#613: scope review of the #276 critical-point bridge — all five items discharge (proof note, notes only)

[LightChainr]

Closes nothing; delivers #613's bounded proof/literature task. Notes only — no STATUS edit, no merge, no ticket close.

Deliverable

notes/proof-613-bridge-scope-review-20260908.md — an independent verification of the five #613 items against the proof in docs/astra/ANSWER-610-20260907.md (Q4) and the repository's digital Alexander duality note.

Verdict

Not falsified. All five items discharge:

1. Lift/embedded-ball union bound — proved with the explicit conservative radius R_N = floor(ell_N/4), valid for both NN (|y|_1<=R => |y|_2<=R) and NN+NNN (diameter <= 0.708 ell_N < ell_N) adjacency; union bound gives C(p) N exp[-(c/4) ell_N] -> 0 iff ell_N/log N -> infinity.
2. Site exponential decay — cited: Duminil-Copin–Tassion Thm 1.1(3) + their §1.2 'Site percolation' remark (a remark, not a numbered site theorem; the Aizenman–Barsky/Menshikov route covers site directly). No unqualified bond theorem used.
3. Amenable matching relation — cited through Grimmett–Li, Hyperbolic site percolation (arXiv:2203.00981): p_u(G1)+p_c(G2)=1 plus p_c=p_u for amenable G (Burton–Keane). Citation precision fix: the RSA 2024 paper (DOI 10.1002/rsa.21226) states (1.1) but proves only the strict-inequality criterion; the equality lives in the companion. van den Berg's counterexample caveat recorded.
4. Scope — repository lemma, not a new theorem. Duncan–Kahle–Schweinhart (arXiv:2011.11903, AIHP 2025) already holds the giant-cycle sharp-threshold mechanism for the square-bond torus; the square-site matching instance + digital Alexander duality are the repository's own additions. No novelty claim.
5. Failure modes — thin tori, unbounded extrapolation weights, u-grid endpoints all correctly excluded; non-claims (no rate, no value, no shape) restated.

Scope corrections recorded

• ell_N/log N -> infinity is sufficient, not proved necessary, for the fixed-quantile conclusion (matches the audit clarification on the issue).
• Item 3's citation should route through the Hyperbolic site percolation theorem with the van den Berg caveat attached to any generalization beyond amenable matching pairs.

No Monte Carlo; no simulation artifacts touched; single notes file per convention.

[LightChainr]

Verification comment, not a merge request. Ticket stays open.

Notes-only PR, five items each have a carrier, no STATUS, no Monte Carlo. Structural check against #613:

1. Explicit R_N = floor(ell_N/4) for both NN and NN+NNN, with the graph/Euclidean conversion written down.
2. DCT Thm 1.1(3) + §1.2 site remark, labelled as a remark, plus Aizenman–Barsky/Menshikov. No unqualified bond theorem.
3. Citation split is the useful part: RSA 2024 (10.1002/rsa.21226) states (1.1); the equality is routed through Hyperbolic site percolation (arXiv:2203.00981) + Burton–Keane, with the van den Berg counterexample attached to anything beyond amenable matching pairs.
4. Scoped as a repository lemma; DKS already has the giant-cycle mechanism for square-bond tori. Square-site matching + digital Alexander are the repo additions. No novelty claim.
5. Thin tori, unbounded weights, u_N → {0,1} kept outside. Non-claims (no rate, no value, no shape) restated.

Two scope corrections travel with the lemma: ell_N/log N → ∞ is sufficient not proved necessary; item-3 citation must not be the RSA paper alone.

I did not re-read the Grimmett–Li / DKS PDFs in this pass — this is a scope/structure verify of the note, not an independent legal-review of the citations. Do not close #613 / #276.