Research · Papers · The quadratic hull and its defects · MF-024
Rank-one-target deficiency of stationary full covers on GF(2)⁵
The condition of using all ten spare codewords holds if and only if delta ∈ {4, 20}
Published 2026-08-29
For everyone
Plain summary
MF-024 tests fixed shifts applied to five-bit code labels. A shift adds the same five-bit offset to every codeword. Only shifts 4 and 20 preserve both required parity conditions while using all ten spare codewords. Pairing each shift with the two possible K polarities produces four stationary full-cover cases. All four fail the required rank-one target, scoring 2/5 or 0/5, so the register classifies every case as dead. The cubic phase selector for these covers is exact and degree-minimal. This proves the stationary full-cover family cannot work. Transient relations and selector-chosen relations lie outside this result. No corrections exist, and the register notes no prior art.
Result
Let delta range over five-bit translations preserving both required semantic parities affinely. The condition of using all ten spare codewords holds if and only if
delta ∈ {4, 20}.
Across the two K polarities for each delta, all four stationary full-cover cases are rank-one-target deficient, recording outcomes of 2/5 or 0/5. The register classifies all four cases as dead.
The cubic phase selector for these covers is exact and degree-minimal. Independent pinning, vertical, and flip scans corroborate the classification. The theorem covers stationary full covers; transient or selector-chosen relations remain outside its scope.
Setting and definitions
A five-bit translation is an affine shift delta on GF(2)⁵ applied to each five-bit codeword. Semantic parity conditions are the two affine parity constraints preserved under translation. Spare codewords are the ten reserved codewords available for the cover.
A stationary full cover is generated by a fixed translation delta using all ten spare codewords. The construction admits two K polarities. A rank-one target requires supplying the target relation at rank one; the recorded outcomes for the four cases are 2/5 and 0/5.
The cubic phase selector is the cubic polynomial choosing the phase for the cover. Exactness denotes realization of the selected phase relation in the construction; degree-minimality means no polynomial of lower degree realizes it. Transient relations and selector-chosen relations form distinct classes outside stationary full covers.
Method
Classification and deadness certificates are provided in this paper's downloadable evidence pack.
Independent pinning, vertical, and flip scans corroborate the certificate on the two surviving deltas, the four stationary full-cover cases, their rank-one-target deficiency, and the exclusion of transient or selector-chosen relations.
Discussion
MF-024 proves that among five-bit affine translations preserving the semantic parities, only delta ∈ {4, 20} use all ten spare codewords. Evaluating the two K polarities yields four stationary full covers, each rank-one-target deficient at 2/5 or 0/5. The associated cubic phase selector is exact and degree-minimal.
The boundary holds at stationary full covers: transient relations and selector-chosen relations remain unconstrained. Pinning, vertical, and flip scans confirm the results. No corrections, retractions, restorations, or scope flags exist (Corrections: NONE), and no prior art is recorded.
For everyone — the takeaway
What this means
MF-024 tests five-bit shifts to see if any can build a working fixed cover using all ten spare codewords without breaking parity rules. Only shifts 4 and 20 use every spare codeword. Testing both K polarity options for each shift yields four cases, and all four miss their rank-one targets. This rules out the entire stationary full-cover design. Search efforts can skip this family and focus elsewhere, though temporary shifts and selector-driven choices remain open. The result is verified by a certificate and three independent scans.
Register references
MF-024; Package-3 phase_translation_cover_certificate.json; CONT affine_translation_covers_delta2_vert_flip.json; prior art: the register does not record this.
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 1 of 2 receipt files bundled (2 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.