[petalas]

Summary

Finite certificate with a scoped minimality proof: share zxv's PR #6072 paired denominator-13 forward graphs while preserving every ordered coin tape's output and first stopping depth.

- 44,928 internal states; 101,088 bytes for 144 separate resident tables
+ 22,482 internal states; 84,578 bytes including all 144 entry pointers

The new quotient saves 16,510 payload bytes. Its state count is minimal for those fixed ordered-coin functions when all residual information is in the counted controller state. Moore minimization is classical; this exact quotient, complete certificate and packed-bank construction are the contribution. The graph definition, sampling law, expected-bit optimum and sharing question remain credited to zxv. Produced by Nick Petalas.

Only research/shared-residue-dispatcher/ is added. The graph is specified and reconstructed locally, so replay needs neither pending Git objects nor predecessor imports. Canonical main and overlap were refreshed at 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8. Advance claim.

Evidence

• Before: PR Joint denominator-13 decisions need fewer expected fair bits than separate calls #6072 stores 312 internal states and 702 packed bytes per configuration. The comparison here is a resident bank of all 144 configurations. Its on-demand single-table alternative still needs only 702 bytes.
• After: a complete homomorphism checks all 89,864 original edges including artificial terminal loops, all entries and output separation. Separately sorted signatures check every refinement split; reachability establishes the scoped state lower bound. The partition stabilizes at depth nine, confirmed at depth ten.
• Exact packing and unpacking give 84,308 transition bytes plus 270 entry bytes. All 36,864 eight-coin tape controls agree, and five semantic certificate corruptions are rejected.
• CPython 3.12.15 on macOS arm64: both warning-strict registered modes pass, timed locally at 0.358 s normal and 0.356 s optimized. The direct verifier also passes and generates a source-hash-bound receipt outside the checkout.
• Package metadata, committed bytes, source ancestry, file modes and export integrity/input coverage pass. Read-only Git-free normal and optimized export replays pass with all eight export files unchanged.

From the repository root:

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

Expected PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES and PASS_SHARED_RESIDUE_DISPATCHER. Elsewhere, substitute a local CPython 3.12 executable. README.md gives optional packed-bank emission and receipt commands; PROOF.md gives the self-contained graph and proof.

Limitations

This is a forward logical inventory improvement. The state minimum fixes output order and every coin-tape function; it is not a ROM byte minimum or an optimum over all samplers with the same probability law. A parameter oracle or extra uncounted state is outside the model. Interpreter/header costs are omitted on both sides.

Physical selection, lookup, random-tape movement, stopping records, reversible cleanup, native/caller integration and their full bill remain open. The quotient does not transfer PR #6102's predecessor-uniqueness condition. No bounded stopping time, runtime speedup, new kappa or selected-frontier change is claimed. Physical/native, selected scientific and formal proof-kernel replays were not run; all five framework assumptions remain.

Open next questions

• Price one actual nonadaptive caller's selection, packed lookup, outputs and tape movement against on-demand table generation.
• Determine the retained discriminator and cleanup bill needed for reversible use of a merged residual state.

Related open work

• Joint denominator-13 decisions need fewer expected fair bits than separate calls #6072, zxv, 9011bf83e85ecd0443c789cf03259442f750877d: defines the 144 ordered-coin graphs and asks about sharing. This builds a new shared bank, with the same inherited per-configuration behavior and unchanged expected draws.

• The denominator-13 inverse needs bias, aborts or unbounded fair-bit sampling #6068, zxv, 689e28282fd5259dec7f75b83904617dab2d7b3a: bounds finite-cap sampling choices. This retains unbounded stopping and claims no new bias/abort tradeoff.

• Classify actual /52 inverse residues and denominator13 applicability #6096, JoyboyBrian: identifies actual inverse residues and a conditional selector adapter. This improves the logical table inventory only, without implementing or repricing that adapter.

• Reverse the paired DDG with a resident transcript and clean stopping counter #6102, JoyboyBrian: reverses the literal (1,1) DDG using a resident transcript and counter. This certifies forward sharing only; its reversal tariff and unique-predecessor premise do not transfer.

• Remove the paired DDG stopping counter with one owned tape cell (no new κ) #6127, Nick Petalas: removes the stopping counter for reversal of the original fixed (1,1) DDG using an owned adjacent tape cell. This PR shares forward residual states across configurations; reversal of the shared quotient and transfer of Remove the paired DDG stopping counter with one owned tape cell (no new κ) #6127's tariff are not established. No same-quantity numerical comparison is made.

CI notes

CI is uncertified at commit 9ae1d82f4e248dd830e32c4d2a0330b62cddc1dd. In run 40692500348, plan failed during Onboarding helper tests and the package replay was skipped. The package replay did not run in CI.

The failure is unrelated to this package and is inherited from main commit 721c03822c421b7f7fcbb2598993b947c4d88419, which admitted PR6030 with [skip ci] and updated the selected record without matching the onboarding documentation:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the exact marker **0.003125804** in docs/research/current-status.md. The document instead bolds 0.003125804 = 781451/250000000 together, so that marker is absent.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

The failed-job log reports 678 tests, two failures and 23 skips. PR #6120's run 40691776027 fails the same two tests, confirmed from its failed-job log. Both check statuses were read once for this PR; no scientific CI success is inferred.

All package checks listed above passed locally, but this hosted run certifies none of them. The PR needs a maintainer re-run after main's documentation is fixed. No new commit or push was made for this inherited failure.

Merge danger

Two-way change with an isolated research-package scope. Shared implementation, frozen evidence and the selected result are unchanged.