[1337isnot1337]

Scope

Cycle seed 0d762194a82c097da60f154a08625527; task complex-new-helper-inventory; base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

This PR packages a conditional local mechanism for the selected width-73 complex word. It asks whether two tagged one-dimensional register-frame quotients can be served by one rank-two recursive child at the ordinary rank-two price. The native word and selected result are unchanged. The proposed J2 interface has not been admitted or physically hosted, so this is not a new multiplication bound and claims no new κ.

Evidence

The pinned source word decompresses to SHA256 c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8, matching the selected slot output pin. The exact construction pairs 574 A/B same-gate climbs, 16 scheduled early-cleanup climbs, and 163 terminal pairs. It recomputes the rational gate shear and inverse, all tagged quotient dimensions, and the virtual profile.

• Baseline profile: {1:2412,2:418,3:432,4:142,5:82,8:8}.
• Conditional profile: {1:906,2:1171,3:432,4:142,5:82,8:8}.
• Rank mass remains 5,586; the exact fixed-word ceiling improves by at least 5.241623654% under the zero-overhead J2 assumption.
• At the 5% threshold, the exact model permits at most 11 added rank-one children per local call when charged against fractional rank-51 idle mass. The relaxation at a=1/99 remains above one.
• Four mutated recipes are rejected: missing pair, duplicate pair, changed rational gate coefficient, and invalid quotient vector.

Reproduce

python3 -B research/complex-joint-child-seed0d762/build.py
python3 -B research/complex-joint-child-seed0d762/verify.py
python3 -I -O -B research/complex-joint-child-seed0d762/verify.py
python3.12 -B scripts/research_package.py --package research/complex-joint-child-seed0d762

The direct verifier passed locally with Python 3.14.7 in normal and optimized modes. Python 3.12 is unavailable in this container, so the registered replay was not run locally; package metadata and source-pin checks passed.

Open obligation

The package identifies all cost categories that still need an implementation: J2 supplier admission, source access, copies, phase and precision transfer, restoration, and complete-call hosting. No owner map, global assembly, or selected frontier is changed. A fresh overlap search covered all 435 open PRs and 396 Discussions with complete pagination and no warnings. It found only one broad expired match, Discussion #1328; no exact joint-register child match appeared. The scope and expiry are recorded in CLAIM.md and OVERLAP.json.

Substantial OpenAI Codex assistance is disclosed in the package.