DONE: exact mutable connector cost with sequential delivery cuts (PR1780)

[RossRichardson]

DONE / HANDOFF — exact mutable-connector delivery optimum; CLAIM RELEASED.

Focused PR: #1780 . Published and locally verified head 15a91b5d634023a3a14129cbb240854c4b188111, base 0b1cccf86e2f5132ed286a26ef0a681a495740ea.

The generic full-alphabet minimum is 2 sum a_i + min_{p0=pk=0} [sum b_i(2-(p_(i-1) XOR p_i)) + sum d_i p_i], plus mandatory source updates and actual caller overhead. A coefficient/parity proof covers arbitrary call multiplicities; the typed word attains it for arbitrary compatible F2-linear maps. Uniform source prices reduce to maximum-weight matching on adjacent requests; unequal prices can favor a longer open span. Allowing prior correction calls in later request intervals at unchanged prices does not lower this optimum.

The literal three-output pilot uses 52 versus 64 scalar XORs, including source updates and dirty connector correction. This is a fixed macro-tariff optimum, not unrestricted XOR synthesis. The 43-to-31 pair and BC correction retain woklerx PR1439 credit; the chronological-cut model retains woklerx PR1442 credit. Ross Richardson PR934/943 and their dirty-commutator lineage, Cinna Davis PR1739's changed-map pair, and tennisbowling PR1524's distinct freely mixed-source/offline tariff are credited in the package.

Registered CPython3.12 replay passes: 518 tariff cases and 3280 uniform matching comparisons; separate Dijkstra agrees on all tariff cases, settling 67023 cut-local and 184183 delayed-adapter states without a depth cap. All 16384 2x2 map/path instances pass, and each three-output schedule passes all 131072 dirty inputs at every delivery. Six semantic controls, six package test groups, source/receipt corruptions, selected-record integrity, and both implementations in a read-only committed Git export pass. Both implementations have the same assisted author; no independent human or Lean proof review is claimed.

Final overlap at09:45–09:47UTC covered446 open PRs and418 Discussion records, with complete pagination and no warnings; all31 returned matches plus a synonym search were inspected. The PR is open; hosted checks are separate from these local passes. Neither codex nor codex-automation label is available upstream, so no such label was applied.

Next: extract one source-pinned physical caller's exact output cuts, source updates, B/BC prices, connector lifetime and hosting/precision/tape obligations; reject adoption if added costs exhaust the saved source calls. No selected occurrence, new recursive profile, new kappa, full selected cold replay or outer theorem is supplied. No merge, closure or selected-frontier action was taken. Substantial OpenAI Codex assistance; Apache-2.0.

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Original scope and advisory claim (now released):

CLAIM — exact mutable-source connector cost under delivery cuts

Question: for k sequential outputs y_i += A_i B x_i, with mandatory independent updates x_(i+1)=x_i+C_i u_i, what is the minimum fully charged query price when y_i can be written only within its own request interval? One dirty connector is carried; B and A_i are complete additive calls; correction D_i=BC_i is available only at update cut i. No extra root, free source change, cross-output writes or combined query is admitted.

Base: 0b1cccf; selected #1094 record unchanged. Catalogue nonce d75d8f45c97a449ba6ec24ee203b7c12, recursive-interfaces / persistent-boundary, revision 3, refined scope. Credit woklerx PR1439 (mutable source) and PR1442 (delivery cuts), and Ross Richardson PR934/943. PR1739 changes the B map in a pair; this fixes B and closes the chronological chain price. The live helper-476 claim and Z4 echo compositions are distinct.

Method/domain: exact coefficient/parity lower bound allowing arbitrary call multiplicities, a matching attaining word, independently implemented Dijkstra quotients for k<=4 and positive rational-price cases, and literal arbitrary-dirty three-output replay. Proposed bound is the two-state path minimum sum_i b_i[2-(p_(i-1) XOR p_i)]+sum_i d_i p_i, with p_0=p_k=0, plus all readouts, mandatory mutations and additional physical charges. The uniform pilot targets 52 versus 64 scalar XORs; this is not a selected supplier or kappa claim.

Entry overlap: 442 open PRs and 408 Discussion records, complete pagination, no warnings, plus local package/ledger evidence. Expected artifact: a source-pinned proof/checker package or a precise falsification. Stop after verifying this interface; physical caller/hosting/precision/analytic admission remain open. Substantial OpenAI Codex assistance.

Claim expiry: 2026-10-14T11:35:00Z.

[RossRichardson]

FINDING / DONE — signed dependent-source delivery. Claim RELEASED.

Public artifact: PR3069, exact reviewed/published head cd02d94c5eb3c0e1452cc94aeb2231c4d704a684. The PR head, title, body and all eleven public Git blobs were read back and match the self-reviewed commit. The scoped task is complete; no new kappa or selected-supplier admission is claimed.

The written characteristic-zero theorem prices all fixed signed unit-query words under strict delivery and final dirty-connector restoration by 2 sum a_i + min_{M^T t=0} sum b_i phi(t_i) (phi(0)=2, otherwise abs(t)). For sources x,x+u,x+2u, the balance (1,-2,1) uses 4B+6A rather than 6B+6A: two whole completed B calls disappear. Its complete comparison retains all source acquisition, imposed updates/return and noncommon storage/lifetime, precision, controller, routing and hosting charges. The actual supplier occurrence remains unknown.

The full nine-dirty-port calibration is 32 versus 40 ADDs, with all four restoring source mutations paid, but direct fusion is cheaper at 14. Primitive-prefix maxima are candidate/closed 9/8, and inverse maxima 8/9; a common bidirectional capacity already needs 9 for either word. These are scoped scalar-cell counts and exact range bounds, not a physical speedup.

Registered CPython 3.12 and optimized 3.14/Git-free/read-only replay pass: 117 primary optimizer cases, 130 separately implemented unrestricted signed-query cases, 19, 683 dirty grid states, all 81 coefficients per pilot and every delivery cut, 68 primary controls, 5 independent controls and 4 portability groups. Self-review found assertions disabled by -O; those checks were invalidated, replaced with explicit gates and rerun, including injected checker failures. Evidence is a written all-parameter proof plus finite exact checks/separate AI review, without a formal-kernel or independent-human theorem claim.

Credit Adam Gall PR2060 and the Ross Richardson PR1780/934/943, woklerx PR1439/1442 and martinxxx9 PR1088 lineage. Exact source pins and full credit remain in the package. Prepared by Ross Richardson with substantial OpenAI Codex assistance; Apache-2.0.

HANDOFF: extract one named actual complex recursive caller with this affine source trajectory, exact signed child interfaces and legal reserved connector; compare complete child histograms, best fused alternative, physical lifetimes and compiled precision. Stop further toy expansion until that occurrence is supplied. Primary scientific base d8e0f83f00191df62174b641e9fff055c357f641, priority 6, unchanged at the 21:38:13 UTC refresh. Overlap corpus at21:36–21:37 UTC fully paginated 305 open PRs including drafts, 1,675 Discussion records and 2,440 closed PR bodies; non-atomic metadata coverage is not novelty proof. Hosted CI remains separate; no rerun/dispatch/retrigger was performed or requested.

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Original scoped claim, retained for provenance (now RELEASED):

CLAIM — signed dependent-source connector under sequential delivery

Status: CLAIM, 2026-10-16 21:29 UTC. Next: derive the signed all-word tariff and independently verify a three-delivery dirty-input witness. Advisory expiry: 2026-10-16T23:29:00Z. Evidence currently local-only.

Question / changed restriction: Adam Gall PR2060 (3555161) prices binary dependent source updates. Does a characteristic-zero interface with signed unit source/readout queries support an exact integer-balance tariff, while retaining each output's request cut and final dirty-connector restoration? Fix x_i=sum_j M_ij v_j for nonzero rational rows M_i, arbitrary exact linear B,A_i, and positive sign-symmetric complete prices. No correction queries or intermediate connector observer are admitted. This is a different algebra from the F2 theorem; it does not reopen that theorem.

Smallest decisive test: sources x,x+u,x+2u and signed balance (1,-2,1). The proposed word uses four complete B calls and six reader calls rather than six B and six reader calls. Pay every imposed source update and terminal source restoration when required. Prove the lower bound for arbitrary query multiplicities and test the literal inverse and dirty-state behavior. Signed intermediate growth, source/coefficient access, private cleanup, routing/frames/head returns and persistent connector capacity remain charged. Supplier admission requires any extra interface cost to fit below the two removed complete B children; no selected caller or new kappa is claimed.

Base: d8e0f83; priority revision 6; selected record SHA256 bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129. Credit Adam Gall PR2060, Ross Richardson PR1780/934/943, woklerx PR1439/1442 and martinxxx9 PR1088's signed complete-price precedent. Same-source static signed-star and binary correlation results retain their original scopes.

Overlap: helper snapshot fully paginated 303 open PRs including drafts (21:24:23 UTC), 1,805 Discussion records (21:25:13 UTC), and 2,440 closed PR bodies (21:25:49 UTC), plus current local packages/ledger. Exact predecessor head was reread. This is non-atomic keyword coverage, not a novelty proof. Batch member05 reserved only this obligation; other members' work is untouched.

Expected artifact: one scoped proof/checker package or a precise handoff. Stop after this three-epoch mechanism and full macro-cost comparison; no large search, selected-record change or supplier expansion. Prepared by Ross Richardson with substantial OpenAI Codex assistance, including a separate mathematical checker; Apache-2.0.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "connector-alphabet-change",
  "scope": "Signed characteristic-zero dependent-source sequential delivery with integer-kernel restoration",
  "base": "d8e0f83f00191df62174b641e9fff055c357f641",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/3069"
}