[aliromman]

This PR was merged at 09dd0965711ca7eeb169057a420abab1c9052137 on 2026-10-25. This description covers that merged construction. Later commits on the source branch are a separate follow-up.

Change and mathematical scope

This construction raises complete conditional κ to 0.00310045, a gain of 0.00001525 (0.4943%) over PR5565. It changes the independent helper target inventory while retaining the source's paid Gaussian calls. The selected frontier is unchanged.

Proof · Exact cuts · Complete receipt · Source pins · Validation.

At a common-frame cut, a helper-only prefix functional and a helper-only suffix direction have unit pairing. Explicit scalar input, cut and output bases let one independent helper finish early and let a fresh independent input use the remaining program. The 13 splits add 342 scalar events and no Gaussian call. The first split replaces H9 by independent H5 and H4 targets with 18 ADDs and four sign SCALEs.

The local word has 748 public inputs, 540 helpers and eight owned lifetimes. The complete p16/a7 caller has 66,726 paid calls, 13,048 productive fields and rank surplus 760. Every helper normalizer, main adapter, scalar operation, route, COPY/CLEAR path, query and idle is charged. The added fields enter the completion-tail bound. Tail decay is 39/40 and the compiler logarithm power is 1024.

The exact raw crossing is (0.00310045630724, 0.00310045630725); the claim is tested at 0.0031004505. An exact negative derivative bound on [0,0.0034] gives the raw ceiling for this fixed invoice. The final finite continuation screen finds no further tested basis pair in 52 helper groups. Neither check proves a global optimum.

The five selected framework assumptions and the helper-endpoint contract remain. The huge uniform word and complete rounded multiplier are not materialized. These are author-side checks; independent scientific review and hosted CI are pending.

Task and overlap

Base: canonical main 5e0052ca0aa3036a3afa8b0438927377c8d97d53, priority revision 7. The consumed PR5565 word is pinned to 9af814164b9dfc21f19ec737959d67cb71488c95. The verifier also replays PR5580 and checks the exact paid and helper changes supplied by PR5565.

Canonical coordination records the scope. The earlier fifteen-split PR5580 component is frozen at 31a4de233bbdd65c04c63dc3398ecf0eed8e1375. Two cuts are absent from the PR5565 composition, but its lower paid cost improves the complete claim by 0.00000206. No new commuting-contact search is introduced here.

The parallel coupled-basis claim owns six separate paid-boundary changes and is testing them with our earlier component. Our public handoff releases that component. This PR credits the shared mechanism and claims neither exclusive method novelty nor a combined bound with those paid changes.

Task: complex-new-helper-inventory; scope changed. The donor requested a different mechanism and a complete kappa gain. The scalar word and independent target count change the frozen-provider premise. Their complete caller bill supplies the acceptance test. TASK.json preserves the original card and scope reason.

Validation

python3 -B scripts/research_package.py --package research/p104-helper-subspace-splits

The direct isolated normal replay passes. It checks all 13 prefix/suffix witnesses, every intermediate arbitrary-input interface, 571,536 transpose coefficients, 2,923 exact native maps, the physical caller, two rational moment engines, compiler conditions and 54 strict plus four functional outer conditions. Mutation controls reject false support, missing basis operations, extra target credit and missing paid calls. Thirteen live inputs are authenticated before imports.

Registered normal and optimized replay, committed-byte checks and export integrity passed for this commit. Hosted CI passed for this merged head. A standalone read-only Git-free replay was not run for this composition; the previous component's checks are not substituted for it. Full-history preflight was not run in this shallow checkout. The retained source copies are pinned to ancestors of this branch; replay needs no cached unmerged-branch objects.

Credit

Pierre-Loic Doulcet / hexapode supplies PR5565's commuting-contact construction. Credit Ali Romman (PR5580 and PR5544/5545), Joel Pulikkan / GamingPuzzled (PR5569 and PR5472), DreamingOfClouds (PR5561/5568), thu-nmrc (PR5551), Gareth Davidson, David Arias, Cinna Davis, Jacob Alan Sussman, zxv, Cao Yifan and all retained scalar, native, geometry, BIT and analytic notices. The elementary rank-one basis identity is not claimed as a new general theorem. New work is by Ali Romman with substantial OpenAI Codex assistance; the same assistant constructed and checked it. No subagents. Apache-2.0.

Open next questions

The first follow-up is independent review of the first split and complete certificate. The parallel claim is testing composition with its paid-boundary changes. Further composition must use an exact public word and a new full native caller and invoice; no combined gain is assumed here. Structured next questions.