[petalas]

Summary

Scoped counting proof with an exact computer-assisted finite component. For three corank-one requests over F2 with pairwise orthogonal rank-one kernels and one uniformly weighted complete GL_m(F2) header:

- Known: Pr(E1 E2 E3) < Pr(E1)^3
+ New:   Pr(E1 E2 E3) < Pr(E1 E2)Pr(E3), every m>=3

This answers the first bounded step in Open next question 1 of #4767. The two-block comparison is stronger because the pair is already negatively correlated. The new proof cites #4275's triple law, #4261's pair law and #4227's marginal law. It derives the grouped ratio, proves a spectral tail for m>=43 and covers the remaining forty dimensions exactly. At m=4 the joint-to-block-product ratio is 40/189. No predecessor verifier is imported or replayed.

Only research/corankone-triple-block-correlation/ is added. This contribution was produced by Nick Petalas. The counting laws and question retain Nick Petalas's predecessor credit; Cinna Davis supplied the BIT request context; source527, commdata2338 and Jose Manuel retain the normalizer lineage, and Chris R / Clear1nsight retains the earlier #3950 counting context. No new kappa or selected-frontier change is claimed.

Advance CLAIM. Base and refreshed canonical main: 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. TASK.json records the explicit change from the interface-formalization card to this counting side question and the limitations of overlap coverage.

Evidence

• Before: Prove all-field negative correlation for corank-one pairs (no new κ) #4767 explicitly leaves the pair-versus-singleton comparison open. After: PROOF.md and the exact finite component establish it for every m>=3. This is a new consequence of cited formulas, not an audit of their proofs.
• All 40 finite proof cases, 3<=m<=42, have strictly positive integer gaps. The generated receipt enumerates every case and its exact rational ratio. A quantified tail argument proves every larger dimension, using the checked integer inequality 57*19^42 < 21^42.
• 86 further exact comparisons, 43<=m<=128, corroborate the tail. Seven mutation/domain controls detect grouped independence, a reversed leading spectral sign, a missing finite-GL correction and four invalid replay dimensions. At m=4 the cited probabilities are p1=9/40, p2=1/80 and p3=1/1680, giving ratio 40/189.
• Both registered commands pass locally with warnings as errors. Direct command times were 0.022 s and 0.021 s, below their 120-second limits. These are check durations. Normal and optimized receipts are byte-identical; explicit checks remain enabled under optimized Python.
• Metadata/source pins, six-file committed-byte/source-history preflight, official export integrity, both Git-free read-only export replays and unchanged exported bytes all pass at d161261eaf899e0d15036a312aac0022aa1a4ab6. The generated stdout receipt binds all six package files. No receipt is handwritten or committed.
• Ruff 0.14.0 E/F/B/UP and AST syntax/function-size inspection pass. The longest function is 42 lines. The standard-library verifier has finite fixed bounds and no recursion; the managed-allocation deviation is documented in README.md.

From the repository root, using CPython 3.12 in place of this exact local interpreter if needed:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/corankone-triple-block-correlation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/corankone-triple-block-correlation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/corankone-triple-block-correlation
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/corankone-triple-block-correlation --check-source-history

Expected markers: PASS_CORANKONE_TRIPLE_BLOCK_CORRELATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier writes no files and runs without Git; the provenance and source-history commands need full Git history.

Limitations and unrun obligations

The result assumes F2, three orthogonal kernel lines and complete equally weighted GL headers. It does not establish negative association for arbitrary increasing functions. The union-versus-singleton comparison, odd fields, nonorthogonal kernels, larger blocks and restricted header distributions remain open.

No request producer, header selection/emission, dispatch, dirty restoration, stock, precision or complete paid physical caller is supplied. Those costs are unknown. No supplier admission or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected cold replay were not run. All registered local package obligations ran and passed.

The historical card orientation command scripts/research_baseline.py raises KeyError: 'bit_lower' on the selected certificate schema. Its documented replacement scripts/research_selected_baseline.py prints PASS_RESEARCH_BASELINE_BINDINGS_ONLY at selected kappa 60403/20000000. The historical reporter is unchanged; the passing replacement is source-binding orientation only.

Related open work

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 19:45:46 UTC were refreshed. Eight open PRs were returned in that time range, including this PR. Titles and stated scopes were checked; the related header/correlation bodies were read. This supersedes the initial publication snapshot. Sparse local-note and unpublished coverage remains incomplete.

• #4778, head 9997ae7d1c99ecf5eb915aad28eb7b61e778956c, proves all-field negative correlation for two one-way-coupled kernels, UV nonzero and VU=0. This PR uses three orthogonal kernels and a pair-versus-singleton split. At q=2,m=4, exact evaluation of this verifier and Prove all-field negative correlation for one-way full guards (no κ) #4778's stated marginal expression gives the same 9/40. Their joint fractions describe different events; no equality is asserted.
• #4774 studies least Schur-commutator dimensions for distinct nonzero scalar pivots. F2 has no distinct nonzero pivot pair, so its parameter domain has no overlap with this binary theorem, and it gives no common numerical probability.
• #4785 studies least dimensions for scalar leading blocks and prescribed products of commuting idempotents. It supplies no three-request complete-header fraction, so there is no overlapping numerical probability to compare.

The other new account PRs, #4776/#4780/#4781/#4784, study right-vector reachability, factor costs or coefficient-algebra counts. They do not study this split probability. No conflicting overlapping number was found. The merged predecessor values p1=9/40, p2=1/80 and p3=1/1680 at m=4 agree under exact evaluation here.

Open next questions

1. Does the full three-event Bernoulli vector satisfy negative association? Prove or refute 2p2-p3 < p1(2p1-p2), the comparison of E1 union E2 with E3. This is the remaining increasing-event type on a two-coordinate block.
2. Does the split inequality hold over odd finite fields? Start with the all-field k=3 transfer and a uniform grouped bound.
3. Can a physical producer use a shared header with a favorable complete paid invoice? Pin its kernel lines and header domain before charging selection, emission, dispatch, restoration, precision and stock.

CI notes

Both hosted checks passed on contribution head d161261eaf899e0d15036a312aac0022aa1a4ab6 in run 39843751712: plan in 3m18s and the registered research/corankone-triple-block-correlation replay in 1m14s. The run's head SHA matches the published commit. The registered replay executes both normal and optimized commands. All registered local commands, committed-byte/source-history checks, export integrity and Git-free replay also passed. Hosted CI certifies the stated package checks, not independent human or formal proof-kernel acceptance of the quantified proof.

Merge Danger

Door: two-way. This adds one isolated research package.

Blast Radius: research. Existing scientific sources and the selected record are unchanged.