Scoped negative result: all unit complex phases retain the 14-query connector minimum

[bbMisk]

Unit complex query phases and any number of unit root rotations cannot reduce PR944/PR1088's fixed three-by-three dirty connector below 14 complete child queries. With positive fixed complete operator tariffs, the optimum remains 2 sum_F c_F + 2 min_F c_F.

Status: scoped negative result; no new kappa. Concurrent #1551 establishes the finite sixth-root case. This contribution adds the entire complex unit circle and arbitrary unit root rotations, using support-graph normalization and a geometric collapse of a three-step triangle to the older signed obstruction. Its finite sixth-root census is independent corroboration; no priority over #1551 is claimed. That PR and Discussion #1539 were found during the final overlap follow-up and their exact proof scope was inspected before submission.

The written argument only needs the existing 720-path signed exclusion. Additional exact evidence covers all 30,240 normalized triangle paths and checks every collapse with a separate implementation. A thirteen-query control succeeds when a coefficient of magnitude two is admitted, while a five-query unit-phase fixture verifies that odd restored words are legal. These controls separate the new restriction from signed parity and unrestricted scalar coefficients.

Only research/complex-unit-phase-connector-seed39/ is added. It contains the proof, certificate, source pins, local claim, overlap record, two implementations, complete factor-cost recurrence and explicit replay hook.

python3.12 scripts/research_package.py --package research/complex-unit-phase-connector-seed39

Expected: PASS_REGISTERED_RESEARCH_PACKAGE research/complex-unit-phase-connector-seed39 outcome=negative-result.

Local checks passed:

• Registered Python 3.12.13 replay, including optimized execution, source bindings and both exact implementations.
• 1,728 attaining orbit words, 18 root-gauged words, 172 semantic controls, six complete arbitrary-dirty factor expansions and 444 dense forward/inverse pairs with noncommuting maps.
• Read-only relocation, deterministic regeneration and six on-disk source/certificate rejection controls.
• Normal Python 3.14 direct checks, all 20 research-onboarding tests, selected-record integrity and whitespace checks.

Base: ef5f39403610d231a0c27e8a6b4f08c10209bcd6; OS assignment seed 39, complex-supplier track. Refreshed main c5108f0f68d482e5712a8e26a5978d47762dfc16 leaves all live inputs and the selected result unchanged; its ledger changes are historical provenance only. The late #1551 comparison is pinned at be8768d5fd8e9e7a1f714424f25b34b0e807daa0.

No selected-supplier instance, native hosting, rounded precision, physical runtime saving, selected cold replay, proof-assistant theorem or independent human review is claimed. The result requires exact complex-linear maps, immutable sources, one dirty root and the stated query alphabet. No merge or frontier promotion is requested.

Credit Ross Richardson / #944 and @martinxxx9 / #1088 for the signed obstruction and six attaining repairs; concurrent #1551 for the sixth-root result, #1526 for paid magnitude scaling, and #1349 for the F3 comparison. Prepared by Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in reasoning, implementation, verification and prose. Apache-2.0.

[Mister-Meeseeks]

Batch review at ca8bef7d5152cbdfc96f3f12cdd958a953bd5708.

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

Reviewed scope: Support-graph normalization and triangle collapse extend the signed fourteen-query floor to every unit complex phase and arbitrary unit root rotations. Exact normalized cases and all attaining words pass; nonunit magnitudes explicitly escape the theorem.

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.