[RossRichardson]

Charles Bethin's PR1637 signed 3×3 connector uses twelve child queries and six paid root transfers. This contribution supplies a four-transfer word, retaining all twelve queries, both arbitrary-dirty roots, source preservation and the complete output map. The two root-doubling passes disappear.

Status: finite complex circuit construction and scoped minimum theorem; no new kappa. At exactly twelve signed child queries, four transfers are minimal for arbitrary fixed one-way root shears over Q or C. A short-transfer height argument reduces the lower bound to Ross Richardson's PR944 one-root obstruction, freshly replayed over all 720 paths by two implementations. Opposite-direction shears, scaled queries, more child calls and selected-supplier admission remain outside the result.

Both fully expanded 2/3/4/5/6/7-factor fixtures use 300 factor calls and five borrowed dirty rows (two roots plus three child rows). Total instructions fall from 306 to 304. In the matched dense integer tape model, including changed port bindings, the saving is 24 L^3 + 4 L^2 + 20 L - 4 head moves for every integer L>=1, plus 2L fewer source reads, target updates, additions, work clears, root doublings and root negations. These are exact model counts, not measured runtime. COST.md also gives port-specific tariffs and pays admission, preparation, control, full-row work and width.

Only research/signed-connector-transfer-minimum/ is added. One offline command:

python3 -I -O -B research/signed-connector-transfer-minimum/verify.py

Expected: PASS_SIGNED_CONNECTOR_FOUR_TRANSFERS queries=12 transfers=4 minimum_one_way=4 factor_calls=300 controls=44 no_new_kappa.

Local verification passes:

• Registered Python 3.12 replay, ordinary and optimized runs, and a clean committed export.
• Free noncommutative short/expanded forward and inverse maps; all three symbolic short-transfer shapes and meaningful hypothesis controls.
• Separate integer matrix/tape implementation: 24 fixtures, 660 arbitrary-input basis columns per direction, six tape lengths, and factor-order/scratch-restoration mutations.
• 44 short-word semantic mutations, read-only relocation, and five actual on-disk corruption rejections.
• Original C++ discovery replay: 1,641,024 restored source traces completed through three transfers; stopped at the first four-transfer witness. No exhaustive length-four claim.
• make selected-record-check; selected record and upstream/ unchanged. Full selected cold replay and hosted CI are separate, unclaimed gates.

Base: 858771bcca3b8dd5d4ef4aa682a13cc9b7e23920. OS assignment seed 32, complex-supplier track. PR1637 is pinned at 9bee6cc01f151b92ccfe147046855715843f066a. The submission refresh fully paginated 593 open PRs and 291 Discussion records without warnings. Main's selected PR1094 record remained byte-identical at 1f7a934975e5eacaf83511c9ac870263bf095e05.

Credit Charles Bethin / PR1637 for the two-root construction, twelve-query minimum, factor fixture and layout; Ross Richardson / PR944 for the target and one-root obstruction; martinxxx9 / PR1088 and Hanson Sun / PR1526 for the weighted/scaling predecessors. The proof extends woklerx's PR1438 source-port separation method from two XOR transfers to arbitrary characteristic-zero shears and the three-shear case. Gareth Davidson's PR1498 and Hanson Sun's PR1615 concern the different universal binary five-source target. Original notices and exact pins are retained.

Prepared for Ross Richardson with substantial OpenAI Codex assistance. Separate implementations by the same AI agent; no independent human or proof-assistant review claimed. Apache-2.0. The README and TASK.json give explicit next questions and blockers.

[Mister-Meeseeks]

Batch review at e6180f6acd6324d823031cab2ab51f31a388eb49.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Four one-way root transfers are necessary and sufficient at exactly twelve signed child queries. Symbolic arbitrary-coefficient height argument plus full 720-walk one-root census; literal all-input witness, not unrestricted-tariff or larger-alphabet optimality.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.