[Clear1nsight]

Two literal historical C5 callers have total support budgets of 156 and 148. This package proves that eleven pieces spanning six P112 code dimensions need at least 164 support incidences, excluding 5^10,7 and 5^9,6^2 with margins 8 and 16 over every characteristic-zero field. Their quotient dimension is five under the explicitly inherited second generalized-weight bound.

The new argument pairs generalized Hamming weights with root-matroid fibers and singleton/pair incidence. A common-zero correction raises the naive 152 floor to 158; pair incidence strengthens it to 164 + 4z, where z coordinates vanish on the same six-dimensional code subspace. It permits arbitrary coefficients, cancellations and dependent pieces in the general lemma. Actual caller pieces retain their original independence hypotheses.

Evidence:

• Exact rational enumeration of all 4,900 rank-two and 85,680 rank-three root-spanned flats, with closure digests and attaining controls.
• Separate 23-pattern Gram certificate for the rank-three root cap of six lines.
• All 393 positive definite rank-four Gram patterns and 245,625 integral target vectors certify the pair-fiber cap of twelve; Bareiss and permutation-determinant implementations agree on the complete case digest.
• Both named rows pass the unchanged PR #3154 filter. Registered optimized replay, meaningful boundary/actual-code/receipt-mutation controls, committed-source/history checks and read-only Git-free export replay pass.
• A separate agent audited scalar extension, common-zero counting, quotient binding and the pair-incidence inequality. This is assisted-team corroboration, not independent human or proof-kernel acceptance.

Reproduce: python3.12 -I -Werror -B scripts/research_package.py --package research/p112-six-dimensional-incidence.

Credit Nick Petalas/PR #3269 and PR #3300 for the inherited 12/20/36 binary support capacities, DreamingOfClouds/PR #3154 for the historical reduction, and the retained root-simplex, David Arias and joaqcoo lineage. Exact canonical-main inputs and predecessor pins are included. Substantial OpenAI Codex assistance; Apache-2.0.

The dominated 5^11 and 5^10,6 profiles also follow. This is not a claim that these were the only public residues, a first-exclusion claim, or a completed 174-row reconciliation. Full C5, constructive decoder/supplier admission, the multiplication theorem and a new kappa remain open. The selected frontier is unchanged. This focused package creates no dependency on the concurrent common30 or kappa investigations.