Research note: exact minimum frame completions for the h=10 parity DAG

[ToikY]

Change and mathematical scope

This research note distinguishes a degenerate source span from an impossible nondegenerate frame sandwich in the historical h=10 parity decoder. It adds an exact certificate for all 2,271 active nodes: 692 degenerate spans admit minimum-rank completions with one extra dimension, while two nodes have explicit common-span obstructions.

The note supplies the standard arbitrary-span completion argument: under A perpendicular B, a nondegenerate A subset U subset B-perp exists iff A intersect B is zero, with minimum dimension dim A + dim rad(A). A Gaussian producer and independent explicit-vector checker bind the fixed DAG and verify every witness. This does not give compatible nested frames, legal incident residuals, a supplier, or a new kappa.

Base: 007664306fff84fbb1c9a1b4053b69f439764034. Track: structural decoder research, OS seed 47. Only research/frame-completion-seed47/ is added; the selected frontier and upstream/ are unchanged.

Validation

• python3 -B research/frame-completion-seed47/check.py passes all 2,271 nodes, 14,400 signed output coefficients, 595 exhaustive small-space pairs, and 15 negative controls.
• Optimized isolated replay agrees; optimized regeneration is byte-identical.
• Producer independently checked against every tiny-space optimum; four existing parity tests and the selected-record/source-inventory check pass.
• Full repository-wide make verify and the selected cold replay were not run for this additions-only geometry note. Exact scoped receipts and that limitation are retained in VALIDATION.md and VALIDATION.json.

Credit the historical parity-side research, eumemic's #114, DreamingOfClouds' format-limit analysis, and ArnavSwiggy's #920. The last already studies two-source/two-target completion with incident residuals; its alternating-residual counterexample is retained as a negative control. This contribution adds arbitrary-span pointwise minima and the full h=10 node census, not a replacement for #920's stronger legality test.

Prepared for ToikY with substantial OpenAI Codex assistance in mathematics, implementation, verification and writing. Apache-2.0. Finite replay is not formal verification of the complete multiplication theorem.