[petalas]

Summary

Prove the exact bill for arbitrary positive fixed complete-call prices in the original five-source F2 dirty-connector alphabet, with all 31 nonzero-mask outputs:

base = twice the sum of all 36 operator prices
optimum = base + twice the cheapest source or size-three/size-five output

The envelope has 16 irredundant linear pieces, each attained by a 74-call schedule. A cheap singleton, pair or four-element output cannot pay the necessary surcharge. The new finite lemma is that every ten-toggle paired source word misses an odd non-singleton mask. This extends Cinna Davis's #1770 fixed 182-price fixture and retains Ross Richardson's #943/#934 contract and construction credit.

Claim type: scoped proof with a complete finite combinatorial dependency. No new kappa, supplier admission or selected-frontier change. Package: research/connector-price-envelope; claim. The connector-alphabet-change assignment is explicitly changed to the general unequal-price decision.

Evidence

Before: #1770 certifies one unequal source-price vector with unit output prices. After: a written all-positive-price proof and sixteen attaining witnesses determine the entire fixed-tariff envelope.

Local CPython 3.12.15 checks passed:

• Both complete word generators and independent prefix implementations agree on 945 canonical paths. Their histogram agrees with Research: sharp four/five-source dirty-connector boundary (no new kappa) #943; every path misses a size-three or size-five mask.
• All 16 schedules pass free noncommutative action checks, 1,184 two-bit-row basis cases with inverse replay, 512 mixed dirty cases, 20 tariff cases and seven rejection controls.
• Both registered commands pass, timed individually at 0.14 and 0.13 seconds locally. These are single check durations, not performance claims.
• Committed-byte/source-history preflight passes on all nine package/input files, with base 12fa8c5c844be21bfefb38a19503fbd867a11244 reachable from the publication commit.
• The official committed export passes integrity checking. Ordinary and optimized Git-free checks pass in 0.149 seconds each; all ten exported files, including the export receipt, remain byte-identical. A corrupted predecessor input is rejected by its SHA256 pin.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/connector-price-envelope
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/connector-price-envelope --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B research/connector-price-envelope/verify.py --show-witness A31

Any CPython 3.12 executable can replace that local interpreter path. Expected marker:

PASS_CONNECTOR_PRICE_ENVELOPE paths=945 witnesses=16 basis=1184 dirty=512 price_cases=20 controls=7 no_new_kappa

Generated receipts bind all package files and declared live sources. The witness digest is 5a6e1c58b42488081d44bb5914904d7af4fd9ed6df2c2f8493e995e8ad0280a8.

Limitations and unrun obligations

The exact optimum assumes one dirty connector, immutable sources, generic F2-linear maps, separately paid complete calls and fixed per-operator tariffs. Word-dependent routing, hosting, setup and dispatch require a separate caller ledger. No selected-supplier occurrence, recursive-moment improvement, physical tape implementation, selected cold replay or formal-kernel proof is supplied. The finite obstruction is exhaustively recomputed; a structural proof replacing that census remains open. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Extract all 36 complete tariffs and output demands from one pinned selected-supplier caller. Does it meet this exact contract after admission and instruction costs?
2. Prove structurally that every restored paired five-label walk misses an odd non-singleton mask, replacing the 945-path dependency.

Related open work

• Prove the exact 108-call unsigned F3 five-source connector optimum #4225 studies the same five-source, all-31-mask, single-dirty-connector interface over unsigned F3, attaining three calls per operator. Both interfaces have 36 operators; checked unit-price totals are 74 here and 108 = 3 × 36 there, so the optima differ across characteristics.
• Add five-source arity-four connector cost bound #3909 studies the same five-source/all-31-mask interface with an added paid arity-four source operation. Its original-alphabet baseline 74 calls / 148 dispatch-plus-addition units agrees with this formula at prices one / two, checked numerically. Its extended-alphabet 149-unit lower bound is outside this proof and was not rerun.
• Research: exact F3 minimum for the one-source additive connector #4189 studies unsigned F3 rather than F2 and leaves the five-source F3 batch open. There is no shared numerical optimum to compare; its six-call one-source result is not re-certified here.
• research: audit minimal joint-output connector charge #3897 adds a joint-output operation on a one-source/two-output boundary. Its dispatch-fusion quantity differs from this fixed 36-operator tariff; no overlapping invoice is claimed.
• Audit dirty direct-call supplier gap #3919 studies an added direct-product operation on the one-source boundary. This theorem keeps the original alphabet; no shared five-source invoice is reported by that work.
• Research: transfer-coupled (5,1,1) 7q readout obstruction (no new kappa) #2817 studies a five-source mutable common-B, two-connector transfer protocol. Its 452-versus-436 ledger is a different contract; no overlapping fixed-source quantity is asserted.

CI notes

Hosted run 39483387328 is green on head f3a717b2040076ce69e7b569fd57d0cf411b35b3: planning passed in 2m33s and the package replay passed in 1m27s. Both ordinary/optimized hosted scientific receipts match the local committed export exactly, including all source hashes and the witness digest. All registered commands, direct verification, export integrity, Git-free replays and source-history checks passed locally. No unrun local obligation was delegated to CI.

Merge Danger

Door: two-way. Blast radius: isolated. Only the new research package is added.

Produced by Nick Petalas. Apache-2.0. Ross Richardson and Cinna Davis retain the predecessor credit described above and in the package.

[petalas]

Closing as a duplicate. This envelope (2·sum of all 36 prices + 2·min over sources and weight-3/5 outputs, 16 minimal call-count vectors, 74-call attaining words) is already proved in Adam Gall's merged #2063 (research/connector-all-tariff-optimum), which this PR missed and did not cite. Credit for the theorem belongs to #2063 and its predecessors (#943/#934, #1770).