Research · Papers · Direct sums, wedges and the p14 frontier · MF-074
A counterexample to the laminar multiplication-tree normal form
Laminar multiplication-tree normal form does not hold on n=6: p4 circuit has q-rank 4, unrestricted [1,2,1,1,1] vs laminar [1,2,1,1,0]
Published 2026-08-29
For everyone
Plain summary
Some proposed circuit-building rules assume every step can reuse at most one earlier multiplication result alongside input signals. This paper gives a concrete six-input circuit where that rule fails. The circuit needs four multiplications to hit its four target directions, relying on cancellation across earlier results. An unrestricted solver finds the circuit, while a solver restricted to the one-prior-gate rule stalls at the final step. A positive control circuit passes under both rules. The witness disproves the proposed tree-like normal form for these circuits.
Result
On n=6, the explicit p4 mixed-cancellation circuit with factor-mask pairs (95,21), (6,119), (79,505), and (941,783) has q-rank 4. Unrestricted rank-tight flags succeed with step profile [1,2,1,1,1]. Laminar-only factors—affine XORs containing at most one prior gate—fail at the final step with profile [1,2,1,1,0]. A positive control of two disjoint ANDs succeeds under both policies. The laminar multiplication-tree normal form does not hold in this model.
Setting and definitions
Bit 0 is the affine constant, bits 1 through 6 index inputs, and bits 7 and above index prior gates. Each pair defines the two factor masks for one product. Rank-tight means four products match target quotient rank q-rank 4.
Method
An independent audit evaluated the factor masks under unrestricted and laminar-only factor policies, recording both progress profiles alongside the positive control. Receipt: CONT lens_r2c_independent_audit.json.
Discussion
The witness disproves the laminar normal form within this model without addressing the separate support-cage question recorded in ML-044. This is a counterexample, not a novelty claim; the register records no prior-art position.
For everyone — the takeaway
What this means
Restricting circuit factors to tree-like dependencies cuts off valid designs. Here, four multiplications suffice only because later factors recombine multiple earlier products through cancellation. Forcing each factor to reference at most one earlier product prevents synthesis of the final step. Solvers that assume the laminar normal form will miss valid minimal circuits.
Register references
- Entry: MF-074.
- Receipt artifacts: CONT lens_r2c_independent_audit.json.
- Prior-art works: 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 1 of 1 receipt files bundled (4 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.