What this programme is about
When a circuit or verification system checks arithmetic over GF(2), degree-two equations like A·B=C are the cheapest non-linear constraints it can enforce. The quadratic hull of a relation is the set of all input and output states that satisfy every degree-two equation vanishing on valid transitions. Spurious solutions that satisfy these quadratic equations are called defects. This programme asks whether low-degree equations can pin down complex relations—such as carry chains, adders, and hash step transitions—without letting false points slip through. For engineers designing zero-knowledge circuits or SAT preprocessors, a sound quadratic hull means verification costs stay small. When a quadratic hull leaks false points, the defects measure the exact cost of plugging the leak: whether a designer must add extra auxiliary wires, insert an OR-chain lift, or pay for additional AND gates.
What has been settled
Matroid analysis MF-087 links the quadratic hull to degree-two Boolean closure and the Crapo–Rota critical exponent. Classical Reed-Muller bounds prove relations with fewer than 2^(n-d) forbidden points have full-space quadratic hulls MF-089, ruling out static degree-two hulls as general SAT preprocessors for width-k clauses with k ≥ 3 ML-052. For AND graphs, the positive false point has separator degree r MF-088. An OR-chain lift closes width-k clauses with cover number k − 1 MF-090.
Deterministic state-only lifts cannot repair carry relations. The c0c2 lift spans all 256 state functions yet leaves 23,808 defects MF-031, whose fibre defects have rank two locally and rank three globally MF-033. Censuses eliminate all 8,184 affine data-split pins [MF-036, ML-024], all 8,040 acyclic p6 catalyst models [MF-052, ML-024], 491,040 stationary additive carry phases ML-034, and full-rank Quadratic-Atlas charts ML-016. Ghost-P5 admits a two-column false path 8 -> 0 -> 0 MF-025, and five-bit covers are rank-one deficient MF-024. Conversely, retaining garbage coordinate g0 or g1 closes the Z-counter hull with three minimal rows [MF-009, ML-028]. Forcing 32 gauge residuals requires exactly 16 quadratic equations via Chevalley-Warning MF-065.
In multiplicative complexity, a master packing filtration identity MF-144 extends gate-span techniques of Schnorr, Boyar, and Peralta. Linear spaces of quadratics satisfy packing floors MF-145—corrected to qMC for dimensions d ≥ 3—and deficit laws MF-146. Exhaustive checks settle MC(W) = qMC(W) for three-dimensional quadratic spaces on at most 5 variables MF-165, settling a problem recorded by Boyar and Find, while single AND gates destroy 3/8 of additive quadruples MF-151.
What is still open
Four structural questions remain open across this programme. The Boolean ramification polytope equality MF-132 is an unproved conjecture because its formulation leaves the zero-variable boundary V=0 undefined, though its costed lower bounds stand. Whether concrete algebraic pinning obstructions account for the one-missing-direction wall across every relation family is also an open conjecture ML-000. In relaxed SHA step relations, tested five-code encodings produce linear fresh-defect growth across columns one through six, leaving sublinear or bounded-defect amortisation conjectural ML-032. Finally, while the specific Ghost-P5 ITER19 allocation and K0 trellis adapter fail soundness across all 131,072 replayed assignments ML-022, alternative allocations, higher lifts, and distinct relational presentations remain unresolved.
How to read the evidence
Exhaustive checks dominate this programme, especially for impossibility results across finite candidate spaces. When a ledger entry reports hundreds of thousands of eliminated configurations, that search space is completely closed. Certified proofs and machine receipts back the complexity identities and lower bounds, giving them high reliability. Paper proofs supply the broader structural theorems. Treat negative verdicts on finite spaces as absolute, and treat global claims as bounded by their verified hypotheses.