[petalas]

Summary

Finite certificate with a written proof of its finite interpretation, produced by Nick Petalas. This answers the proposed six-block search in Jacob Sussman's PR #6706, open question 5.

Every shrinking-chain rank inequality on the six bit-permutation orbits admits seven single-bit calls for two target arrays. The exact minimum cost-to-demand ratio is 27/10. Six calls also pass with 23/10. Thus the proposed relaxation alone cannot exclude the two-array seven-call schedule. These are formal schedules; physical programs remain open. No new kappa.

35 exterior plane blocks -> 6 bit-permutation orbits
729 nested endpoint pairs -> exact rectangular ranks
all shrinking chains -> minimum ratio 27/10 at seven calls

Only research/bit-permutation-chain-limit/ is added. The new contribution is the complete coarsened rank table and its exact limitation. Sussman's decomposition, character-block equivalences and original chain principle are cited inputs. His exclusions and predecessor executables were not replayed as this contribution. Credit eumemic and the network-coding/temporal-cut lineage retained in the README.

Evidence

Before, #6706 proposes a search using the six surviving pieces when bit permutations are free. After, this package supplies a passing schedule against every inequality in that search.

• Exact Gaussian-integer construction and integer elimination cover all 729 nested endpoint pairs for both call and target matrices.
• Dynamic programming computes every minimum endpoint cost. Separate direct enumeration covers all 262,144 six-call chains and 531,441 seven-call chains, 793,585 in total. Their exact minima agree. Positive-demand counts are separately enumerated.
• The five-call ratio 19/10 fails for two targets. Three false mathematical conclusions and a dependent-row rank control reject in normal and optimized Python.
• Both registered commands passed locally. Each direct normal/optimized replay is about two seconds, below its 120-second limit. Complete registered replay passed in 3.61 seconds on CPython 3.12.15, macOS arm64.
• Committed-byte and source-history checks passed at 5de51eac590de356e81085c2596dfe3f7c0d2327. The official eight-file export and input-coverage check passed. Both read-only Git-free export runs passed with every exported byte unchanged.

From the repository root, reproduce with the exact local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/bit-permutation-chain-limit/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/bit-permutation-chain-limit/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bit-permutation-chain-limit
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bit-permutation-chain-limit --check-source-history

Any CPython 3.12 interpreter may replace this installation-specific path. The verifier prints a generated receipt bound to the eight package hashes and writes no files.

Limitations and unrun obligations

The claim concerns chains of unions of the six plane orbits in the explicitly defined finite model. Other information constraints and representations can reject the schedules. The stronger kept-subspace bound already excludes two arrays in six actual calls. Passing rank inequalities supplies no physical construction.

The engine-game interpretation retains #6706's cited representation premises. Independent scientific acceptance of those premises, native delivery, dirty restoration, precision, stock, the complete supplier invoice and outer admission remain separate. No selected cold replay, Lean or Docker run was performed. All five selected assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Find a bit-permutation invariant that excludes this seven-call schedule. Start with target-specific information subspaces or a different representation.
• Couple rank chains to the actual information delivered by a proposed program. A concrete two-array seven-call program and its intermediate subspaces are missing.

Related open work

• #6706 supplies the block-four exterior model and excludes schedules without address maps. This contribution addresses its distinct open question with free bit permutations and proves a limitation of the coarsened chain relaxation. It does not recompute its 35-plane exclusions.

The initial overlap refresh checked the complete cached open inventory, live REST metadata and the unchanged exact #6706 head. Additional initial GraphQL refresh reads were rejected for rate limit; that historical coverage gap is retained in TASK.json. The completion refresh read the nine other open account PRs created since the advance CLAIM: #6795, #6796, #6797, #6798, #6800, #6801, #6802, #6803 and #6804. Their tape, dirty-word, binary algebra, Word RAM, prefix-walk and centre-route objects differ from the six-orbit exterior rank-chain quantity here. No additional related open work or conflicting same-instance number was found.

CI notes

Both hosted checks passed at exact head 5de51eac590de356e81085c2596dfe3f7c0d2327: plan and replay (research/bit-permutation-chain-limit), in run 42229123778. The completed check states were read once. Local registered normal/optimized replay, publication checks and read-only Git-free export checks also pass. Hosted success covers execution of the registered finite replay. Independent human and formal proof-kernel review remain unclaimed.

Merge Danger

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

Blast Radius: package. It registers two bounded commands and changes no shared implementation or selected-frontier file.