Research · Papers · The quadratic hull and its defects · MF-009
Quadratic hull closure of the Z-counter transition relation
|H₂(R_∂) ∩ W_same| = 24; exactly three canonical quadratic rows A·B=C are necessary and sufficient
Published 2026-08-29
For everyone
Plain summary
MF-009 tests whether simple rules formed by multiplying two Boolean terms can capture the Z-counter's transition relation without error. The set of inputs and outputs that satisfy every degree-two rule valid on the true transitions is called the quadratic hull. When given only boundary variables, 192 candidate states share a valid input but produce an invalid output. Twenty-four of these bad states survive the quadratic hull, and each one flips the highest output bit.
Retaining an auxiliary garbage bit changes this behavior. Retaining g0 or g1 eliminates all 24 spurious states, closing the hull completely. Retaining g2 fails to close it, leaving the defect intact. For the canonical row format A·B=C, an exhaustive search across 524,800 candidate factor pairs against 21 graph-vanishing equations proves that no one-row or two-row system can exclude all false states, whereas explicit three-row systems do. The register notes no prior-art attribution.
Result
Let R_∂ be the boundary-only Z-counter transition relation, H_2(R_∂) its quadratic hull, and W_same the 192 false same-input points. Then
|H_2(R_∂) ∩ W_same| = 24,
and every point in this intersection flips the high output bit.
For presentations that retain an auxiliary garbage coordinate:
- The hull closes under g0.
- The hull closes under g1.
- The hull does not close under g2.
Within the canonical quadratic-row class A·B=C, exhaustive enumeration rules out all one-row and two-row covers and identifies explicit three-row covers. Exactly three canonical quadratic rows are necessary and sufficient for the covered cases.
Setting and definitions
The primary object is the Z-counter transition relation in boundary-only presentation R_∂. A wrong same-input point is an input-output pair matching a valid transition on its input coordinate while carrying an invalid output coordinate. The set W_same contains the 192 such points evaluated here.
The quadratic hull H_2(R) is the vanishing locus of all degree-two polynomials that vanish on the transition graph R. Retaining coordinate g_i means including that specific garbage bit in the transition presentation before computing H_2(R). A canonical quadratic row has the form A·B=C. A cover is a set of canonical rows whose joint solution set contains the intended graph R and excludes all points in W_same.
Method
The closure profile was computed by exhaustive search on R_∂ and on the three retained-garbage presentations. The boundary evaluation isolated the 24 surviving points in W_same and verified that every survivor is a high-output-bit flip. Exact quadratic-hull closure was then evaluated for the g0, g1, and g2 representations.
Row minimality was established by exhaustive enumeration of canonical rows A·B=C. The search tested 524,800 factor pairs per case against 21 graph-vanishing equations. No one-row or two-row cover exists; explicit three-row covers were extracted. Replay transcripts are available in this paper's downloadable evidence pack.
Discussion
The boundary-only defect is concentrated entirely in the high output bit across 24 specific points of W_same. Retaining a single garbage bit resolves this defect conditionally: g0 and g1 achieve quadratic closure, whereas g2 preserves the boundary defect. Hull closure is therefore sensitive to the specific garbage coordinate retained.
The three-row minimum applies strictly to the canonical form A·B=C evaluated against the 21 graph-vanishing equations. It does not establish bounds for non-canonical quadratic constraints, alternative transition representations, or other counter families.
The entry holds STATUS: PROVED with no corrections, restorations, or scope modifications in the record. No prior-art claim is registered.
For everyone — the takeaway
What this means
Keeping an extra internal bit can determine whether a circuit transition can be checked using simple degree-two rules. Without internal bits, the Z-counter rules let through 24 incorrect states, all with the same top-bit error. Keeping g0 or g1 fixes every error, but keeping g2 leaves the errors in place. When writing constraints in the standard product format A·B=C, two rows are never enough and three rows always suffice.
Register references
- MF-009
- CONT census/certificates
- 019ff22c-{16d8,8a92}
- The register does not record this.
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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 0 of 0 receipt files bundled (1 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.