Research: classify signed 3x3 connector obstructions and a 14-query repair (no new kappa)

[RossRichardson]

The binary two-calls-per-operator result in #934 needs a new argument for signed integer coefficients. This research construction and exact finite classification supplies that boundary in the same one-connector, three-source/three-output query alphabet. No new kappa.

Of all 19,683 matrices in {-1,0,1}^(3 by 3), 19,395 attain the support bound, including all 512 unsigned patterns. The other 288 form one signed/permuted orbit and need exactly 14 queries. A written reduction turns every possible 12-query word into one of 720 closed source paths; exhaustive exact exclusion and even query multiplicities establish the lower bound. An explicit 14-query word attains it and restores arbitrary dirty input. This is not a lower bound for general circuits.

For the canonical obstruction, complete recursive lowering of factor lengths 2 through 7 uses 308 leaf calls versus 388 for row stars (676 for column stars), with four dirty arrays throughout. The note charges every child and inverse, source/coefficient access, routing, head returns, storage and instruction work. A dense exact realization and bit-width bound make the local bill concrete. These are operation counts, not measured runtime or a supplier improvement.

Only research/signed-connector-obstruction/ is added. It contains the proof/reduction, exact certificate, independent verifier, complete cost recurrence, retained predecessor notices/source pins, and local sprint claim. No predecessor implementation is imported. Supplier integration, physical frames/owners, approximate-child precision, all-size compilation and outer analytic obligations remain open.

Original experimental base: 91a2094f2e022930069e8cfdc810a3ea1b81cd09. Refreshed submission base: 47481c89fd30a43f8f882eb4b391e89c29273fd0. The refreshed selected result is #890 at 1661957774647029/10^18; it and upstream/ are unchanged.

Validation (Python 3.14.6, macOS arm64):

python3 -I -O -B research/signed-connector-obstruction/verify.py
python3 -I -O -B research/signed-connector-obstruction/test_package.py

Expected markers: PASS_SIGNED_CONNECTOR_OBSTRUCTION and PASS_SIGNED_CONNECTOR_PACKAGE.

• All 720 source paths, 19,683 free noncommutative operator maps and inverses, and the literal recursive programs.
• 330 dense cases, 22 mathematical controls and 15 independent tape trace/formula comparisons.
• Normal and optimized receipts agree. Standalone relocation and byte-identical regeneration pass; all nine corrupted packages reject.
• Four retained files match pinned GitHub blob identities and SHA256. Selected-record integrity and all five selected-result tests pass on both bases; staged whitespace is clean.

No full repository regeneration, supplier cold replay, hosted-CI success, proof-assistant verification or independent human review is claimed.

Sprint: recursive interfaces, OS seed 53. No pinned coordination issue exists. The final full overlap refresh covered 711 open PRs through #940, followed by arrivals through #943. #934 is the directly credited binary predecessor; #938 counts minimum words in the distinct one-source fanout alphabet. Concurrent #943 studies the four/five-source binary boundary; this contribution instead changes the coefficient contract and supplies a signed three-source obstruction and repair. Predecessor heads #889 and #934 remain pinned and unchanged.

Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance. Credit to CubeJerry's #889 fanout interface, Ross Richardson's concurrent #934 binary walk, Chafik Boukhalfa's #628 dirty commutator, and the selected construction's cumulative authors. Original notices and Apache-2.0 license are retained.