[petalas]

Summary

Scoped written proof with an exact finite certificate. All 144 nonzero ordered denominator-13 residue pairs are served by 21 unchanged canonical DDG tables, a one-byte selector per pair, and output exchange/complement logic.

- 144 separate resident source tables: 101,088 transition bytes
+ 21 canonical tables: 14,742 transition bytes
+ 144 selectors: 144 bytes
+ total payload: 14,886 bytes

The requested independent product laws and the original stopping time on every coin tape are preserved. An individual tape's output label may change. This changes the fixed coin-to-label contract of the older #6124 bank, which reports 84,578 bytes; the new payload is 69,692 bytes smaller under that changed contract. No byte minimum, physical runtime saving, supplier or new kappa is claimed.

The new bank, packed representation and proof were produced by Nick Petalas. zxv's #6072 supplies the unchanged 312-state source graphs, probability law, expected-bit optimum and sharing question. Those results are cited, not audited. Classical DDG methods and output symmetries retain their credit. JoyboyBrian's inverse/residue/reversal lineage, Nick Petalas's earlier stochastic-prefix and fixed-function sharing work, and ToikY's source reviews are credited in the package.

Advance CLAIM.

Evidence and reproduction

At head e8832b7fd18458ff82a96e2c9a061e9df66943cc, CPython 3.12.15 and the standard library pass:

• All 144 selector laws, eight output permutations and 89,856 packed selector/state/coin lookups.
• 576 terminating paths, 576 unfinished prefixes, 144 phase-wrap continuations and 15 exact-reason rejection controls. A seven-coin example explicitly distinguishes the changed output function while keeping its stopping depth.
• Normal and optimized receipts agree. Direct registered-command equivalents took 0.094 s and 0.174 s locally, below their separate 60-second limits. These are timeout checks, not a speedup measurement.
• Registered metadata and both commands, static compilation and function-length checks pass. Nine committed package/input files, modes and declared source ancestry pass. The official export passes input coverage; read-only Git-free normal and optimized replays pass with all ten exported files unchanged.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/signed-swap-residue-bank
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/signed-swap-residue-bank --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/signed-swap-residue-bank/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/signed-swap-residue-bank/verify.py

Expected markers: PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES, PASS_SIGNED_SWAP_RESIDUE_BANK. Substitute a local CPython 3.12 executable elsewhere. Optional bank and source-hash-bound receipt emission is documented in README.md and requires an output path outside the checkout. No receipt was hand-written.

Limitations

This is a forward logical payload result. Code, headers, selector preparation, physical table lookup, tape movement, output storage, stopping records and reversible cleanup are additional costs. The mathematical stopping time remains unbounded; the finite-prefix helper's 4096-coin fixture limit is not a draw cap. A single on-demand source table still uses only 702 bytes.

The written covariance argument supplies the universal claim; finite checks alone do not. No original probability/mean catalogue replay, physical/native or selected cold replay, Lean, Docker, independent human review or formal proof-kernel acceptance ran. All five selected framework assumptions remain. Selected #7000 and pinned predecessor files are unchanged.

Open next questions

• Bind an actual nonadaptive caller and price selector preparation, packed lookup, output and random-tape movement against on-demand generation. Begin with its reached residue pair and owned tape layout.
• Retain the selector through a reversible lifecycle, prove its predecessor map, and price its stopping record and cleanup. Forward law preservation does not transfer Reverse the paired DDG with a resident transcript and clean stopping counter #6102/Remove the paired DDG stopping counter with one owned tape cell (no new κ) #6127's reversal tariffs.

Related open work

No open PR studying this paired DDG bank was found in the complete cached title/body scan and refreshed live open metadata. The merged #6072 supplies the source tables, #6124/#6749 concern fixed coin functions, and #6102/#6127 concern reversal of an original graph. This result changes output labelling and supplies no physical reversal tariff. Unpublished work and arbitrary diffs remain outside the overlap search.

The completion refresh checked this account's nine other open PRs created since the CLAIM: #7178, #7179, #7180, #7181, #7182, #7183, #7184, #7185 and #7186. Their titles and relevant body text concern multiplicative shells, A2 tariffs, Gaussian inverse norms, Clifford response sectors, cube-zero lifts, radical delivery, rational headers and block signatures. None studies this paired residue bank or computes a competing value for the same object. No extra scientific computations were run to check another contribution.

CI notes

CI did not certify this package. At head e8832b7fd18458ff82a96e2c9a061e9df66943cc, run 42993299068, plan job 130431131041 failed in Onboarding helper tests. The single failing test was test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with an AssertionError that `make frontier7000-verify` was not found in the CI verification document. The log reports 760 tests, one failure and 24 skips. The package replay job was skipped because plan failed; no hosted package command ran.

The failure is inherited from canonical main e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json requires make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. The test, selected record and CI document are byte-unchanged by this PR. The base mismatch was inspected directly without rerunning the test or job.

Failed-job logs were also read for 42905094314, 42903313790, 42992088329, 42992679183, 42993082934 and 42993110077. Each reports that same sole failing test, the same missing-command assertion, 760 tests and 24 skips.

All package checks in Evidence passed locally at the published head. No fix, retrigger commit, push or CI rerun was made for this inherited failure. Hosted package certification remains outstanding pending the maintainers' repair of main.

Merge danger

Two-way change, confined to seven new files in research/signed-swap-residue-bank/. Shared code, frozen evidence and the selected frontier are unchanged.