[aliromman]

Arbitrary simultaneous source/capacity inclusions now have an exact finite-field completion reference. It counts matrices of every rank, constructs an isomorphism when one exists, and otherwise constructs an exact maximum-rank witness. Targets may overlap, repeat or be nonnested. This answers the general finite-field count obligation left open by PR6489.

The proof compiles the full joint contract into affine equations, counts solutions that kill each kernel subspace, and inverts the resulting triangular Gaussian-binomial system. These counting tools are classical. The new repository application includes exact maximum-rank construction and fixed-injection quotient compression. Fixing u independent directions reduces the kernel lattice from dimension n to n-u while retaining every coupling to the fixed image space. In the 12-dimensional fixture with ten fixed directions, this uses five subspaces instead of 488,176,700,923. This is an exact inventory comparison, not a timing claim.

The four-line F5 control is decisive: every triple has four invertible completions, while all four contacts force the zero map. Every corresponding subset sum/intersection dimension agrees. Separate feasibility and even these complete dimension data cannot certify a shared map. The familiar projective obstruction is credited, not claimed as new geometry.

Both warning-strict CPython 3.12.13 modes pass 6,653 literal-map contract comparisons, 72 arbitrary affine systems, 167 rank witnesses, 104 compressed/uncompressed comparisons, 164 compressed rank witnesses, nine independent dimension-four count formulas, two dimension-four witnesses, six larger compressed witnesses, eight separate large basis maps, 38 subspace-census comparisons, fourteen invalid inputs and an actual source corruption. The retained nested example reproduces 4/12/0. All returned maps are checked against original vector inclusions, not just compiled equations. Receipt.

python3.12 -B scripts/research_package.py --package research/nonnested-capacity-rank-count

All 11 committed files pass byte/mode and official-export checks. Both read-only Git-free, empty-PATH replays agree and preserve all 12 export files. Hosted planning and replay passed at the published head: run 41572916354. The source proof is byte-exact and checked before use. No pending Git object or network is required. Full-history ancestry is unclaimed in this shallow checkout. Independent human and proof-kernel acceptance are unclaimed.

The method is exponential. The uncompressed executable supports dimension at most four; the compressed API supports ambient dimension at most twelve with at most four unfixed directions. Both execute over F2/F3/F5; the written proof covers every finite field and dimension. This is a GL interface, not a symplectic/signed native map, phase anchor, chronological caller, physical restoration invoice or new kappa. Precision, stock, tape, compiler, outer admission and all five selected assumptions remain. The next task is an actual emitted caller with a small unfixed quotient and a full paid implementation. Lean, Docker and the selected cold replay were not run.

Only research/nonnested-capacity-rank-count/ is added. Advance claim. Canonical main 7c49fe3, priority 8/selected 6030 remained unchanged at submission. The selected binding example passed. Live handoffs, current PRs, main searches and the 479-PR/6,467-Discussion cache were checked. Nonnested G20 coframes and the new scaled-transport κ proposal have different contracts. Sparse history, nested diffs and unpublished work remain coverage limits.

Ali Romman with substantial OpenAI Codex assistance; no subagents. Credit Nick Petalas/6489/4919, Brian/JoyboyBrian/5020, Cao Yifan/intqwq/4610, codyw912/4735, Cinna Davis, Dugongue, zxv, Jacob Alan Sussman and the retained native/compiler lineage. Kernel-lattice inversion, Gaussian coefficients, basis extension and cross-ratio geometry are classical; Stanley's text is cited for historical context. No global literature-priority claim is made. No merge, closure or frontier change is requested.

Verified head: efa89adc2c71f1ef57228111639e1c8fc48393e3.