Research · Papers · Direct sums, wedges and the p14 frontier · MF-097
Decomposability of the extremal top form in acyclic XOR–AND circuits
If C is an acyclic XOR–AND circuit with r AND gates and f is an output of degree r+1, then top_{r+1}(f) is decomposable
Published 2026-08-29
For everyone
Plain summary
MF-097 sets a boundary rule for acyclic XOR–AND circuits. If a circuit has r AND gates and an output hits degree r+1, its highest-degree part must factor into a single decomposable form. We replayed the original MF-001 certificate; it returned unsatisfiable, matching its 50-monomial degree-six support, Plücker relation test, and generic determinant identity. This resolves a warning raised after MF-094 built a counterexample at lower degree (eight gates, degree six, below the maximum degree nine where terms can cancel). Because MF-001 has five gates and degree six—right on the r+1 boundary—the scope flag was withdrawn and MF-001 stands as filed. A direct full-domain synthesis run at five products is still running. We make no separate prior-art claim.
Result
Let C be an acyclic XOR–AND circuit with r AND gates, and let f be an output of degree exactly r+1. Then the degree-(r+1) homogeneous component top_{r+1}(f) is decomposable.
MF-001 targets r=5 and degree 6=r+1, so the extremal lemma applies directly. The MF-094 counterexample occupies a non-extremal regime: r=8, extremal degree 9, and degree-six top form.
Setting and definitions
deg f denotes polynomial degree, and top_d(f) denotes the homogeneous degree-d component. Decomposability is the single-wedge top-form condition tested by the relevant Plücker relations. The extremal case is deg f=r+1; deg f < r+1 is non-extremal. Circuit acyclicity ensures gate j has degree at most j+1 in the induction.
Method
We located and replayed the original certificate. The 2026-08-27 replay returned status=UNSAT. It reproduced output degrees [1,2,4,6], the 50-monomial degree-six support e₄(x₀,x₂,x₄,x₆,x₈)·e₂(x₁,x₃,x₅,x₇,x₉), a Plücker relation value of 1 (where decomposability requires 0), and the generic 6×10 determinant identity vanishing over GF(2).
The replay and induction establish complementary parts of the proof. The replay evaluates the concrete degree-six target, finding all 50 monomials and the forbidden Plücker value 1, while the generic determinant identity confirms the relation over GF(2) independently of this instance. Induction proves the decomposability condition holds for every extremal degree-(r+1) output.
The certificate inducts on gate count: replacing the first AND L·R with an indeterminate z yields F=F₀+z·F₁ with r−1 gates. Achieving degree r+1 after setting z:=L·R forces deg F₁=r−1 and deg F=r. By induction, top(F) is decomposable; contracting by z shows top(F₁) is decomposable; wedging with top(L) and top(R) yields top(f). Only the final gate can carry degree r+1, so no competing gate contribution exists at that degree to cancel it.
The non-extremal comparison audit is provided in this paper's downloadable evidence pack.
Discussion
MF-094 refutes decomposability only when deg f < r+1. Its r=8 example produces a degree-six form by cancelling two degree-seven gate contributions, an interaction allowed only because 6<r+1. At r=5 and degree six, no such cancellation can occur, preserving the extremal argument.
The earlier scope flag incorrectly asserted that MF-001 relied on the inference refuted by MF-094. The register recorded the flag as WITHDRAWN on the same day; MF-001 remains PROVED as filed with its certificate verified.
A direct full-domain synthesis of the tile at p=5 remains in flight. The p=6 control returned SAT in 1,139 seconds. An UNSAT result at p=5 would independently corroborate MF-001 without the Plücker argument. MF-097 makes no separate prior-art claim.
For everyone — the takeaway
What this means
MF-097 separates two circuit regimes. When a circuit hits the maximum degree its gate count allows, its top-degree part must follow a strict factoring rule. At lower degrees, internal cancellations break this rule. Because MF-001 operates at five gates and degree six, it sits right on the maximum boundary where cancellations cannot happen. The recovered certificate confirms MF-001 remains valid. Direct synthesis will provide an independent check once finished.
Attribution and prior art
Prior art: This result makes no separate prior-art claim. It validates the extremal lemma underlying MF-001 and distinguishes it from the non-extremal counterexample in MF-094.
Register references
- MF-097
- zkgolf-pro-corpus-three-pack-2026-08-14\staging\01-attack-frontier\zkgolf-sha256\scratch\schedule-p5-alt\plucker_certificate.py
- README.md
- plucker-certificate.json
- 04-andcount-optimization/audit_topform_counterexample.py
- 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 4 of 4 receipt files bundled (12 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.