[petalas]

Result

Scoped written proof with exact finite corroboration. No κ change. This answers the open question of #3321: can two Q5 calls, or a mixed Q5/DUO/TRIO alphabet, create a new priced point on the five-source, 31-row connector batch?

In the #943/#3321 interface, with every nonempty source subset available as one complete call (one dispatch plus |S| source additions, no surcharge):

• Theorem 1. Every correct schedule, over any toggle alphabet and any number of uses, has at least 72 queries and at least 74 additions. The DUO word of Research: two-source toggle DUO for the five-source connector — 72 queries at 74 adds (no new kappa) #1531 attains both, so the minimum price is 72o + 74p for all o >= 0, p > 0, and (72, 74) is the only Pareto point. No mixture and no number of Q5 calls improves it.
• Theorem 2. Every schedule that contains a weight-5 call has at least 78 additions, so its price is at least 72o + 78p, the DUO price plus 4p. The bound is attained by a word with one Q5, two DUOs and seven singletons (ten calls, 16 additions). Two or more Q5 calls give at least 80 additions.

So a mixed Q5/DUO alphabet does give a priced point that beats the singleton witness (74, 74), exactly when o > 2p. No alphabet beats the DUO point at any price. With a per-multi-source-call surcharge sigma, the DUO word wins over singletons exactly when sigma < o, and the Q5 word is dominated for every sigma >= 0.

New here: the partial-coverage form of the support bound (T + 2u >= 10, where u counts rows no boundary pair realizes), the expansion bound A >= 74 for every alphabet, the single-point frontier, the weight-5 floor 78 and its attaining word. The full-coverage support bound for steps of weight 1 to 5 is #4137. The 945-pairing census and singleton minimum are #943, the DUO word and the 99,200 ten-point covers are #1531, and the Q5 model and one-Q5 census are #3321. These are cited inputs, recomputed as controls, not audited.

Evidence and reproduction

Interpreter: /Users/nick/git/integer-mult-bounds/.venv/bin/python3.12. Any CPython 3.12 works; only the standard library is used. From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/five-source-toggle-frontier/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/five-source-toggle-frontier/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/five-source-toggle-frontier
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/five-source-toggle-frontier --check-source-history

Run locally at head 7ef416e83b8220d0a2f91cb34ea0c3bb9d892154: both verifier modes pass in about 17 s each (PASS_FIVE_SOURCE_TOGGLE_FRONTIER nine_point_covers=0 ten_point_covers=99200 ... weight5_witness=(72,78) no_new_kappa), the package runner prints PASS_REGISTERED_RESEARCH_PACKAGE research/five-source-toggle-frontier outcome=negative-result in 33 s, --check-only prints PASS_PACKAGE_METADATA_AND_PINS, and the commit check prints PASS_RESEARCH_COMMIT_BYTES.

The verifier recomputes from integers:

• no nine-point set of F_2^5 has P + P = F_2^5 (exhaustive search with an exact waste cut, and the affine-basis normal form); the same search finds 99,200 ten-point covers, the count recorded by research/duo-toggle-connector;
• the 945 length-ten pairing words (coverage histogram equal to Research: sharp four/five-source dirty-connector boundary (no new kappa) #943's, best 31 of 32 masks);
• the 115,920 closed words with one weight-5 step and nine singletons (at most 27 of 31 differences);
• the DUO and weight-5 words as explicit 72-call schedules, each replayed symbolically for arbitrary operators and concretely with random binary maps and dirty data;
• the T + 2u table, the Theorem 2 case checks for T <= 40, 30,000 random closed words over all 31 toggles, a 6,000-step hill climb, and nine rejection controls.

Limitations and unrun obligations

• The price model is the favorable ledger of Research: one all-five toggle has no favorable price (no new kappa) #3321: a multi-source call costs one dispatch plus its additions, with no frame, routing or implementation surcharge. No physical call, compiler or supplier is established.
• The model is the binary-linear five-source, 31-row interface with singleton readouts. Joint readouts, extra connectors, mutable sources, frame changes and other source counts are outside it.
• The reduction to prefix differences is inherited from Research: sharp four/five-source dirty-connector boundary (no new kappa) #943 (necessity by the identity-map specialization). The proofs are written; finite components are exhaustive under the stated reductions. No proof assistant was run. Independent human and formal proof-kernel review are unclaimed.
• Weights 3 and 4 have no analogue of Theorem 2 here.
• Verifier discipline: no recursion and a finite bound on every loop; the cover search enforces a 60,000,000-step limit against the 5,403,753 and 41,511,450 steps it uses. Python allocates integers and containers while it runs, a managed-runtime deviation confined to these bounded searches.

Open next questions

1. Exact minimum price among schedules containing a weight-3 or weight-4 call. First step: rerun the depth-first search used for the Q5 witness with one such call and extra weight 2 to 6. Nothing blocks it.
2. Does the single-point collapse hold for every alphabet at six sources? Research: high-arity connector toggle cost floor #2249 reports r((C2)^6) = 14 and f(6) = 16, which would give (140, 142). First step: rerun the support and expansion arguments with those invariants. The invariants are cited from an unmerged package.
3. Can a readout-side alphabet move the 62-readout floor (62 of the 72 queries)? Toggle-side extensions cannot. Blocked: no priced joint-readout call is defined.

Related open work

• Research: high-arity connector toggle cost floor #2249 studies the six-source analogue (general-arity toggle floor). It uses the same support-plus-expansion idea at six sources; this PR computes only the five-source numbers, so there is no shared figure to compare.
• Add five-source arity-four connector cost bound #3909 bounds the five-source arity-four extension at 149 units against 148 for singletons. This PR's alphabet-agnostic floor is 146 units (72 + 74), attained by DUO, so the two are consistent.
• Six-source one-TRIO path misses full paired coverage at 14 calls #2134 is a six-source one-TRIO census, a different source count.
• research: audit minimal joint-output connector charge #3897 studies a joint-output readout, the readout side that open question 3 leaves open; this PR does not touch it.
• Audit dirty direct-call supplier gap #3919 audits a direct-call interface boundary; it does not study the toggle ledger.
• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079 bounds seven-source connectors of weight at most four, a different source count.
• Prove the eight-query mutable-source tariff is exactly 460 (no κ) #6696 (same author) studies the eight-query mutable-source tariff, a different connector interface (common-B mutable-source words). This PR computes no figure that it also computes.

CI notes

CI did not certify this result. The plan job of run 41721283470 at head 7ef416e83b8220d0a2f91cb34ea0c3bb9d892154 (job 126214604601) was cancelled by the workflow's 10-minute timeout while still in the actions/checkout step (21:23:22Z to 21:33:26Z). The replay job was skipped, so no package step ran in CI. The package did not cause this.

Checked on other PRs: the plan job of #6676, #6691 and #6692 was cancelled at the same actions/checkout step after about 10 minutes. Of the 19 PRs numbered 6676 to 6696 whose checks I read, 13 show a plan failure of about 10 minutes and 6 passed.

The result is certified only by the local checks listed under Evidence and reproduction (both verifier modes, the registered package runner, the metadata check and the commit-byte check, all at head 7ef416e83b8220d0a2f91cb34ea0c3bb9d892154).