[P2 rigorous] Feasibility study for certified finite-size bounds on square-site p_c

[LightChainr]

Motivation

The main project is numerical/operator-focused, but there is a separate route to a theorem-level result: turn finite crossing/wrapping computations into a rigorous bound or rigorous confidence interval for square-site p_c using a published finite-size criterion / renormalization reduction.

This will not compete with 1e-12 numerical threshold estimates. Its value is qualitative: a completely auditable mathematical statement derived from finite computation.

Literature anchor

Riordan & Walters, Phys. Rev. E 76, 011110 (2007), "Rigorous confidence intervals for critical probabilities", use the Balister–Bollobás–Walters finite reduction to obtain rigorous high-confidence intervals for site/bond thresholds of Archimedean lattices. Related finite-size criteria and 1-dependent renormalization methods are standard in rigorous percolation.

Phase 0: feasibility only

Before writing a production codebase, answer:

1. Which published finite-size criterion applies directly to square-site percolation on Z^2?
2. What finite rectangle/annulus/circuit events and numerical thresholds are required?
3. Can our existing exact/Monte-Carlo engines estimate those events with a mathematically valid error bound (binomial tail / exact rational / interval arithmetic)?
4. What lattice size and sample count would be needed for any nontrivial improvement over textbook bounds?
5. Is a deterministic exact enumeration/transfer-matrix variant feasible for the required event, or is the result necessarily a rigorous confidence interval rather than a deterministic theorem?

Deliverable

A short feasibility note with a concrete theorem statement of the form that could actually be certified, including all constants from the chosen finite-size criterion and a power/cost estimate.

Only open a production implementation if the resulting bound would be scientifically meaningful relative to effort.

Possible outcome classes

• Go: an auditable interval/bound can be obtained with modest compute; implement it.
• Methodologically interesting but weak: document the criterion and stop before expensive production.
• No-go: current rigorous constants make the required finite event unrealistically rare/large; close the route without spending server time.

Priority

P2. This should not delay #43/#57. It is included because a theorem-level side result would diversify the project beyond asymptotic model selection.

[LightChainr]

I am claiming a bounded Phase 0 certification/cost gate.

Scope:

1. instantiate the Riordan--Walters block event directly for square-site percolation: two adjacent s x s cells, a unique largest open cluster in each half, and connection of those clusters in the 2s x s union;
2. record why these block bonds are 1-independent and why a successful infinite block path implies an original open cluster;
3. update the legacy p0=0.8639 criterion with the rigorous 2022 bound p_max(Z^2)<=0.8457;
4. implement an exact-Fraction binomial-tail oracle for predeclared family-wise error, reproduce the 2007 N=400,m>=378 tail, and calculate the modern success cutoff/power table;
5. separate a high-confidence statistical interval from a deterministic theorem, including the IID/randomness assumption and the site-matching lower-bound route;
6. give a frozen heuristic scale/cost multiplier table, clearly labelled non-rigorous, to decide whether production is worthwhile.

No production simulation or new bound will be claimed. The parent will remain open; the deliverable is a go/weak/no-go feasibility classification and an executable certification protocol.

[LightChainr]

Phase 0 has been merged via PR 278, squash commit 8433863a6431be4061c2f6b9fcd962316618715f.

Exact method conclusion:

• The Riordan--Walters event applies directly to square-site percolation: two adjacent s x s cells must each have a unique largest open cluster, and those clusters must connect in the 2s x s union.
• Nonincident block bonds depend on disjoint rectangles, giving a 1-independent block model; an infinite block path lifts to an infinite original cluster.
• The 2007 deterministic marginal constant was 0.8639; the rigorous 2022 result improves the usable strict threshold to p_max(Z2)<=0.8457.

Exact statistical results under family-wise alpha 1e-6, two sides, and at most three final attempts per side:

• per-run alpha: 1/6000000;
• legacy N=400: require M>=378, exact tail 1.14899035289400922e-7;
• modern N=400: require M>=373, exact tail 9.51451513592604452e-8;
• one fewer success fails each exact inequality.

At the modern cutoff, power is about 0.01494, 0.20566, 0.77376, 0.95201 when the true event probability is 0.90,0.92,0.94,0.95. Thus statistical sample count is not the main obstacle; the geometric block must already push the event into the mid-0.9 range.

Feasibility class: methodologically interesting but weak. The historical square-site interval already has width 5e-4. Under the explicitly heuristic delta p~s^-3/4 planning law, width 1e-4 costs about 8.55x in linear scale and 73.1x in area work.

Validation: focused 8/8, combined local regression 35/35, full CI green on C++17 and Python 3.9/3.11/3.13, and post-merge audit confirms exactly six files.

Boundary: this is a high-confidence statistical route conditional on fresh independent genuine Bernoulli trials, not a deterministic decimal theorem. The parent remains open for the exact open-boundary event evaluator and any later production decision.

[LightChainr]

I am claiming a second, separate task: the smallest open-boundary block-event exact oracle required by the Phase 0 production gate.

Scope:

1. implement the Riordan--Walters unique-largest-cluster connection event on two adjacent s x s cells;
2. support both square NN site adjacency and the square site-matching NN+diagonal adjacency used for the lower interval side;
3. exhaust every configuration for s=1,2, recording success counts by occupied-site number as an exact reliability polynomial;
4. verify the event decision independently through component labels and union connectivity;
5. freeze boundary, tie, empty-half, and graph-adjacency semantics in tests.

This is a semantics/exact-control task only. It will not simulate production scales, estimate p_c, or reuse exploratory samples for certification. The parent remains open after this slice.

[LightChainr]

Tiny open-boundary event-semantics slice completed and merged in #279 (squash commit 4d6403e4ba3d45c8432d7796b776ca3517d75538).

Exact conclusions:

• Exhausted every site configuration for adjacent open s x s cells at s=1,2, for square NN and square matching (NN plus both diagonals) adjacency.
• Frozen the event ordering: reject an empty half; reject a largest-cluster tie; choose the unique largest component in each half; then test those selected components for connectivity in the full 2s x s rectangle.
• At s=1, both graphs have coefficient vector [0,0,1] and P_{1/2}(E)=1/4.
• At s=2, square NN has [0,0,2,8,19,24,20,8,1], 82/256 successes, and P_{1/2}(E)=41/128.
• At s=2, the matching graph has [0,0,4,20,41,44,26,8,1], 144/256 successes, and P_{1/2}(E)=9/16.
• The square NN unique-largest event is not increasing: there are exactly 28 success-to-failure one-site openings at s=2. Frozen witness: row-major mask 6 opens (1,0),(2,0) and succeeds; opening index 4 (0,1) gives mask 22, creates a left-largest tie, and fails.
• The matching graph has zero such transitions at s=2; this is not claimed as a monotonicity theorem for larger s.
• This nonmonotonicity does not by itself invalidate fixed-parameter locality/1-independence plus binomial certification, but the event must not be used as an increasing crossing observable in coupling arguments.

Verification:

• 9 focused tests and 17 related local regressions passed; generated JSON/Markdown reproduced byte-for-byte.
• Complete CI passed on C++17 and Python 3.9/3.11/3.13; each Python job ran the 484-test full suite plus the main-branch smoke check.
• The initial Python 3.9 run caught use of the Python-3.10-only int.bit_count; this was replaced with a 3.9-compatible equivalent without changing any artifact, then the full matrix passed.
• Post-merge audit confirms main points to the squash commit and it contains exactly the six intended files.

Unresolved boundary: this slice proves only tiny exact semantics. A production-scale implementation, differential validation at larger s, fresh IID final trials, event probabilities near the relevant parameters, and any new certified p_c bound remain open. Keeping the parent issue open.

[LightChainr]

Claiming the next bounded Phase-0/engineering slice: implement a linear-space, linear-time open-boundary evaluator for the frozen two-cell block event, without adding a stochastic sampler.

Scope:

• square NN and square matching (NN plus both diagonals);
• exact empty-half / unique-largest-tie / union-connectivity ordering already frozen by the tiny oracle;
• exhaustive differential comparison against every s=1,2 configuration (520 graph/configuration pairs total);
• deterministic larger-cell controls and explicit operation-count/complexity accounting;
• no probability estimate, no PRNG-independence claim, no production p_c bound.

I checked current main, open PRs, and branches. No open PR touches this intended evaluator/result namespace. I will keep this parent issue open after the independent slice.

[LightChainr]

Deterministic linear-evaluator slice completed and merged in #280 (squash commit 1668bddd9e8b10cbe5c57add79eb61f74861fb4b).

Exact conclusions:

• Added a caller-supplied-configuration evaluator using two disjoint-set forests: full-rectangle connectivity and half-local connectivity with cross-half edges suppressed.
• The frozen order is preserved: empty-half rejection, unique-largest tie rejection, half-local selection, then full-union connectivity.
• Time and memory are both O(s^2) for the 2s x s rectangle.
• Square NN scans exactly 4s^2-3s full edges and 4s(s-1) half-local edges.
• Matching NN+diagonals scans exactly 8s^2-9s+2 full edges and 8s^2-12s+4 half-local edges.
• Exhaustive differential validation covered all 520 graph/configuration pairs at s=1,2, comparing success, failure reason, and selected clusters; mismatch count is zero.
• Deterministic all-open controls at s=64 select exactly 4,096 sites per half for both graphs.
• The square nonmonotonicity witness remains intact: mask 6 succeeds and mask 22 fails with left_tie.

Verification:

• 6 focused tests, 15 exact-plus-linear tests, and 50 locally available regressions passed.
• Complete CI passed on C++17 and Python 3.9/3.11/3.13, including the full repository test suite and main-branch smoke check.
• Post-merge audit confirms main points to the squash commit and the commit contains exactly the six intended files.

Unresolved boundary: the evaluator accepts deterministic configurations only. Randomness/domain separation, independence certification, production event-probability measurements, pre-registered final trials, and any new certified p_c bound remain open. Keeping the parent issue open.

[LightChainr]

Claiming a second bounded Phase-0 audit slice: freeze and validate a pre-registration/trial-ledger contract around the already merged confidence gate.

Scope:

• canonical plan with familywise alpha 1/1000000, two sides, at most three attempts per side, N=400, and modern block marginal threshold 0.8457;
• exact derivation of per-run alpha 1/6000000 and cutoff 373 through the existing exact binomial-tail gate;
• SHA-256 plan digest plus fail-closed validation of result records, attempt/domain uniqueness, count ranges, and exploration/final separation;
• synthetic fixtures only; no generated random trials or empirical probability claim.

Latest main is 1668bddd; no open PR touches the intended Issue-112 protocol namespace. Existing evidence-ledger work belongs to another issue and different paths. The parent issue will remain open.

[LightChainr]

Content-addressed pre-registration/trial-ledger slice completed and merged in #281 (squash commit d433f53432e8540ecbfe93db96617d0b1ed89fc5).

Exact conclusions:

• Froze familywise alpha 1/1000000, upper/lower sides, at most three pre-indexed attempts per side, N=400, null block probability 8457/10000, and per-run alpha 1/6000000.
• Recomputed the exact cutoff as 373 successes; the null tail at 373 is 9.51451513592604452e-8, while 372 does not pass (2.37086590688800581e-7).
• Canonical plan SHA-256 is 8beeb3977457f391993c19012b14bb264f76744052c616fdb9eaf76b09d7e29a; changing any plan field invalidates existing records.
• The validator rejects plan-digest mismatches, non-final records, graph/side mismatches, attempts outside 1..3, repeated side/attempt pairs, repeated record IDs/domains/data digests, exploration-data reuse, and trial/success counts outside the plan.
• Exact Fraction arithmetic determines each decision and is checked to agree with the frozen cutoff.
• The six-record checked-in ledger is visibly synthetic and exists only to exercise both sides and acceptance/rejection paths; it is not percolation data.

Verification:

• 9 focused tests, 17 confidence/preregistration tests, and 59 locally available regressions passed.
• Complete CI passed on C++17 and Python 3.9/3.11/3.13, including the full test suite and main-branch smoke check.
• Post-merge audit confirms main points to the squash commit and the commit contains exactly the six intended files.

Unresolved boundary: SHA-256 detects plan/record substitution but cannot prove genuine randomness or IID Bernoulli trials. No empirical event probability or p_c bound is claimed. Real randomness provenance, fresh final data, and production execution remain open, so the parent issue stays open.

[LightChainr]

Parked for the 2026-08-31 weekly decision: this remains a useful support/long-horizon line, but it is not part of the current #154/#334 validation execution. Please do not prepend new handoff blocks; resume only after docs/STATUS.md explicitly reprioritizes it (cleanup PR #528).

[LightChainr]

Queued for the Go-quota refill (omen-alpha / qwen3.8-flash), not started. Retrieval/feasibility only — Riordan–Walters / Balister–Bollobás–Walters finite reduction vs our wrapping/M tables. No new Monte Carlo, no Huawei, no #636. ASSIGNED_MACHINE when a CLI slot opens. Draft PR against main.

[LightChainr]

ASSIGNED_MACHINE: local

Phase 0 feasibility only. Notes + citations. No production codebase, no new Monte Carlo, no Huawei. Draft PR against main. Do not edit STATUS. Do not close #112.

[LightChainr]

Draft PR opened: #685 — Phase 0 retrieval slice (one notes file + three missing references.bib entries). Verdict: methodologically interesting but weak, matching the #278 classification. Primary sources checked 2026-09-08: RW PRE 76 011110 (arXiv PDF, Lemma 1 p0=0.8639, N=400/M>=378 tail, [0.5925,0.5930] all verified verbatim); BJSS arXiv:2206.12335 v2 (Theorem 1.3 p_max(Z^2)<=0.8457 verified verbatim); BBW CPC 21 bibliographic data via Cambridge Core. Key new finding: the repo's exact torus wrapping/M tables are the wrong observable family (periodic/global vs open-boundary/local block event) and cannot feed the certificate; the merged #278–#281 machinery remains the only valid estimator path. Parent stays open.

[LightChainr]

PR: #685

Phase 0 verdict: methodologically interesting but weak. Wrapping/M tables do not feed BBW. Keep parked; no production. Ticket stays open. Not merging.