[1337isnot1337]

The actual H3-prefix offer uses an original H9 helper as its third chain owner. Reusing that owner through two native FULLs replaces two independent H9 rewards. This follow-up to merged #5137 prices that loss using the pinned native paid/helper receipt, before expanding the one-port prefix into a larger construction.

Even granting112 usable prefixes, free homogeneous charts and boundaries, all other helper rewards, and zero new split/precision/cleanup cost:

• The bare three-chain output 224 H17 +112 H16 +2(H-112 H9) loses at least128 moment units for every 0<=C<1.
• The more favorable, unconstructed bonus output 336 H17 +2(H-112 H9) still loses at least16 throughout that range.

The proof supplies exact nonnegative sum n_r min(r,z) cuts at every breakpoint. Increasing concavity transfers the certificate to every power r^(1-C); no floating root scan is used. This stops the stated full-capacity repeated-H9-owner policy. It leaves callers retaining both distinct helper jobs, changed provider/target censuses, other stage/axis counts and broader operation models open. No112-prefix construction, fresh native replay, new kappa or supplier admission is claimed.

The package authenticates the3772-byte source receipt from one-terminal-h3-prefix, preserving Chris R/Clear1nsight's actual prefix/phase work and Cao Yifan/intqwq/Cinna Davis's source lineage. Three budget mutations reject an excessive margin, double-counted aliased helper jobs, and omitted paid centers. The original all-length controller theorem and its source pins remain intact.

Validation at 8d1a267b708bd30bdd84a3ab47d5a67de7aeb745: CPython3.12.15 normal/optimized direct and registered replay pass; exact committed-byte/source-history preflight, whitespace check, official eight-file export/integrity, and fresh Git-free optimized replay pass. On refreshed canonical a5df48c55ea806d7952b65acc9c76d878ff09b13, all11 targeted onboarding tests affected by the earlier baseline failures pass in22.993s. The old #5137 CI failure was reproduced on its original base and is preserved in that PR; this branch includes the upstream fixture fixes. Hosted follow-up checks are pending. Selected conditional kappa0.00302015 and priority revision7 are unchanged.

Reproduce: python3.12 -B scripts/research_package.py --package research/homogeneous-chain-extraction. The full proof, exact cut tables, scoped stop decision and open questions are in that package. Prepared for Cinna Davis with substantial OpenAI Codex assistance; Apache-2.0. No independent human or proof-kernel acceptance is claimed.

Before submission, live overlap refresh covered306 open PRs and4310 Discussion records with complete pagination and complete local-note inventory at the contribution head. No warnings were returned; these term/source checks are not a novelty guarantee. The exact native budget obligation is distinct from the source prefix's unconstructed caller and from #5137's abstract ring optimum.