[petalas]

Summary

Scoped written proof with exact finite corroboration. The generic four-source paid-clean connector has exact complete-map-call optimum, for every positive uniform source price b and readout price a:

- #5372: general-n Gray optimum when a >= b; #6254: three-source frontier
+ four sources, all a,b > 0: min(8b + 23a, 16b + 15a)
+ total-count frontier: (8,23), (10,21), (12,19), (14,17), (16,15)

The eight-call source word 01021323 reads seven outputs once and eight twice. An all-finite-word counting proof establishes optimality. At b=10,a=1, the exact map-call price is 103 versus the Gray schedule's 175; common allocation, zeroing, release and admission costs must still be added. The theorem preserves arbitrary incoming outputs, immutable sources, the untouched dirty connector and final zero restoration for arbitrary compatible F2-linear maps.

This settles the uniform-price specialization of Open next question 2 in older #6254. The source-mask bound, Gray construction and clean-row contract are credited inputs, not new results. No new κ or selected-frontier change. Advance CLAIM.

Evidence

Before, #5372 covers the a>=b region and #6254 leaves four-source prices open. After, PROOF.md proves the uniform-price formula and all five aggregate Pareto pairs, with explicit arbitrary-map attainers.

Local CPython 3.12.15 on macOS 27.0.1 arm64 passed warning-strict normal and optimized direct checks and both registered commands. Direct runs took 3.327 and 3.781 seconds; the combined registered replay took 7.290 seconds. Each registered command has a separate 120-second limit. These are check durations, not performance gains.

Generated SHA256-bound receipts report 190231 walk states, 20769 closed visited sets, 20440 spanning closed sets, two independent XOR-distance calculations per closed set, 147 exact price comparisons, 4880 rectangular dirty-input executions and 174 rejecting controls. The written proof supplies the universal quantifiers. Committed bytes, executable modes and declared source ancestry passed at 6aa446cd8d188db3cdd713c85a99e073b5de1feb. No predecessor scientific verifier was replayed as the deliverable.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/clean-four-source-uniform-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/clean-four-source-uniform-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/clean-four-source-uniform-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/clean-four-source-uniform-tariff --check-source-history --check-modes

Use another CPython 3.12 executable elsewhere. Expected markers are PASS_CLEAN_FOUR_SOURCE_UNIFORM_TARIFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Selected source-binding orientation and the card's 945-pairing positive control passed separately and do not establish the new theorem.

Limitations

The frontier concerns two aggregate counts under uniform prices, not nineteen independent call counts or a particular degenerate map assignment. Exactly one paid clean row, singleton complete calls and the generic all-map requirement are essential. Word-dependent physical preparation, descriptors, routing, map/native lowering, precision/tape, ownership admission, recursive hosting and a full signed supplier invoice remain open. The five selected assumptions are unchanged.

No selected cold replay, full native/caller replay, Lean, Docker, complete multiplication theorem, independent human review or formal proof-kernel acceptance is claimed. The verifier's bounded managed allocation and static-analysis limitations are documented in the proof.

Open next questions

1. Determine the four-source heterogeneous frontier using weighted visited-state paths and target-specific readout costs. The uniform sixteen-call dominance cutoff cannot be reused without proof.
2. Pin one actual recursive caller and price its clean-row lifetime, complete maps, routing, descriptors, restoration and precision before claiming a supplier gain.

Related open work

No open PR studying this four-source paid-clean tariff was found in the complete 502-PR body snapshot refreshed at 2026-11-11 11:22 UTC, the subsequent live open-PR refresh, and the object/formula searches.

• #7206 gives a two-source GF(8) dirty-connector unit-call optimum. This theorem uses four binary sources, a paid clean row and uniform source/readout prices. The field, incoming connector state and demand alphabet differ; no competing same-instance numerical value is computed or checked here.

The completion refresh inspected 47 open account PR records, including the five other contributions created since the CLAIM: #7203, #7205, #7206, #7207 and #7208. Only #7206 concerns this connector-alphabet neighborhood. Merged #5372 and #6254 are the cited predecessors. Sparse notes, arbitrary historical diffs and unpublished work remain coverage limits.

CI notes

Hosted CI did not certify this package. At head 6aa446cd8d188db3cdd713c85a99e073b5de1feb, run 43012524632 failed its plan job in Onboarding helper tests. Its sole failing test was test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with an AssertionError stating that make frontier7000-verify was not found in the CI guide. The log reports 760 tests, one failure and 24 skipped. The package replay job was skipped because plan failed.

This is the unchanged main-base mismatch at e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, but docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both files were inspected directly from that Git commit; this PR modifies neither.

The same sole failing test, exact missing-command assertion, and 760-test/one-failure/24-skipped summary were verified directly in these comparison job logs:

• Run 42905094314, plan job 130140894651.
• Run 42903313790, plan job 130134720799.
• PR #7203, run 43012497455, plan job 130495338611.
• PR #7199, run 43001851217, plan job 130459654862.

No fix, retrigger commit, push or job rerun was made. The passing normal/optimized local registered checks, direct verifier, and committed-byte/mode/source-history preflight remain the verification evidence. No hosted package pass or independent scientific review is claimed.

Credit

This contribution was produced by Nick Petalas. Credit Ali Romman #5372 for the paid-clean contract analysis, source-mask bound and general-n Gray theorem; Nick Petalas #6254 for the three-source tariff and four-source question; Cinna Davis #1954/#2271 for the paid-row extension and related borrowed-source tours; Ross Richardson #943 for the connector lineage. Cube walks, Gray cycles, shortest paths and subset-XOR dynamic programming are classical. All predecessor credit lines remain unchanged. Apache-2.0.

Merge Danger

Door: two-way. Blast radius: package. Only one isolated research directory is added; shared code, frozen inputs and the selected frontier remain unchanged.