Research · Papers · The quadratic hull and its defects · MF-022
A profile for quadratic pinning and defect structure
delta2^flip ≤ delta2^vert
Published 2026-08-29
For everyone
Plain summary
MF-022 provides a diagnostic profile to explain why specific invalid candidate states, called false points, survive degree-two polynomial checks. The diagnostic tracks four core numbers: how many evaluation columns vanish at the point, the independent rejection rank outside the quadratic hull, the remaining outside-hull defect, and the smallest set of graph-vanishing quadratic checks needed to cover the required cases. It also tracks the defect inside the hull across two views: a full vertical view and a semantic flip projection. If an invalid point remains in the hull, the support-cover size is set to infinity. The profile was validated on c0c2, ITER19, AND3, four affine Ghost covers, and two prior separating suites. The register records no prior-art position.
Result
For a selected false point, the diagnostic profile computes:
- h2: the number of zero evaluation columns
- rho2: the outside-hull rejection rank
- delta2_pin = |F_nonhull| - rho2
- kappa2_cover: the minimum graph-vanishing quadratic support cover, where kappa2_cover = infinity if a hull point remains
The profile complements the repaired in-hull filtration, where delta2^vert measures H/F_fibre and delta2^flip denotes its semantic projection. Universally,
delta2^flip <= delta2^vert.
The register records no corresponding comparison for rho2, and explicitly distinguishes kappa2_cover from Batch-B's added-product quantity kappa2_flat.
Setting and definitions
The profile operates on selected false points in Boolean graphs and related finite systems. The parameter h2 counts zero columns in the quadratic evaluation matrix at the target point. rho2 measures the rank of independent rejection constraints from false points outside the quadratic hull.
F_nonhull is the set of outside-hull false points used in delta2_pin. The quotient H/F_fibre comes from the repaired in-hull filtration. The superscripts in delta2^vert and delta2^flip denote the full vertical defect and its semantic projection, respectively. The support cover measures the minimal support among degree-two checks vanishing on the target graph; it evaluates to infinity when an uneliminated hull point prevents a finite cover.
The register separates kappa2_cover (the graph-vanishing support cover) from Batch-B's kappa2_flat (an added-product measure). The two metrics maintain distinct definitions across shared datasets.
Method
Validation data is recorded in this paper's downloadable evidence pack. The validation set comprises c0c2, ITER19, both prior separating suites, AND3, and four affine Ghost covers.
The entry specifies the profile metrics, the repaired in-hull comparison, the universal inequality delta2^flip <= delta2^vert, and the separation between kappa2_cover and kappa2_flat. The register specifies no solver name, certificate format, or exhaustive replay script.
Discussion
MF-022 is a METHOD entry defining a diagnostic profile for false points across the specified benchmark suites. It separates outside-hull rejection rank, residual outside-hull defect, graph-vanishing support requirements, and two projections of in-hull defect. The single universal bound recorded between projections is delta2^flip <= delta2^vert, with no equivalent relation given for rho2.
Maintaining the distinction between kappa2_cover (graph-vanishing support cover) and kappa2_flat (Batch-B added-product) prevents conflating distinct obstruction metrics. The method establishes a diagnostic framework on tested instances rather than a universal theorem over arbitrary false-point configurations or repair strategies.
The INDEX corrections section contains no corrections, retractions, restorations, or scope flags for MF-022 (Corrections: NONE). The register records no prior-art attribution.
For everyone — the takeaway
What this means
MF-022 gives engineers an inspection panel for analyzing broken Boolean constructions. Instead of returning a single failure flag, it shows whether an error sits inside or outside the degree-two envelope, measures the strength of the evidence against it, and counts how many low-degree checks are needed to rule it out. It also separates the full internal failure from the portion visible under semantic flip projections. The method has been tested on c0c2, ITER19, AND3, affine Ghost covers, and earlier benchmark suites.
Register references
MF-022; CONT quadratic_pinning_profile_live_results.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 1 receipt files bundled (5 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.