Issue 642: torus wrapping retrieval — cite-or-gap Q1–Q5 (notes only)

[LightChainr]

Theory input for #642. Notes only plus a bibliography patch; does not touch docs/STATUS.md, does not close any ticket.

What this is

One notes file, notes/torus-wrapping-retrieval-20260908.md, answering Q1–Q5 of #642 at first hand, plus eight entries in references.bib.

Headline answers

• Q1 (cite). Pinson 1994 (J. Stat. Phys. 75, 1167–1177 — Short Communication; page range sometimes quoted as 1167–1190 is wrong; no arXiv version exists, hep-th/9309029 is a different paper) computes homology-resolved torus crossing probabilities (subgroups of H₁(T²)), not Newman–Ziff coarse sectors, via theta functions of the modulus τ. Continuum Q=1 FK theory; transfer to square site is by universality (confirmed numerically for site and bond by Pruessner–Moloney 2004). Gap within Q1: no published matching-lattice table of the sector probabilities.
• Q2 (gap — a real result). No published joint law of primal and matching wrapping on a torus, no exact equal-p difference, no complementarity-defect value. The entire published record is Mertens–Ziff (PRE 94, 062152 (2016), arXiv:1603.07289 — corrected venue): the exact finite-size Sykes–Essam identity M_L(p) = R^x_L(p) − R̂^x_L(1−p) (at (p, 1−p), not equal p), per-configuration iff complementarity for the cross event, exact zeros only for self-matching/self-dual pairs. So [P1 exact] Wrapping-type-resolved census: the exact table #635 has to test any sector decomposition against #640 is the frontier, not a rediscovery. The bond dual_fail = 118133 is marked as an implementation artifact (PR Repair #632 bond laboratory; wrapping-type census L=3,4 (#640) #653), not physics; the site M(p) + M(1−p) ≠ 0 fact stands as the discrete shadow of the gap.
• Q3 (cite). The per-configuration matching function D(C) ∈ {+1,0,−1} is introduced in MZ 2016; the Scullard–Jacobsen "all equals none" criterion (2012 papers) is the conjectural-estimator ancestor. Proved: monotonicity, unique root, exact zeros for self-matching pairs. Asserted, not proved: p*ᴸ → p_c. Empirical: p*ᴸ − p_c ~ L^(−w), w ≈ 4 (w = 4 exactly is a conjecture attributed to Jacobsen). Confirms [P2 long-horizon probe] After Q_N(u)→p_c: which rates are theorems from named inputs? #618: no proved polynomial rate exists in the literature.
• Q4 (cite + literature comparison only). R_L(p_c) − R_∞(p_c) ~ L⁻² (Newman–Ziff, empirical); the famous L⁻¹¹ᐟ⁴ is the estimator correction 2 + 1/ν, not 1/ν + θ. ω = 0.970 ± 0.031 compared as literature only: L⁻² is the canonical analytic-correction exponent of this observable class; ω = 1 is consistent with it. No published treatment of corrections to a difference of wrapping probabilities (gap).
• Q5 (theory number exists). The exact aspect-ratio dependence is computable in closed form (Pruessner–Moloney eq. (9)); the Pinson π({1,0}) ratios are 2.969244784 (r=2) and 5.052487215 (r=4) — named here prospectively as a fourth hypothesis for the aspect ladder. Inside the frozen Fieller interval [2.39, 27.47], so it does not resolve the three-way ambiguity this run, but converts the ladder to scoring against an exact continuum theory number. The existing non-claim (nothing identifies P4[S′] with a Pinson wrapping) is restated.

No exact threshold is claimed; scripts/threshold_claim_intake.py run as a boundary check only.

Routing: Q2/JL-census → #640; Q3 → #567, #613/#276; Q4 rate inputs → #618/#622; Q5 → #567/#573/#575/#577. #637 and #601 deliberately untouched.

Closes nothing. Refs #642.

[LightChainr]

Q2 overclaims a gap. (\hat R(1-p)) is wrapping of the vacant matching colouring at black density (p) — equal configuration, not a second independent ensemble. Mertens–Ziff §II already states the pairing as deterministic iff's (no black wrap ⇒ white cross-wraps; single wraps pair; cross-wrapping exclusive; spirals match). That is the joint support of the repository observable.

The unpublished pieces are a CFT/modular formula for the joint at (p_c), and (R(p)-\hat R(p)) at equal occupancy of the matching lattice. Those do not make #640 a rediscovery.

Axis L=5 (2^{25}) and diamond L=4 K2 (2^{32}) now repeat the same five cells with both-two = 0 and Bernstein tripwire pass (comments on #640/#651). Bibliographic extras here (Pinson 1167–1177, Arguin, ZLK, MDS-A, Scullard–Jacobsen) are folded into PR #654.

Prefer #654 as the retrieval object. This PR can close as duplicate once you are happy the bib merge is complete.