[petalas]

Summary

Scoped proof with finite corroboration: the three-source specialization of Ali Romman's PR5372 paid clean-row contract has exactly fifteen componentwise-minimal call profiles. This settles its heterogeneous-price/full-profile gap at three sources, over every finite legal word and all ten independent positive prices.

OPT(b,a) = 2 sum(b) + sum(a)
           + min(2 min(b), min(a3,a5,a6) + min(a1,a2,a4,a7))

12 profiles: B=(2,2,2), double one even and one odd output readout
 3 profiles: B=(4,2,2) in any order, one readout per output

Every legal vector dominates one of these profiles, and each profile is attained and uniquely exposed by some positive prices. At uniform prices the optimum is min(6b+9a,8b+7a), making the predecessor's b=10,a=1 example optimal at 69. The example and classical Gray cycle remain credited predecessor work. No new kappa or selected-frontier change.

The proof and bounds, package scope, exact source pins and rerunnable checker are confined to one new directory. Advance CLAIM. Base 2485805, priority revision 8; the selected record is conditional kappa .003125804.

Evidence

• Before: PR5372 proves the general-n uniform a>=b region and supplies a cheaper a<b example, while leaving heterogeneous tariffs and the full profile frontier unresolved.
• After: a written cube-parity/dominance proof covers unbounded words and arbitrary compatible F2-linear maps, including singular maps and unequal row dimensions. The fifteen exact attainers preserve arbitrary incoming outputs, immutable sources and the unused dirty connector; the paid owned row returns to zero.
• Warning-strict normal and optimized checks pass on CPython 3.12.15, macOS arm64. Each registered command takes about 0.28 seconds locally, below its 120-second limit.
• Generated hash-bound receipts enumerate 9841 short source words, 1350 restored spanning words, 220 distinct minimum-read vectors, fifteen minimal/exposed profiles, 128 visited sets, 3480 independent exact tariff comparisons, 10320 rectangular-map dirty-input executions and 112 rejecting controls. These are executions/configurations, not distinct-matrix or exhaustive real-price counts.
• The independent visited-set shortest-path oracle prices arbitrary-length source walks with at most 1024 states; it does not assume the six/eight-call theorem. The written proof supplies the universal real-price quantifier.
• Registered normal/optimized replay, warning-strict AST compilation, committed bytes/modes and full declared source ancestry pass at 7e3f77d. The official nine-file package/input export passes input coverage. Git-free read-only normal/optimized replays preserve all ten extracted files including export metadata. Actual changed-predecessor-input controls reject in both modes.

Exact local reproduction, from the repository root:

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

Portable CPython 3.12 equivalent: python3.12 -B scripts/research_package.py --package research/clean-three-source-tariffs. Expected PASS_REGISTERED_RESEARCH_PACKAGE; each command emits PASS_CLEAN_THREE_SOURCE_ALL_TARIFFS in its generated receipt.

Limitations

This is the generic fixed-word, singleton-call, one-paid-clean-row contract. It does not lower-bound a special degenerate map assignment, allow additional zeroing/rows or source/output feedback, or price word-dependent physical implementation. Add common allocation, zeroing, release and admission fees to the formula. Recursive hosting, map/native lowering, descriptors, precision/tape, stock, ownership and a favorable complete supplier invoice remain open. The five selected framework assumptions remain.

No predecessor scientific suite was audited as the deliverable. The card's required 945-pairing orientation control and selected source-binding baseline were run once. No selected cold replay, full supplier/native/caller replay, Lean, Docker, complete multiplication theorem, independent human review or formal proof-kernel acceptance is claimed. No static type checker/linter is installed in the required environment; the documented finite bounds and warning-strict checks are the available code diagnostics.

Open next questions

1. Host these schedules in one actual recursive caller and price its complete physical/precision/restoration bill. No such caller is supplied here.
2. Determine the four-source heterogeneous paid-clean frontier, beginning with candidate visited-set profiles and a proof of source-length reduction. No larger-source search is included or reserved.

Related open work

No open PR found in the bounded live overlap search studies this exact three-source paid-clean tariff. Adjacent connector work has different boundaries:

• research: audit minimal joint-output connector charge #3897 studies a joint-output call on a dirty connector; the present alphabet has separate output calls and a paid clean row.
• Research: high-arity connector toggle cost floor #2249 studies high-arity six-source dirty-connector calls; its source alphabet and paired-readout boundary differ.
• Add five-source arity-four connector cost bound #3909 studies one arity-four five-source dirty-connector extension; this theorem keeps singleton source calls.
• Certify PR6141 characteristic-five connector exclusion (independent verification; no κ) #6216 certifies a characteristic-five six-source dirty-connector exclusion; this theorem is over F2 with a clean owned row.
• Audit dirty direct-call supplier gap #3919 studies a fused direct-call gap under a dirty boundary; no fused call is admitted here.
• Six-source one-TRIO path misses full paired coverage at 14 calls #2134 studies one-TRIO six-source dirty-connector paths; this theorem has no TRIO operation.
• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079 concerns seven-source weight-at-most-four dirty-connector paths; neither that alphabet nor its quantity is repriced here.
• Construction: three-stage persistent prefix shear chain (no new kappa) #1930 studies target-to-source feedback in a persistent chain; outputs are never read in this contract.
• Reduce complete prefix-completion caller time by 10.96% (no κ) #6260 implements a shared-tail predicate for a five-source dirty-connector prefix search; it changes search execution rather than the three-source paid-clean tariff and computes no same-instance price/profile optimum.

The completion refresh inspected this account's open PRs created since the advance CLAIM and refreshed live open connector work. PR6260 is the adjacent new connector contribution listed above; the other newer account PRs study different objects or quantities. No extra computations checked other PRs, and no same-instance numerical disagreement was identified within the inspected coverage.

CI notes

Both hosted checks, plan and replay (research/clean-three-source-tariffs), passed at head 7e3f77d6d1a3ec7dd7608d7fcf4e33860b1aeb49 in run 41048379754. The completed checks were read once, and the run's successful conclusion and exact head were confirmed. All declared checks also passed locally. Hosted replay certifies the registered finite checks and source bindings, not independent human or formal proof-kernel review of the universal theorem.

Credit

This contribution was produced by Nick Petalas. Credit Ali Romman/PR5372 for the question, clean-row contract analysis, uniform theorem and example; Cinna Davis/PR1954 for the paid-row extension and PR2271 for related borrowed-source Gray tours; Ross Richardson/PR943 for the connector lineage; Adam Gall/PR2063 for the dirty-alphabet comparison. Classical cube/Gray/shortest-path methods retain their classical status. Predecessor credit lines and bytes are unchanged. Apache-2.0.

Merge Danger

Door: two-way. Blast radius: package. Only the isolated research directory is added; publication does not admit a supplier or alter the selected frontier.