[aliromman]

Count every invertible finite-field completion of one prescribed injection with two arbitrary simultaneous capacities. The closed formula has at most n+1 summands and retains the coupled prescribed-image condition. It replaces PR6533's exponential kernel-lattice count for this restricted case.

PR6535 already proves feasibility over every field; PR6533 already counts general finite families at all ranks. This result answers PR6535's released counting question. It is not a new feasibility theorem or a general all-rank algorithm.

The proof counts common-quotient injections, disjoint subspace images, compatible lifts through the fixed image, and the remaining basis extension. The formula is valid for every finite field and dimension. Its basis extractor uses prime fields; a literal F4 oracle supplies extension-field corroboration. Classical Gaussian coefficients and subspace graph counts retain credit.

Evidence at 96c63f994af628f574c71dca0d7ed219cbfc70c2: exact counts agree for 287,431 complete normalized contracts, including 64,525 positive counts, over F2 dimensions 0–3 and F3/F4/F5 dimension 2. A separate affine kernel-lattice algorithm agrees on 78 arbitrary-image contracts. Sixteen dimension-120 formulas and two actual dimension-48 vector contracts agree with independent product counts. Six named obstruction/nonzero-delta fixtures, four omitted-factor controls, seven invalid inputs and three source corruptions pass.

Warning-strict CPython 3.12.13 normal/optimized registered checks, committed bytes/modes, official export and both sealed read-only, Git-free, empty-PATH replays pass. Hosted planning and replay passed at the published head: run 41576080101. The shallow checkout does not certify full-history ancestry. No source scientific suite, selected cold replay, Lean or Docker run is claimed.

Start with PROOF.md, RESULT.json, and the canonical claim. Main 7c49fe3, revision 8 and selected6030 were refreshed. Full pagination covered 488 open PRs and 6,479 Discussion records; submission refreshed live handoffs. Sparse local notes, unpublished work and non-atomic changes remain limits.

Scope is unrestricted GL counting. Singular-rank counts, uniform sampling, native/symplectic realization, signed phase, an actual selected contact, causal access, physical restoration and the complete precision/stock/tape/compiler/outer bill remain open. No kappa or selected-frontier change follows. All five framework assumptions remain.

Credit ToikY PR6535 at 3b3275b; Nick Petalas PR6489/4919; Ali Romman PR6533; Brian/JoyboyBrian PR5020; Cao Yifan/intqwq PR4610; codyw912 PR4735; Cinna Davis, Dugongue, zxv, Jacob Alan Sussman and retained authors. Exact source bytes and notices are retained. New count, proof and checks: Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. Independent scientific review remains open. Maintainer edits disabled; no merge or closure requested.