Literature sweep: torus rank observables, exact finite-size site polynomials (do not merge)

[LightChainr]

What this is

A bounded prior-art / novelty sweep around PR #733 and the #711/#717 dictionary
correction, executed 2026-09-12. One new file:
notes/literature-sweep-torus-rank-observables-20260912.md. Retrieval only — no
data reanalysed, no result promoted, no STATUS change, no acquisition requested.

Headline finding

The repository's central observable

r(omega) = rank im[ H_1(K(omega)) -> H_1(T^2) ]

is the inclusion-induced map on first homology that has been studied, under
other names, since 1994:

• Langlands, Pouliot, Saint-Aubin (1994): crossing probabilities on compact
  Riemann surfaces via phi: H_1(X_s) -> H_1(S) and the subgroup probabilities
  pi_G.
• Pinson, J. Stat. Phys. 75, 1167 (1994): analytic expressions for those
  probabilities as functions of the torus modular parameter.
• Arguin, arXiv:hep-th/0111193: extension to FK clusters, closed forms in Jacobi
  theta functions for Q = 1 (our case) and Q = 2.
• Morin-Duchesne & Saint-Aubin, arXiv:0812.2925: thin-torus asymptotics of
  pi({1,0}), exponents from Kac-table weights including half-integers — i.e.
  the fixed-width/long-length regime we work in.
• Duncan, Kahle & Schweinhart, arXiv:2011.11903 (AIHP 61, 2235, 2025): call the
  image elements giant cycles, prove a sharp transition via Friedgut–Kalai
  and p_c = 1/2 in the middle dimension, and cite LPSA94 / Pin94 / MDSA09 as
  the 2D precursors.

The note writes down the implied dictionary:

P0 <-> pi({0})                          P2 <-> pi(Z x Z)
P1 <-> sum_{coprime (a,b)} pi({a,b})    A_top = P2 - P0,  E_top = P2 + P0

If that is right at the event level, the scaling-limit values of both
coordinates are already known in closed form, and what we have been calling a
finite-size structure is the correction to them. That is a stronger sentence
than the one we currently make, and it is better coming from us than from a
referee.

Other entries that change what we should say

• Akhunzhanov–Eserkepov–Tarasevich, J. Phys. A 55, 204004 (2022),
  arXiv:2204.01517 — exact site-percolation polynomials on plane / cylinder /
  torus, reaching L = 12 on the torus. This is the competitive frontier for
  exact finite-size work. Their events are direction-based
  (R^{(e)}, R^{(1)}, R^{(b)}, R^{(h)}, R^{(v)}); ours are ambient-rank. The
  map between the two is a dictionary exercise we have not done.
• Bobrowski & Skraba, PRE 101, 032304 (2020), arXiv:1910.10146 — numerical
  evidence that zeros of the expected Euler characteristic approximate
  homological-percolation thresholds. Directly adjacent to our use of the
  Sykes–Essam matching polynomial and of M_B(p) as a threshold functional.
• May & Wierman (2005) and Jacobsen, arXiv:1401.7847 (explicit
  "role of the symmetries and the embedding") are the closest published
  relatives of the D4 / strong-lumping reduction. The specific state counts in
  Width-four site sources and exact rank conditioning (do not merge) #733 found no match in this sweep — a weak statement that needs re-checking
  against the transfer-matrix literature before any novelty use.
• No match found for the two-row site-source mixed response or for
  final-rank-conditioned exact sampling. Per the note's stated search boundary
  this is not evidence of novelty, only a reason to write those up first.

Deliberate omissions

• references.bib was not modified. BibTeX for the ten new entries is in
  §7 of the note so that adding them is a separate, explicit decision.
• Not searched: MathSciNet / zbMATH / Scopus / Google Scholar citation graphs,
  non-English literature, books, theses, patents; no forward-citation chasing of
  Pinson or Arguin. Several entries are known only through secondary quoting and
  are flagged in the note.
• No issue was closed or edited; [Grok retrieval] Cylinder/TM literature vs width-4 all-p matching identity #711 and Lit retrieval: cylinder/TM literature vs width-4 all-p matching identity (#711, do not merge) #717 were left untouched (per the
  previous handoff choice to route through [P1 reserve] Source-visible topology and all-width closure beyond the completed width-four laboratory #636 only).

[LightChainr]

Targeted correction to section 1a after reading arXiv:0812.2925v2 itself (PDF page 5, immediately before equations (3)–(7), plus the beginning of section 2; rendered page checked). KEEP the historical identification of the ambient-homology observable. CORRECT the claim that its continuum cusp asymptotics are the same limit as microscopic fixed width 3/4 and length m→∞.

The paper first specifies the thermodynamic/mesh→0 limit at critical temperature, then takes the modular parameter to the cusp. With continuum periods 1 and ir and mesh δ, microscopic width is w=1/δ and length is m≈r/δ. Fixing w=4 never takes that inner limit. The fixed-width rank-1 concentration in #716 holds at every fixed p∈(0,1), not only at criticality; it cannot by itself assign the paper's Kac labels to finite-width decay rates.

A precise conditional guard: if log λ_w=−γ/w+a/w^(1+ω)+o(w^(−1−ω)), then at m=wr the ratio to exp(−γr) is exp[a r/w^ω+o(r/w^ω)]. Relative convergence needs control of r/w^ω and of the remainder, not just w→∞ at each fixed r. No square-site exponent is asserted by this calculation.

Also restrict the AET2022 L≤12 statement to that paper's reported reach, not an independently verified 2026 global frontier. Its specified-direction wrapping marginal does not itself determine P0/P1/P2: the both-projections event includes rank-one spirals. The repository already has this dictionary correction, so do not redispatch it as a new unknown.

Primary: https://arxiv.org/pdf/0812.2925 (page 5); https://arxiv.org/html/2204.01517v1 (abstract, event definitions, section 2.1). No merge, STATUS change, or new computation request.