[JoyboyBrian]

Independent tests for separate prelude lines can all succeed even when no shared native map exists. This adds the exact conditional replacement for PR4610's single-line map helper: an arbitrary partial injection h:U→U′ extends to G∈GL(V) with G(A)⊆C iff h(A∩U)⊆C and dim A−dim(A∩U)≤dim C−dim(C∩U′). The written all-field proof constructs a completion and counts every realization over finite fields. The discriminating binary three-dimensional fixture has 8/8/0 completions for the two separate/joint constraints.

The source-bound adapter authenticates the retained PR4610 caller and checks all 144 directional native-map contracts, including its twelve prelude operators 47–58. It compares the old one-line predicate on all 106,296 actual-source/capacity-catalogue cases. Augmented two-line examples are explicitly calibration fixtures; the original emitted word and source files remain unchanged.

Validation: complete independent literal GL counts match all 174,389 partial-injection/source/capacity cases over F₂³ and F₂²,F₃²,F₅², plus the zero-dimensional case; 20,131 returned matrices are checked literally. Six mathematical failure cases, six malformed-input cases and consistent dependent-vector controls pass. Registered normal/optimized replay, committed-byte/source-history checks, official Git-free exports and source/typed-receipt corruption controls are checked before submission.

This is a scoped written proof with finite corroboration. The theorem covers arbitrary fields; the reference implementation deliberately covers only prime q≤31 and n≤12. General simultaneous late capacities, symplectic/absolute-phase/native physical realization, a changed paid profile, routing/precision/stock/compiler/outer admission and new κ remain outside scope. Selected κ stays 0.00302015. No independent human or proof-kernel acceptance is claimed.

Credit Cao Yifan/intqwq #4610, Nick Petalas #4919, codyw912 #4735, Cinna Davis, Dugongue, zxv, Jacob Alan Sussman and the retained native/compiler lineage. Prepared for JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. Canonical claim: #1166 (comment) . Base bee95ce, priority 7. Initial overlap covered 310 open PRs/4,096 Discussion records/6,690 local notes with complete reported pagination/inventory; no exhaustive novelty claim. Final refresh and exact validation results are reported with publication.

Exact local validation at head a34db2858255a9a2a5efd1577f52b809a173dfdb: Python 3.12.2 registered normal/optimized replay passed; 12 committed/exported files and source ancestry passed. Official Git-free normal/optimized replay passed in 70.127/59.511 seconds. Actual exported source-byte mutation, integer-to-Boolean receipt mutation and occupied-output controls all rejected; every exported byte was restored and checked against the original archive. These are automatic checks, not formal proof acceptance. Three meaningful commits separate the constructive proof/API, independent literal/source checks, and reproducible package/receipts.

Submission refresh (2026-10-23 06:22–06:24 UTC): canonical main f259dabc7d70e0db8d6d36bfda888f008bbfecc1 was fetched; governing guidance, selected record and declared scientific inputs are unchanged. Scientific source base bee95ce88f9f19802f0613f92838895cd0e14a1f is intentionally retained; changed paths are disjoint. Refreshed overlap covered 304 open PRs and 4,114 discussions; local inventory covers all 6,690 notes at the scientific base. Coverage is advisory, not a novelty certificate.