[SHIN09-git]

Result

For corank-one complement guards whose rank-one kernels share an image direction, four is the smallest field order that admits an all-pass probability below the product of the marginals. Every family of at least two members over the two- and three-element fields lies strictly above its product. Over every field of order at least four, three distinct directions on an affine line eventually lie below it. Three independent normalized covectors stay above the product in every allowed dimension.

A small exact example uses five directions over the five-element field. In GL_2(F_5), 336 of the 480 headers pass each guard and 272 pass each pair. Only 80 pass all five, giving 1/6 < (7/10)^5, although every proper subfamily of size two through four beats its product. For the same five directions, dimensions three and four lie above the product and every dimension from five onward lies below it.

Review entry points — field-size boundary, arbitrary dependency spaces, pencil signs, independent triples.

Relation to prior work

This follows question 3 of ToikY’s #6701, which counted independent binary rows and asked how the law changes beyond that domain. Nick Petalas’s #4227 and #4807 supply the marginal and pair laws. ToikY’s #6703 established pair positivity over all fields. The new calculations determine the genuinely multiway signs, including dependent row spaces and uniform fixed-determinant headers. The exact source heads and original notices are included. The overlap search covered the canonical inventory, open and closed PR metadata, and Discussion metadata.

Verification

python3.12 -B scripts/research_package.py --package research/common-image-guard-pencils

• All seven registered commands passed on commit 465a0060fbd236a5e202619de4a45a694d44970a. Normal and optimized results match. The tests reject 21 actual mutations in both modes.

• Literal determinants verify every named subset in the complete three-dimensional affine-plane families over fields of orders two, three and four. The four-element field alone contributes 65,536 subsets. Exact arithmetic also checks 232 pencil windows against 18,560 finite parameter values. Eleven raw-prefix runs independently recover the actual passing-tuple counts at the crossings, including both triples in dimension 61.

• A conditional sampling argument for independent triples has 17 finite exceptions to its sufficient tail bound. Each is counted both by the subspace law and directly by subtracting collinear triples in coordinate boxes. The five remaining ternary quartet configurations are also recounted by a raw-prefix program.

• The repository’s source-history check and export integrity check passed on a nonshallow canonical history. The exported archive contains 44 package files, uses only the Python standard library, and has SHA256 637bc002471a6fc192a07f0bfce2cbb6e7db909c2cc1fe0470b0d5e72118b3d9.

• The larger fixtures are reproducible with python3.12 -I -W error -B extra_censuses.py from the package directory. All 1,488,000 literal headers of GL_3(F_5) give 99,200 all-five passes. In GL_4(F_3), 1,516,320 representatives account for all 24,261,120 headers by free diagonal scaling. All 64 subsets of six directions agree with the exact law.

• The official export passed all seven commands with read-only inputs and Git absent from PATH.

• Ubuntu 24.04 WSL2 passed the seven registered commands and source-history/mode checks. The unmodified onboarding suite ran 711 tests with 23 skipped; the selection matrix contains this one package.

• Both hosted jobs passed on this head. The seven hosted science receipts match the local receipts exactly. GitHub Actions run

The claim is a scoped written proof with exact finite corroboration. A physical header producer, its costs and a changed multiplication bound are outside the result. SHIN09-git prepared this contribution with substantial OpenAI Codex assistance. The alternate counting methods share that authorship.