Research · Papers · The quadratic hull and its defects · MF-087
Matroid invariants of the quadratic hull of a Boolean relation
h₂=false-loop count, ρ₂=false-syndrome rank, δ_2,pin=nullity after deleting loops, κ_2,cover=Crapo–Rota critical exponent
Published 2026-08-29
For everyone
Plain summary
A Boolean relation is a set of allowed yes/no assignments. Its quadratic hull contains every assignment that satisfies all degree-at-most-two equations holding on the allowed set. This entry maps four project defect measures to standard objects in matroid theory: a false-loop count, a false-syndrome rank, the nullity remaining after deleting loops, and the Crapo–Rota critical exponent. Two fibre-sensitive measures inspect slices defined by fixed coordinates and flips. Those two measures lack established literature names and represent the only plausibly original packaging. The register marks this entry IMPORTED with a CAVEAT verdict. External writing must replace h₂ with ℓ_d(R;F) to prevent collision with the affine Hilbert function, and must avoid the phrase "2-nilpotent closure" because the Boolean function ring contains no nonzero elements whose product with themselves is zero.
Result
Let R ⊆ 𝔽₂ⁿ be a Boolean relation. The project's quadratic hull is the degree-2 finite-degree Zariski closure, or equivalently the closure in the column matroid of the degree-2 Boolean evaluation matrix.
Contracting by the columns of the honest relation identifies the defect suite with standard matroid invariants:
- The project quantity h₂ is the false-loop count.
- The quantity ρ₂ is the false-syndrome rank.
- The quantity δ_2,pin is the nullity after deleting loops.
- The quantity κ_2,cover is the Crapo–Rota critical exponent (critical number) of the binary false-syndrome matroid.
The fibre-sensitive quantities δ_2^vert and δ_2^flip have no prior names in the literature and constitute the plausibly original packaging. External text replaces h₂ with ℓ_d(R;F), where F is the false-point set.
Setting and definitions
Let R ⊆ 𝔽₂ⁿ be the honest relation and F = 𝔽₂ⁿ \ R the false points. The degree-2 Boolean evaluation matrix tabulates all Boolean monomials of degree at most 2 across the domain. Its column matroid defines the dependence structure.
The quadratic hull is the degree-2 Zariski closure. The defect suite is evaluated after matroid contraction by the columns corresponding to R. The terms loop, rank, nullity, and critical exponent follow standard matroid definitions.
Because the Boolean function ring is reduced and every element is idempotent, the ring has no nonzero nilpotents.
Method
The result matches the project's evaluation-matrix definitions and contracted defect suite against established matroid and algebraic closure concepts. Full details on the quadratic hull, its formal derivations, and the mapping of standard equivalents alongside the unmapped fibre-sensitive measures are available in this paper's downloadable evidence pack; no separate computational certificate is recorded.
Discussion
MF-087 is an IMPORTED entry serving as an instrument dictionary under a CAVEAT verdict. The quadratic hull, false-loop count, false-syndrome rank, loop-deleted nullity, and critical exponent are standard constructions. Only the packaging of the fibre-sensitive measures δ_2^vert and δ_2^flip is plausibly original.
The nomenclature corrections are mandatory. External text must use ℓ_d(R;F) rather than h₂, which clashes with the standard affine Hilbert function. The term "2-nilpotent closure" is mathematically erroneous: the Boolean function ring is reduced and idempotent, so it admits no nonzero nilpotents.
For everyone — the takeaway
What this means
When a system defines valid yes/no configurations, its quadratic hull consists of all configurations passing every pairwise check satisfied by the valid set. MF-087 shows that the project's defect bookkeeping mostly rediscovers standard matroid theory under new names. The only novel packaging lies in two measures that track coordinate slices and coordinate flips. The entry acts as a translation guide, adds an originality CAVEAT, replaces the notation h₂ with ℓ_d(R;F), and forbids using nilpotent terminology for reduced Boolean rings.
Attribution and prior art
Prior art: Most concepts and terms used here follow established literature; only delta2^vert and delta2^flip represent original formulations introduced in this work.
Register references
MF-087.
Receipt artifacts: 07-quadratic-hull-proof-complexity/REPORT.md §2–§3 (Theorem 3.1), §9, NOTES-LIT.md.
Prior-art position: the register records most concepts and names as established, including the Crapo–Rota critical exponent (critical number) of the binary false-syndrome matroid. Only δ_2^vert and δ_2^flip are identified as plausibly original packaging. The register does not record a separate prior-art work.
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 2 of 2 receipt files bundled (22 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.