Research: optimal 3x3 binary connector queries (no new kappa)

[RossRichardson]

Restoring a dirty connector between product stars repeats full factor queries. This research construction carries partial source sums between readouts: every binary 3-by-3 product pattern has an explicit schedule using exactly two calls per active operator, attaining a proved lower bound in the specified one-connector query alphabet.

For the six-edge cycle, the complete schedule uses 12 factor queries instead of 18 for either row or column stars. A literal recursive lowering with factor lengths 1 through 6 uses 198 leaf calls instead of 228 for the better star comparator, with the same four borrowed dirty arrays. The package pays child and inverse calls, coefficient/source access, storage, instruction work, frames and tape returns under explicit interfaces. These are operation counts; no new kappa, selected-supplier integration, general circuit optimum or measured speedup is claimed.

Only research/binary-connector-walk/ is added. The README gives the exact contract, lower-bound argument, full cost recurrence and unresolved obligations. The certificate contains all 512 literal words and the 198-leaf expansion; the independent verifier imports no search/lowering code and does not use reported BFS distances to establish optimality.

Base: 91a2094f2e022930069e8cfdc810a3ea1b81cd09. OS assignment seed 51, recursive-interface track. PR889 is the direct one-source fanout predecessor, pinned at e2021d1c9e06948528cbc0a8fd7079b671da981d, with its README and notice retained byte for byte. Credit also goes to Chafik Boukhalfa's PR628 dirty commutator. PR822's tree storage reuse, PR522's coordinate sector and concurrent PR929's grouped cyclic inverse have different contracts; their savings are not combined here. The recorded prepublication inventory reached #931; the immediate submission refresh also checked #932 (X4 scalar target forests) and #933 (native bit rewrites), which leave this abstract multi-input query model distinct. No pinned sprint coordination issue exists; CLAIM.md records the local claim and completed result. Attribution and substantial OpenAI Codex assistance are disclosed in NOTICE.md; Apache-2.0.

Validation on macOS / Python 3.14.6:

python3 -I -O -B research/binary-connector-walk/verify.py
python3 -I -O -B research/binary-connector-walk/test_package.py

Expected: PASS_BINARY_CONNECTOR_WALK targets=512 dense_cases=23040 controls=11 cycle=18->12 lowered=228->198 and PASS_BINARY_CONNECTOR_PACKAGE relocation=1 regeneration=1 corruptions=4.

• All 32,768 search states and 196,608 directed edges exhausted; all 512 target patterns independently replayed, along with both star comparators and literal inverses.
• Every witness attains the per-operator support bound, covering arbitrary positive fixed query prices.
• 23,040 dense cases; 11 obligation-specific controls, including a valid but nonoptimal word and counterexamples outside the binary/linear contract; 120 independent closed-form tape checks.
• Exact certificate regeneration and standalone relocation pass; four corrupted package files reject. Normal and optimized Python summaries agree. Retained SHA256 and Git blob identities match.
• make selected-record-check, all five existing selected-result tests and staged whitespace checks pass. The selected frontier and upstream/ are unchanged.

Full repository regeneration and the expensive selected-supplier cold replay were not rerun for this standalone package. No hosted-CI, formal-theorem or independent human-review pass is claimed. Physical supplier realization, compatible frames and ownership, whole-row charging, compiler/tape integration and outer analytic constraints remain uninstantiated.