Research · Papers · The quadratic hull and its defects · MF-036
A census of one-block affine data-split pins for c0c2 hull repair
The corrected census spans all 8,184 pointwise-distinct cases.
Published 2026-08-29
For everyone
Plain summary
This result closes a one-block repair route for c0c2. The pins p=y*ell and q=y*(1+ell) introduce only one effective coordinate because q=y+p. A corrected census evaluated 8,184 configurations: 1,023 nonzero ten-bit local-data parts across eight state-addition classes. Of these, 8,177 left all 24,576 false hull points intact; seven state-free choices using y=c1 reduced the count to 20,480. False hull points are candidate states that pass quadratic tests despite violating the relation. No configuration cleared the hull. Every case showed flip-defect measure delta2^flip=4, and the best profile yielded kappa2_cover=infinity, meaning no finite quadratic cover exists. Other boundary laws remain UNKNOWN. The register records no prior-art comparison or novelty claim.
Result
The corrected census spans all 8,184 pointwise-distinct cases. None eliminates the 24,576 false hull points: 8,177 retain all 24,576, while seven state-free y=c1 anchors retain 20,480. Every case yields delta2^flip=4, with the best profile at kappa2_cover=infinity. The anchor class fails the instrument gate, licensing no solver.
Setting and definitions
Let y be the data bit and ell an affine local-data expression. Because q=y+p, the pins add one coordinate. A state addition is a carry-state function. The census evaluates 1,023 nonzero ten-bit local-data parts across eight pointwise-distinct (y,state addition) classes.
Method
An exact counterexample refuted quotienting arbitrary state additions to ell. A second audit showed quotienting c2 for y=c0 holds on the honest graph but fails across 262,144 ambient candidates. A corrected replay produced the four named receipts.
Discussion
This exhaustively closes the one-block full-affine route for hull repair. Path-dependent, relational, cyclic, and unspecified next-boundary laws remain UNKNOWN. Corrections: NONE. No prior-art position is stated.
For everyone — the takeaway
What this means
Although the two pin formulas seem to add two separate choices, the identity q=y+p forces one to determine the other. Every allowed state and data combination leaves bad candidates that satisfy quadratic tests while violating the target relation. Seven state-free choices lower the error count from 24,576 to 20,480, but none removes every false point, leaving delta2^flip=4. This closes the specified route. Approaches relying on history, relational checks, cycles, or alternative next-boundary rules remain UNKNOWN and unresolved in the register.
Register references
Entry: MF-036
Receipts: CONT pro3_c0c2_data_split_state_addition_audit.json, pro3_c0c2_data_split_full_affine_census.json, pro3_c0c2_data_split_c2_gauge_extension.json, pro3_c0c2_verification_report.md
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 4 of 4 receipt files bundled (80 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.