[ToikY]

For a prescribed injection h:U→W and two arbitrary, nonnested source/target inclusions g(A)⊂C and g(B)⊂D, prove an exact seven-condition feasibility criterion over every field. The three prescribed-image compatibilities and three quotient capacity inequalities need one additional coupled-kernel inequality. A three-dimensional oblique counterexample passes the ordinary dimension checks but fails that last condition; a positive nonzero-delta fixture prevents replacing the kernel by the whole common quotient.

This is a complementary closed two-target all-field criterion. Ali Romman's earlier PR6533, head efa89ad, already counts arbitrary simultaneous finite-field capacities at all ranks by exponential affine/kernel-lattice methods. Its README asks for faster restricted nonnested criteria. We do not claim the first nonnested completion result, a general count or a new algorithm for arbitrary contacts. The prepublication refresh discovered this adjacent claim; the initial commit/export is preserved and the correction changes documentation/credit only.

PROOF.md gives the exact quotient sequence, the common-lift kernel test, residual capacity extension and gluing/basis completion. The all-field theorem rests on that written proof. Two separately written engines corroborate 280,228 normalized finite decisions (62,153 feasible) over F2 dimensions0–3 and F3/F5 dimension2, with all16 rank streams and six concatenated transcripts agreeing. One engine uses RREF/annihilators and the criterion; the blind-first oracle uses literal vector sets and complete GL actions, without the criterion or predecessor imports. Seven actual single-condition omission controls, five invalid domains and oblique/nonzero-delta fixtures pass. The q5 dimension3 named fixture is not a full-domain census.

Evidence: both normal/optimized emissions and strict receipts pass; eight external optimized corruption controls reject; registered replay, all14 committed bytes/modes and declared canonical-base history pass. Official export integrity and four read-only, Git-free, empty-PATH replays pass unchanged. Final public head 3b3275bce5f051464392d15fdd4d61dbf76eda58 has exactly tested/exported tree 15f0da9bbf9b609aa33b3c69469cd04409da5705. Source6489 documents are authenticated and inert. First receipts/schema-only oracle alignment and a corrected comparison-harness projection are preserved candidly in VALIDATION.md and the session evidence. Both hosted jobs passed at this exact head in run41574072331:711 helper tests and the registered replay. Completed logs were read once; run metadata matches the head and the synthetic test tree equals the public/tested/exported tree. No actual merge.

Reproduce with CPython3.12 standard library:
python3.12 -B scripts/research_package.py --package research/two-capacity-prescribed-completion
Local registered replay is under one second; timing is not a gate.

Scope: unrestricted GL feasibility for TWO targets. No general completion constructor/count formula, three-target closed criterion, signed/symplectic native realization, phase anchor, actual selected contacts, causal access, dirty restoration, basis/delivery, paid physical/precision/stock/tape/compiler/outer invoice or new kappa. All five selected framework assumptions remain. Separate AI checks are not independent human or formal-kernel acceptance. Selected supplier replay, Lean and Docker were not run.

Task formal-interface-lemma, scope636f26dec98e26b296e8, explicitly changed to PR6489's nonnested question. Claim/handoff. Main7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority8, selected6030 and all eight guidance pins unchanged at intake and submission fetch. Original478-body cache plus current Hub, bounded all-state queries and fresh100 latest PR bodies checked; arbitrary diffs, hidden/unpublished work and complete historical coverage remain unknown. Maintainer edits and auto-merge disabled. No merge or frontier change.

Credit Nick Petalas/6489 nested theorem and explicit next question; Ali Romman/6533 general finite-field count; Brian/JoyboyBrian/5020 one-capacity contract; Cao Yifan/intqwq/4610 caller context; Nick Petalas/4919, codyw912/4735, Cinna Davis, Dugongue, zxv, Jacob Alan Sussman and retained native/compiler lineage. Classical quotients, linear lifting and basis extension retain credit; no general literature-priority claim. New result by ToikY with substantial OpenAI Codex assistance and separate AI implementation/static review, Apache-2.0.

Open next questions: simpler two-target finite-field count avoiding the general kernel lattice; a closed all-field three-target criterion; or one actual shared native contact with its complete invoice. No continuing scope is reserved.