Research · Papers · The quadratic hull and its defects · ML-028
Quadratic hull closure for the canonical Z-difference counter interface
For the canonical Z-difference counter interface A*B=C, retaining garbage coordinate g0 or g1 closes the quadratic hull.
Published 2026-08-29
For everyone
Plain summary
ML-028 tests three helper bits—g0, g1, and g2—for a counter that tracks state differences. The test interface uses the multiplication rule A*B=C and quadratic equations, which are checks built from pairs of values. Keeping g0 or g1 eliminates every false solution. Keeping g2 alone leaves 24 false solutions intact. No one-row or two-row quadratic check works; explicit three-row setups succeed, making three rows the exact minimum. This is an interface simplification only. The register records no route from here to a score improvement and lists no prior art.
Result
For the canonical Z-difference counter interface A*B=C, retaining garbage coordinate g0 or g1 closes the quadratic hull. Retaining g2 alone leaves a 24-point obstruction. In the boundary-only version, this obstruction comprises 24 of 192 incorrect same-input points, always manifesting as a high-output-bit flip.
Under the canonical A*B=C presentation, no one-row or two-row quadratic cover exists. Explicit three-row covers attain closure, establishing three quadratic rows as the canonical minimum. The ledger records this strictly as an interface simplification with no score path.
Setting and definitions
The Z-difference counter designates the arithmetization route under ML-028, featuring canonical product relation A*B=C and auxiliary garbage coordinates g0, g1, and g2.
The quadratic hull is the solution set defined by all degree-at-most-two relations that vanish on the interface graph. A quadratic row is one such relation. A quadratic cover is a set of rows whose joint constraints eliminate all spurious assignments from the hull. The 24-point obstruction is the residual set of wrong same-input points that survive when a coordinate choice fails to close the hull.
Method
The canonical A*B=C interface was exhaustively enumerated across 524,800 factor pairs per case against 21 graph-vanishing equations. Boundary-only transitions were tested first, isolating the 24-point obstruction among 192 incorrect same-input points. Coordinate choices g0, g1, and g2 were then evaluated at the quadratic-hull level.
Row-count minimality was checked directly: every one-row and two-row candidate cover fails to eliminate the full obstruction, while explicit three-row covers achieve complete elimination. The underlying certificate records are cataloged under MF-009.
Discussion
The canonical A*B=C interface yields a sharp trichotomy: g0 or g1 retention closes the quadratic hull, whereas g2 alone preserves the 24-point obstruction. The three-row minimum applies specifically to this fixed presentation and its 21 graph-vanishing equations.
Closure provides only an interface simplification. It yields no score path, asserts no consequences for alternative counter interfaces, and carries no prior-art position in the register.
For everyone — the takeaway
What this means
When using degree-two checks on this counter interface, the choice of retained helper bit determines whether false states survive. Keeping g0 or g1 gives enough information to rule out every bad state; keeping g2 alone misses 24 of them. The math also proves that three equation rows are both necessary and sufficient to clear all false states. The simplification applies only to this exact interface, with no claimed score path or broader optimization theorem. The register records no prior art.
Register references
ML-028. Related register reference: MF-009. Receipt artifacts: CONT census/certificates; transcripts 019ff22c-{16d8,8a92}. Prior art: the register does not record this.
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Evidence pack
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Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.