P104 free helper endpoints: uniform normalizers and exact frame-pair costs

[aliromman]

DONE / RELEASED — Prove the sharp P104 price floor for all fixed-contact frame paths, exact head 627160c5e39501b69603c3a1e03bbfdc50a53a86.

The PR5580 word already attains the best recursive moment for every frame path and public helper endpoint with its scalar contacts fixed, including arbitrary nonnested subspaces. The bound holds for every power tariff r^t with 0<t<1. Further endpoint-only or intermediate-frame detours cannot improve this word's kappa. No new kappa is claimed.

The exact native frame price is the subspace distance dim(U)+dim(V)-2dim(U intersection V). Triangle inequality and concavity give a sharp cost floor between consecutive contacts and a sharp helper endpoint credit bound. All 3,453 nonzero contact intervals attain the complete signed-profile floor. The only unused helper, register 711, contributes a rank-five target and paid job that cancel. Main endpoints and scalar/owned contact frames are fixed. Changed contacts, scalar realizations and joint query interfaces remain outside the model. Canonical coordination is Discussion 5589.

Registered Python 3.12 normal and optimized checks pass. The credited scalar checker replays all 735 arbitrary public inputs and eight owned lifetimes. New checks bind all 2,921 source frame pairs, exhaust 4,770 small ordered subspace pairs, check three nonnested recipes including global phase, and reject changed scalar/contact and false dimension-difference controls. The interval certificate SHA-256 is 544092a4c732d94b7895689745b92c4ea0cafda665cdd6652b9bee352ffbb378.

Next: change a scalar contact or word and use this floor to reject unprofitable candidates before physical compilation. Respect the active changed-contact and basis claims. The selected frontier is unchanged. No new search is reserved. The selected result is unchanged. Prepared for Ali Romman with substantial OpenAI Codex assistance.

Original claim

CLAIM / scope correction — fixed-contact P104 ceiling, including arbitrary nonnested paths and helper endpoints.

PR5567 at 0888d33 already resolves the nested endpoint question in the initial claim. It is credited and not counted again. The new question is whether arbitrary nonnested excursions can improve PR5580 with all scalar/owned contacts fixed. The proposed proof uses the exact canonical-frame rank d(U,V)=dim(U)+dim(V)-2dim(U intersection V), triangle inequality and concavity. A full literal contact certificate will test whether the source attains this floor for every power tariff. This takes no changed-contact, event-order or basis search.

Base 5e0052c, priority 7; source PR5580 head 6efa248. No new kappa or global P104 optimum is claimed. Credit Matarener, Chris R/Clear1nsight and the retained native/P104 authors. Ali Romman with OpenAI Codex assistance. Expiry 2026-10-25T23:45:00Z.

Superseded initial claim

CLAIM — close the free helper endpoint contract for P104 frame search

Question: do the free helper starts and finals used by PR5568 and PR5561 define independent, correctly priced Gaussian jobs in the complete P104 caller? I will bind the exact helper endpoints to monomial normalizers, prove the general frame-pair rank rule, and check the current words. This answers PR5568 README Open next question 1. It is an interface proof/audit, not another event-retiming or basis-resynthesis search.

Base: canonical main 5e0052c, priority 7. Sources: PR5568 toolkit, PR5561 free endpoint word, PR5569 priced handoff and PR5580 event contact, with the PR5545 executable closure. The target is a uniform helper-interface lemma, explicit normalizers, full helper counts and exact source checks. Field maps must be charged; a rank count alone is insufficient.

The test will cover all helper rows of the pinned successor words and exhaust all frame pairs in small dimensions. The proof must extend to every positive tensor width, preserve scalar coefficients outside tensor powers, and identify the cost and all inherited compiler assumptions. Stop if arbitrary independent input or endpoint normalization fails. No new kappa, selected frontier update, or new word is claimed. The active searches in Discussions 5125 and 5575 remain separate.

Prepared for Ali Romman with substantial OpenAI Codex assistance. Credit DreamingOfClouds, GamingPuzzled, thu-nmrc, Gareth Davidson, David Arias, Cinna Davis and the retained native/compiler lineage. Claim expires 2026-10-25T23:45:00Z.