[petalas]

Result

Scoped exact result with a finite certificate and a written lower bound; no new κ. The universal five-source dirty connector of Ross Richardson's #934 contract costs exactly 74 complete queries for the all-31-nonzero-mask batch. Adding one priced operation and changing one boundary premise

Z:  z := 0        one complete query; the connector is consumed, its exit
                  value is unconstrained

gives exact optimum 63 = 2^6 - 1: one reset, 31 source calls along a reflected-Gray walk of the 5-cube, 31 single readouts. The matching lower bound 2^(n+1) - 1 holds for every n >= 1, so the closed form is not a five-source accident.

The saving is 11, not the 31 that a "one readout per mask" comparison suggests. The readout term 2(2^n - 1) is identical under both contracts; only the source walk changes, and the saving is exactly T(n) - 1 where T(n) is the restored minimum source-call count: 7 at four sources (38 -> 31) and 11 at five (74 -> 63). So connector restoration is worth T(n) - 1 queries, at least 2n - 1 in general, and the dominant readout term is irreducible in this alphabet.

The reset is essential, not decoration. Keeping the connector one-way but forbidding resets still leaves every readout dirty, so every row needs an even read count and the cheapest such design costs at least 9 + 62 = 71: the package exhausts all 70,846 simple eight-step paths from 0 in the 5-cube and finds pairwise coverage at most 28 of 32, which excludes a nine-vertex difference basis and forces a ninth source call.

Claim type: scoped written proof plus exact finite certificate. It is not a κ improvement, not a supplier admission, and not a complete multiplication theorem.

Evidence

PROOF.md proves 2^(n+1) - 1 in two cases. For r >= 1, the counting lemma R >= 2(2^n - 1) - |S| >= 2(2^n - 1) - T makes the total independent of T, so a longer walk buys no discount. For r = 0, parity gives R >= 2(2^n - 1) and T >= 1, with strict inequality for n >= 2.

verify.py replays the literal schedule in a free noncommutative symbol model (arbitrary linear, possibly noncommuting A_i, B_j, arbitrary dirty connector and outputs), checks the three counts, recomputes the complete optimum for n = 2, 3 from every visited vertex set and its cheapest readout assignment, runs the eight-step path census, and applies six rejection controls (missing reset, one readout short, paired-restored cost, truncated 30-step walk, duplicated readout, optimized mode).

Reproduction

~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev research/one-way-connector-reset-batch/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev -O research/one-way-connector-reset-batch/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/one-way-connector-reset-batch
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/one-way-connector-reset-batch --check-source-history

Expected from the verifier:

PASS_ONE_WAY_CONNECTOR_RESET exact_n1..8=3,7,15,31,63,127,255,511 optimum5=63 restored5=74 saving5=11 reset_free5_floor=71 nine_point_coverage=28 paths=70846 controls=6

Locally observed: both verifier invocations pass in about 0.1 s; research_package.py reports PASS_REGISTERED_RESEARCH_PACKAGE outcome=completed; check_research_commit.py reports PASS_RESEARCH_COMMIT_BYTES with source_history_commits = [e54523517d0f03e58352e647c403a5af3e525a57]. Both registered checks are far below the 10-minute cap.

Limitations and unrun obligations

• Unit query prices only. No nonuniform c_Z, c_B, c_A crossover is computed.
• The changed premise is that the caller may discard the shared connector. A caller that must return the old value pays a separate save and restore, which is not priced here; the favourable margin is only 11 units at five sources, so that wrapper has little room.
• The 74 and 38 values and the restored source-call counts T(5) = 12, T(4) = 8 are cited from the predecessor. Its 945-word census is not replayed or audited here.
• No physical arity-four/arity-five source operator, routing, storage or frame cost is claimed, and no kappa or selected-frontier update follows.
• No proof-assistant formalization was attempted.

Open next questions

• Does a charged save-and-restore wrapper keep the gain? Price one clean width-L row plus copy, reset, readout and restore on the same literal instance.
• With arbitrary positive c_Z, c_B, c_A, what is the exact crossover between the Gray design (31 c_B + 31 c_A + c_Z) and the paired design (12 c_B + 62 c_A)?
• What is the reset-free floor for general n? Five sources give 71 by a complete path census; the general minimum traceable difference-basis size is open here.
• Does a bounded number of resets help a sparse requested mask set rather than the complete batch?

Related open work

• research/universal-connector-boundary/ (PR Research: optimal 3x3 binary connector queries (no new kappa) #934 lineage): supplies the contract, the 945-pairing census, the restored 74 and the four-source 38. Cited, not replayed.
• PR Add five-source arity-four connector cost bound #3909 (open): a five-source arity-four source event, charged one dispatch plus four additions, with every such schedule at least 149 units. That enlarges the source alphabet; this enlarges the connector contract. Complementary, neither implies the other.
• PR Research: high-arity connector toggle cost floor #2249 (open): a high-arity six-source toggle floor with f(6) = 16, again a larger alphabet and arity. No number is compared.
• PR Realize the exact 140-call two-source GF(8) connector (no κ) #7206 (open): realizes an exact 140-call two-source GF(8) connector in a different model. Same general object (connector batches), no overlapping count.
• PR Classify the six-source centre in every characteristic (no κ) #7318 (open): classifies the six-source connector centre in every characteristic. Same connector object, different quantity, no shared number.
• PR Bound distinct adjacent pairs of a length-28 universal word (no kappa change) #7321 (open): bounds adjacent pairs of a length-28 universal word on the same card family. Different quantity, no shared number.
• research/two-connector-transfer-boundary/: two connectors with a paid transfer, explicitly excluding connector resets. This analyzes the excluded operation at one connector.

CI notes

Head fd573cfe4e9714706b01f0534584e6f5032dc858, unchanged: no fix or retrigger commit was pushed.

Run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/43106965056 (job 130812696406): the plan job failed and replay was skipped, so CI certified nothing for this package.

The failure has one cause, a repository-level fixture mismatch on main rather than anything in this branch. The "Onboarding helper tests" step failed exactly one test out of 760 (24 skipped):

test_current_ci_section_matches_selected_record
  (test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record)
tests/test_research_current_status.py:58
AssertionError: '`make frontier7000-verify`' not found in '# CI verification ... Current selected-family replay: #6030 ...'
FAILED (failures=1, skipped=24)

On main e54523517, certificates/selected-result.json carries "reproduce": "make frontier7000-verify", while docs/ci-verification.md still documents make frontier6030-verify in its "Current selected-family replay: #6030" section. This branch touches neither file: git diff --name-only origin/main...HEAD lists only research/one-way-connector-reset-batch/. Running that single test locally from this branch on main's own file contents reproduces the identical AssertionError, which confirms the mismatch predates the branch.

I checked other runs for the same failure rather than assuming it:

• run 43105433200 (this account, head c5de3d6b23): same test, same FAILED (failures=1, skipped=24);
• run 42905094314 (another contributor, head d14f787dc0): same test, same signature.
  Also failing at the plan job: runs 43105694304, 43105875821, 42903313790; I confirmed their conclusions but did not inspect each log line.

Because replay was skipped, the registered VERIFY.json commands were not run by CI. Locally both pass on CPython 3.12 from the repository virtual environment in about 0.1 s each, with research_package.py and check_research_commit.py also passing. Those remain local results, not a hosted certification.