[ToikY]

For binary rank-one idempotents U_i=u v_i with one nonzero common image and independent covectors v_i u=1, prove the exact uniform complete-header probability that every proper leading minor of each corank-one complement I-U_i passes. This supplies the k>=3 shared-direction continuation of Nick Petalas's PR6694 Q3. The k=1 marginal and transpose-dual k=2 law remain predecessor credit.

With A_j=2^j product_(r=0)^(k-2)(2^j-2^r) (zero for j<k-1), the probability is sum_j binom(m-1,j) A_j / ((2^m-1) A_(m-1)), for all m>=2,1<=k<=m. The proof establishes transitivity, constant complete-header fibers, the all-prefix binary criterion and the rank/support count. At (7,6), six distinct noncommuting common-image kernels have joint probability 22/2667 > (729/8128)^6. Each individual marginal remains 729/8128; I7 passes all 36 proper minors.

Validation: two blind-first engines agree on 135 exact finite-domain rows and 20,334 complete literal GL2/3/4 header records. The independent engine uses rank/support DP and a separate 374-span computation at (7,6), then compares every submitted small-header record as data in normal/optimized modes. Primary four strict modes and separate canonical sidecar regeneration pass. Both scientific codes, first receipts, frozen inputs and retained sidecar remain unchanged. All 21 external rejection controls, four registered commands, committed byte/source-history/mode checks, official export with its declared sidecar, and four read-only, Git-free, empty-PATH cold commands pass. Equal-histogram header-mask swaps and same-mask duplicate headers reject. Only the validation prose/run records changed after the science freeze. Final tested tree: 8015ddfb608fff82a6f4051befc0a7ba0a523e26. Hosted CI succeeded on original run 41738361999: 711 helper tests (24 skipped), all four registered package commands including both independent actual 20,334-header comparisons, and both staged/unstaged drift checks passed. Synthetic test commit f1b9966bdeec904e4223584105b9fe47600a0f09 has the exact verified tree and expected main/PR parents; no actual merge occurred. Mode, source-history and export checks above were local.

This is an all six common-line family, not the mixed one-coupled-pair-plus-four-orthogonal reading of PR6515 Q2, and not PR4240's rank-one first-pivot predicate. No GL7 census, prior spectrum replay, odd-field/restricted-header law, physical header producer, stock/restoration/precision/caller invoice, supplier admission or new κ/frontier claim. Canonical main and selected record remain unchanged; no merge requested; maintainer edits disabled.

Credit: Nick Petalas6694/4807/4275/4227/4261, unpaidmoon PR6515 (original Cursor/Grok disclosure retained), Ryanson Jonathan4607 and retained4599/5146/context/source/compiler lineage; classical rank-one determinants, orbit-stabilizer and finite-field rank counting. Exact 8 inert source objects and all credit are retained. Prepared for ToikY with substantial OpenAI Codex assistance and separate AI primary/oracle/static review; Apache-2.0. Human/formal-kernel acceptance unclaimed.