principle cancel that analytic part, leaving the slower universal piece or the faster cluster-number scale — this is a **conjecture**, not a published statement. + +**Comparison to the measured ω ≈ 0.97 (literature comparison only, no new fit).** The measured shape-flow amplitude `A(N) ~ N^−ω, ω = 0.970 ± 0.031` (`docs/ROADMAP.md`; `notes/p582-amplitude-law-20260906.md` — one-exponent law rejected at χ² = 277.4/3df but misfit ≤ 3.8%, ω = 1 not excluded; N⁻¹ = L⁻² on a square torus): + +- ω = 1 ↔ L⁻² is **the leading analytic correction exponent of NZ's wrapping probability** — the same number the literature reports for R_L(p_c) − R_∞(p_c). The comparison is admissible *as a literature number*: the repo's own ROADMAP wording ("a coincidence of numbers, not a named percolation exponent") remains correct, because the measured object is a projected shape amplitude, not R itself; but the retrieval says the number is not unattached either — **L⁻² is the canonical correction exponent of this class of observable** (analytic background), and unity-excluded ω = 0.970 is consistent with it at well under 1σ. +- ω ≈ 1 is **not** 11/4 (that is an estimator-rate exponent, different observable), **not** Ω = 72/91 (leading irrelevant, cluster observables), **not** the matching-function w ≈ 4 (a root-convergence rate, a different composite). Whether the difference observable cancels its L⁻² analytic part is unknown (gap above), so the literature neither confirms nor denies that a difference-type shape amplitude should sit at exactly L⁻²; what it says is that L⁻² is the *prior* exponent for wrapping-type amplitudes, before any projection argument. + +--- + +## Q5 — does the exact aspect-ratio dependence predict the N=580 aspect-ladder ratio? + +**Cite.** Pinson 1994; Pruessner–Moloney 2004 (explicit Z_{m,n} form, eqs. (1)–(2), (9), (15)–(16)); Ziff–Lorenz–Kleban 1999 (shape-dependence statements); frozen repo values `predictions/p156_pinson_arguin_baselines_20260829.json` and `notes/literature-officer-20260905-issue576-wrapping.md`. + +**The formula exists (verified, and independently re-evaluated this pass).** With τ = ir, Pruessner–Moloney eq. (9): +```text +P̂((a,b), ≥1, r) = (1/η(e^{−2πr})²) · [½ Z̃((a,b); 6, r) − Z̃((a,b); 8/3, r) + ½ Z̃((a,b); 2/3, r)], +Z̃((a,b); g, r) = √(g/r) Σ_l exp(−l² π g (a²/r + b² r)). +``` +This is a **continuous modular function of r** — not a power law. Evaluating the {1,0} sector (verified to 10 digits against the frozen baseline at r=1): + +| r | P({1,0})(ir) | ratio to r=1 | +|---:|---:|---:| +| 1 | 0.1694154353 | 1 | +| 2 | 0.5030358977 | **2.969244784** | +| 4 | 0.8559693211 | **5.052487215** | + +(The ratios are not tabulated in Pruessner–Moloney; they are derivable from the published formula and were confirmed by independent evaluation this pass and previously in `notes/literature-officer-20260905-issue576-wrapping.md`. Large-r expansion, their eq. (15): P̂((1,0)) ≈ 1 − 2e^{−5πr/24} + e^{−πr/2} + …, relative error < 10⁻¹⁵ at r=4.) + +**Scoring as a theory number against the three Fieller survivors** (`results/aspect-ladder-n580/latest.json`; 3σ interval on A4(4i)/A4(i) = [2.39, 27.47], denominator A4(i) only 3.6σ from zero): + +| law | r=4 prediction | verdict in the frozen ladder | +|---|---:|---| +| **Pinson π({1,0}) ratio (theory number, new)** | **5.0525** | inside [2.39, 27.47]; z ≈ −0.28 against Fieller center would need the covariance redo — see caveat | +| bare aspect ratio | 4.0000 | survives (z = +0.50) | +| weight-4 modular shape Ê4(4i)/Ê4(i) | 10.9908 | survives (z = −2.08) | +| plain area scaling | 16.0 | survives (z = −2.56) | + +- **Pinson's ratio is admissible as a fourth hypothesis and sits comfortably inside the Fieller interval** — closer to the interval's low end than the other survivors. Two honest caveats, both already recorded in the repo: (a) the ladder statistic A4 is the spin-4 projector of the **matching-odd slope**, and the non-claim of `notes/literature-officer-20260905-issue576-wrapping.md` stands — **nothing identifies P4[S'] with a Pinson wrapping**, and N=290's measured 1.880 ± 0.177 was 6σ from 2.969; (b) the Pinson number is a *ratio of wrapping probabilities of a scalar homology sector*, while the ladder measures an orientation-resolved odd-sector amplitude; the modular weight-4 shape Ê4 is the paper's own candidate and Pinson is a competitor, not an identification. Mark: **the formula exists, the theory number is 5.0525 at r=4 (2.9692 at r=2); whether it describes this observable is the same open question as before, now with the number stated prospectively rather than post-hoc.** +- **Aspect-ladder conversion value (the ticket's Q5 ask):** yes — the exact aspect-ratio dependence **exists and is computable in closed form**, so the ladder can be scored against `pinson_pi10_ratio` as a fourth named hypothesis in the next frozen design. But note it does **not** reduce the underpowered three-way ambiguity this run: 5.05 is between bare-aspect 4 and weight-4 11, and the current Fieller interval [2.39, 27.47] excludes none of the four. What it does do is convert "three hypotheses" into "four named numbers, one of them from an exact continuum theory", which is what the ladder needed. Also note the **twist** caveat: ZLK showed the universal values depend on the twist t of the boundary conditions as well as on r; the repo's frozen ladder uses untwisted tori, where t = 0 is the natural labeling — but any future design that changes boundary-condition labeling must re-derive the theory number at that twist. + +**Threshold-intake note:** no exact threshold value is claimed anywhere in this note; the only closed-form-adjacent numbers are wrapping probabilities at known-critical points (p = 1/2 bond, p_c site) and ratios thereof, which are not p_c claims. `scripts/threshold_claim_intake.py` was run as a boundary check (0.592746 survives as expected — it is the established estimate, and the tool confirms, "not a confirmation"). + +--- + +## Ticket routing + +| answer | belongs to | +|---|---| +| Q1 exact formulas, modular laws, sector structure | this note (#642) | +| Q1 continuum→finite-L transfer, homology conventions | #156 / `pinson-arguin-primitive-baseline.md` (already there) | +| Q2 joint law gap, M_L scaling, #640 frontier status | **#640** (finite-L census); this note is the citation | +| Q2 bond dual_fail artifact | PR #653 (already open) | +| Q3 MZ matching function, w ≈ 4 | **#567** and the critical-bridge tickets **#613/#276** (rate remains unproved — unblocks nothing, confirms the gap) | +| Q4 L⁻² vs 11/4 vs Ω vs w reading | **#618/#622** (rate inputs); ω literature comparison stays with the ROADMAP shape-flow line, no new fit | +| Q5 pinson_pi10_ratio as fourth named hypothesis | **#567/#573/#575/#577** (next frozen ladder design) | +| transfer-matrix / critical-polynomial machinery | #637 (untouched here by design) | +| symmetry quotients, selection rules | #601 (untouched) | + +## Not established + +- that the transfer engine's wrapping channel agrees with the Akhunzhanov polynomials (never run); +- any identification of matching-odd P4[S'] with Pinson π({1,0}) (explicit non-claim, restated); +- a published table of Sq8/NN+NNN wrapping (still none); +- a published joint primal/matching wrapping law or equal-p difference (the Q2 gap — the negative result of this retrieval); +- a proof of the convergence rate of M(p_N) = 0 (w ≈ 4 empirical, w = 4 conjectural); +- that the L⁻² analytic correction exponent applies to the projected difference amplitude (plausible prior, unproved); +- anything in the claim ledger. + +## Primary sources read this pass + +- H. T. Pinson, J. Stat. Phys. 75, 1167–1177 (1994), DOI 10.1007/BF02186762 (via Springer metadata + secondary full-text quotes; no arXiv version exists). +- S. Mertens, R. M. Ziff, PRE 94, 062152 (2016), arXiv:1603.07289 (full text). +- G. Pruessner, N. R. Moloney, J. Stat. Phys. 115, 839 (2004), arXiv:cond-mat/0310361 (full text). +- L.-P. Arguin, J. Stat. Phys. 109, 301 (2002), arXiv:hep-th/0111193. +- A. Morin-Duchesne, Y. Saint-Aubin, PRE 80, 021130 (2009), arXiv:0812.2925. +- R. M. Ziff, C. D. Lorenz, P. Kleban, Physica A 266, 17 (1999), arXiv:cond-mat/9811122. +- M. E. J. Newman, R. M. Ziff, PRE 64, 016706 (2001), arXiv:cond-mat/0005264. +- R. M. Ziff, PRE 83, 020107(R) (2011), arXiv:1103.3243. +- C. R. Scullard, PRE 86, 041131 (2012), arXiv:1111.1061; C. R. Scullard, J. L. Jacobsen, J. Phys. A 45, 494004 (2012), arXiv:1209.1451. +- R. K. Akhunzhanov, A. V. Eserkepov, Yu. Yu. Tarasevich, J. Phys. A 55, 204004 (2022), arXiv:2204.01517. diff --git a/references.bib b/references.bib index 9d04315d..f823bac6 100644 --- a/references.bib +++ b/references.bib @@ -129,3 +129,146 @@ @article{ZiffScullard2006BowTie archivePrefix = {arXiv}, note = {Primary text read 2026-09-05 via arXiv (LightChainr/Matching-One#574). Prints the Wierman bow-tie condition, and its Table 3(d) bow-tie is the same relation as the martini-A / A lattice of Ziff2006CellDualCell up to overall sign -- not an independent height-4 datum. Independent generalized bow-ties are Table 3(b) and 3(c), both degree 11 at heights 36 and 35. Quotes: notes/literature-officer-20260905-issue574-quotes.md} } + +@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 = {Short Communication. Exact torus crossing/homology probabilities in terms of + theta functions of the modulus. Bibliographic data and abstract verified via Springer + 2026-09-08 (no arXiv version exists; hep-th/9309029 is a different paper). + Full-text statements quoted at second hand via Arguin hep-th/0111193, + Morin-Duchesne--Saint-Aubin arXiv:0812.2925 and Ziff--Lorenz--Kleban arXiv:cond-mat/9811122. + Quotes: 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}, + primaryClass = {cond-mat.stat-mech}, + note = {Primary text read 2026-09-08 via arXiv (LightChainr/Matching-One#642). Finite-size + Sykes--Essam identity, eq. (4)/(20): M_L(p) = R^x_L(p) - Rhat^x_L(1-p) exactly for all + L and p; matching function D(C) in {+1,0,-1} per configuration; M_L(p_c) = 0 for all L + for self-matching/self-dual pairs; root p*_L - p_c ~ L^{-w} empirical w ~ 4, faster than + wrapping-estimator L^{-11/4}. 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}, + number = {3--4}, + pages = {839--853}, + year = {2004}, + doi = {10.1007/s10955-004-5725-2}, + eprint = {cond-mat/0310361}, + archivePrefix = {arXiv}, + primaryClass = {cond-mat.stat-mech}, + note = {Primary text read 2026-09-08 via arXiv (LightChainr/Matching-One#642). Monte Carlo + (square site and bond) fully consistent with the Pinson/Arguin winding-cluster + predictions; gives the Z_{m,n}(g;r) form of the exact formulas, eqs. (1)-(2), (9), + (15)-(16). Quotes: notes/torus-wrapping-retrieval-20260908.md} +} + +@article{ZiffLorenzKleban1999Shape, + author = {Ziff, Robert M. and Lorenz, Christian D. and Kleban, Peter}, + title = {Shape-dependent universality in percolation}, + journal = {Physica A: Statistical Mechanics and its Applications}, + volume = {266}, + number = {1--4}, + pages = {17--26}, + year = {1999}, + doi = {10.1016/S0378-4371(98)00609-2}, + eprint = {cond-mat/9811122}, + archivePrefix = {arXiv}, + primaryClass = {cond-mat.stat-mech}, + note = {Primary text read 2026-09-08 via arXiv (LightChainr/Matching-One#642). Torus + universality depends on aspect ratio and twist through modular functions; + lattice-vs-dual cross-configuration resolution via Pinson's oriented-boundary + indicator. Quotes: notes/torus-wrapping-retrieval-20260908.md} +} + +@article{MorinDuchesneSaintAubin2009Homology, + author = {Morin-Duchesne, Alexi and Saint-Aubin, Yvan}, + title = {Critical exponents for the homology of {F}ortuin--{K}asteleyn 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}, + primaryClass = {math-ph}, + note = {Asymptotics (critical exponents) of the Pinson homology-sector probabilities; + records that Pinson's computation is rigorous except the continuum-limit step, + which uses Nienhuis' RG argument. Read via arXiv abstract and quoted passages + 2026-09-08. Quotes: notes/torus-wrapping-retrieval-20260908.md} +} + +@article{NewmanZiff2001Fast, + author = {Newman, Mark E. J. and Ziff, Robert M.}, + title = {Fast {M}onte {C}arlo 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/0005264}, + archivePrefix = {arXiv}, + primaryClass = {cond-mat.stat-mech}, + note = {Source of the universal torus wrapping values R^{(h)} = 0.521058290, + R^{(e)} = 0.690473725, R^{(b)} = 0.351642855, R^{(1)} = 0.169415435 at criticality + (crediting Pinson), and of the L^{-2} correction to R_L(p_c) itself vs the + L^{-11/4} = L^{-2-1/nu} correction to wrapping-based p_c estimators. + Read via arXiv 2026-09-08. Quotes: notes/torus-wrapping-retrieval-20260908.md} +} + +@article{Arguin2002Homology, + author = {Arguin, Louis-Fran\c{c}ois}, + title = {Homology of {F}ortuin--{K}asteleyn clusters of {P}otts 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}, + primaryClass = {hep-th}, + note = {Extends Pinson's torus homology probabilities from Q=1 to Q=1..4; numerics agree + for Q=1,2,3. Read via arXiv 2026-09-08. Quotes: notes/torus-wrapping-retrieval-20260908.md} +} + +@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}, + primaryClass = {cond-mat.stat-mech}, + note = {Probabilistic (transfer-matrix) definition of the critical polynomial P_B(p). + The torus wrapping criterion "all equals none" used here is a conjectural + estimator for unsolved lattices, exact only for solved ones -- the reading + recorded in notes/torus-wrapping-retrieval-20260908.md} +}