[ToikY]

This closes Open next question 2 of Nick Petalas's PR4807: its common-covector full-guard pair is strictly positively correlated over every finite field, in every dimension m>=2.

The increment is a modest sign corollary. PR4807 already supplies the exact all-field joint probability, the binary sign and the suggested identity qD-A^2=q-1; PR4227 supplies the marginal. The new written proof cancels the credited ratio and derives J/p^2>L>=1, preserving strictness at (q,m)=(2,2). Four explicit polynomial identities establish the remaining all-dimension inequalities.

The exact model remains independent columns a,c, a common covector b with ba=bc=1, every proper leading minor of the corank-one complements I-ab,I-cb, and complete equally weighted GL_m(F_q) headers. No new count, arbitrary-pair/larger-family/restricted-header law or physical producer/caller is claimed.

Evidence: two blind-first engines agree on all 205 exact parameter rows and four coefficient arrays. The primary uses rational closed formulas and interpolation; the oracle uses integer recurrences, Euclidean normalization and polynomial convolution. Four primary strict modes and independent normal/optimized comparisons of the actual submitted profile pass. Non-prime-power rows are algebra fixtures, not finite-field counts. No predecessor checker or GL census was run. All code, immutable inputs and first receipts remain unchanged.

All 26 later external controls reject for their intended reasons. The original root control harness failed to save its first diagnostic because a slash in the log name became a missing subdirectory; that first subprocess outcome is unknown. Original evidence is retained, only the harness log filename changed, and the corrected full suite is a later run. All four registered commands, committed byte/source-history/mode checks, the final official export/input check and four read-only Git-free empty-PATH cold commands pass. Published/tested tree 5efa6e9e864726c795235bf83dddcd7b3a08d901. Science frozen at23:42:46.895822UTC; only validation prose/run records changed afterward. Hosted CI succeeded on original run 41741796250: 711 helper tests (24 skipped), all four registered package commands including both independent comparisons of the actual205-row/four-identity profile, and both staged/unstaged drift checks passed. Synthetic test commit fa4571380a1d4aa4ac1138806ac595d713ec3811 has the exact verified tree and expected main/PR parents; no actual merge occurred. Mode, source-history and export checks above were local.

Credit Nick Petalas4807/4227/4261/4778/4769/4274, Cinna Davis, source527/commdata2338/Jose Manuel and retained normalizer/recursive-closure lineage. Exact source notices remain intact. Prepared for ToikY with substantial OpenAI Codex assistance and separate AI primary/oracle/static review; Apache-2.0. No human/formal-kernel acceptance, supplier admission, new κ, frontier change or merge requested. Maintainer edits disabled.

Canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, all eight guidance bindings, selected6030, exact4807 head/zero feedback, latest open page and liveHub were refreshed unchanged before publication. The changed independent-moment card's paid/target invoice gate remains unmet. Auto-merge is disabled.