[petalas]

Summary

Certify the exact joint-success curve for a changed binary five-cycle query contract. Each receiver demands its own bit, observes its two original neighboring bits and receives one common M-value message. The encoder and all five decoders may be arbitrary nonlinear functions.

M message values -> disjoint correct classes -> maximum successful inputs

M	1	2	3	4	5	6	7	>=8
Successful inputs out of 32	5	10	15	20	24	28	31	32

In particular, two message bits have sharp joint success probability 5/8. Seven values can succeed on 31/32 inputs but cannot give zero error. Independent public or private random seeds do not improve these uniform-input probabilities.

Claim type: finite certificate with a scoped combinatorial proof. This contribution was produced by Nick Petalas. Only research/nonlinear-cycle-success/ is added. No new kappa or selected-frontier change is claimed.

Credit ywz65 #2541 for the motivating block-fitting interface, Tekalign #3362 and JoyboyBrian #5023 for unequal-width linear capacity, and their retained Sagar Thakrar, Cinna Davis, Adam Gall and P112/query-mask lineage. Confusion graphs, the binary five-cycle maximum guessing fixed-set size five and the zero-error eight-message endpoint are classical; PROOF.md cites Atkins, Rombach and Skerman and Ong. No priority for those facts or general methods is claimed. The new contribution is the complete partial-success table, its literal attainers and the finite obstruction certificate.

This changes linearity and the exact-recovery requirement. Nick Petalas #5873 classifies exact linear fitting matrices at total payload seven; this is a nonlinear partial-recovery contract on five unit bits, not an extension of that classification or a change of payload widths alone. No predecessor scientific verifier or mathematical result is replayed or audited.

Evidence

Before, the cited linear exact-recovery capacities did not specify this partial-success curve. After, a complete finite reduction proves every upper bound and literal encoder/receiver tables attain every entry.

• Enumerate all nonconfusable classes through size six: counts (1,32,256,640,480,32,0). No larger class is possible because it would contain a six-element class.
• Enumerate 3,249 partial size-five packings, all 1,600 disjoint four-packs and 25,280 complementary four-class pairs. These exclude five disjoint size-five classes and complete seven-message recovery.
• A separate receiver truth-table enumeration obtains the identical full class inventory, with digest a39ed8349b302d38efbdacb422c394044ec1c2042364c21f4d01ddda1a555c73. A distinct pair-union check excludes the seven-message endpoint.
• Construct all eight attaining encoders and five receiver tables per encoder; execute each on all 32 inputs. Eight malformed-witness/decoder controls reject.
• All four warning-strict normal/optimized CPython 3.12.15 commands pass locally. Individual durations are 1.043/0.977 seconds for the primary and 0.051/0.050 seconds for the separate implementation, below their 120-second limits. These are check durations, not a performance claim.
• Registered metadata/pins and replay pass. Committed bytes, executable modes and full source-history preflight pass at 06754cb96e8f852eeddc49f099d4a50fa80528d1 for all ten package/input files.
• Official committed export integrity and all three declared-input coverage checks pass. All four read-only, Git-free replays with empty PATH pass. All eleven exported files retain their hashes. An actual changed cited-source input is rejected, then restored byte-for-byte.
• The primary emits a generated receipt bound to all seven package files and three cited inputs. No handwritten result receipt is included. Static syntax and function-size inspection pass; all functions have at most 29 physical lines. No external static analyzer is claimed.

From the repository root, reproduce with the exact local interpreter path, or substitute another CPython 3.12 executable:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/nonlinear-cycle-success/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/nonlinear-cycle-success/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/nonlinear-cycle-success/independent.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/nonlinear-cycle-success/independent.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/nonlinear-cycle-success
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/nonlinear-cycle-success --check-source-history --check-modes

Expected markers are PASS_NONLINEAR_CYCLE_SUCCESS, PASS_NONLINEAR_CYCLE_TRUTH_TABLES, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The theorem fixes five unit binary payloads, one simultaneous common message, the two original neighboring bits and joint success on uniform inputs. It does not prove a tensor/direct-product, adaptive-observation or input-correlated side-information theorem. Message values are not native child ranks or paid costs.

No physical nonlinear encoder, source acquisition, dirty restoration, native operations, ownership, frames, stock/precision, fixed-tape implementation, compiler, outer recovery or complete multiplication theorem is supplied. All selected framework assumptions retain their status. No Lean, Docker or selected cold supplier replay was run. Independent human and formal proof-kernel review are unclaimed.

The card changes to this finite interface side result. Base main remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, and the selected binding positive control passes. Main, merged work, live claims and open PRs were searched before claim and commitment. The final overlap snapshot reports 581 open PRs and 6,655 Discussion records, with complete remote pagination; sparse local notes, arbitrary diffs, unindexed literature and unpublished work remain limitations.

Open next questions

• Can a source-owned physical query layer realize the two-bit 5/8-success witness with a useful complete paid bill? First specify original observations, arbitrary dirty fields and the entire encoder/receiver operator. Nonlinear native operations, restoration and a compatible caller are blockers.
• What is the exact joint-success tradeoff for two copies with coupled encoding? Start with the 1,024-input coupled confusion graph at one message budget. The one-copy packing proof supplies no direct-product theorem.

Research stops at the first verified finite result.

Related open work

• Prove exact block fitting rank under compatible port flags (no κ) #6471 gives compatible-flag linear block-fitting rank bounds and uses a C5 boundary. This result instead allows arbitrary nonlinear functions and partial recovery; it computes no flag minimum or block fitting rank.
• Prove sharp flag-defect budgets for unequal decoder payloads #6528 gives unequal-payload flag-defect bounds and a priced linear C5 word. This result computes no defect rank, physical ADD bill or common-instance rank; it supplies a different nonlinear joint-success curve.

No other open PR in the inspected complete snapshot studies this finite partial-success quantity. No extra computations were run to check these submissions or compare their numerical claims.

The completion refresh read this account's twelve other open PRs created since the advance CLAIM at 2026-11-03T14:04:00Z. They concern a P104 carrier, source-intake searches, transfer-table horizons, B3/C3 root-code terms, composite-axis projections, BW16 clique bounds, partial-Pauli replay, routing ROMs, common-flag reachability, GL2 dirty-role targets, sparse-union headers and weighted fanout. None studies this nonlinear partial-success quantity or supplies a common-instance number computed here. No extra computations were run to check those contributions.

CI notes

Both hosted checks, plan and replay (research/nonlinear-cycle-success), passed at exact head 06754cb96e8f852eeddc49f099d4a50fa80528d1 in Registered research packages run 41651517000. The completed checks were confirmed once with gh pr checks 6645 --repo CrocSwap/integer-mult-bounds; separate run metadata confirms completed status, success and the matching head. The plan job took 4m26s and the package replay took 1m39s.

Hosted replay executes the four registered warning-strict normal/optimized certificate and truth-table commands plus source metadata validation. All local scientific, publication and portable-export checks listed above also passed. No failed or waived check remains. This certifies execution of the finite checks, not independent mathematical or proof-kernel acceptance. Independent human and formal proof-kernel review remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated. Adds one research package with explicit replay inputs; no shared code, frozen predecessor bytes, credit lines or selected result change.

Advance claim: #1166 (comment)