Research · Papers · Direct sums, wedges and the p14 frontier · MF-035
A conjectured rank-two bilinear custom gate for the p14 bridge
MF-035 is a conjecture.
Published 2026-08-29
For everyone
Plain summary
MF-035 asks whether a proof system can handle the cross-copy part of the p14 bridge in a single row. When two expressions draw on separate copies, their mixed component takes the form a tensor d + c tensor b, combining at most two elementary pieces. The conjecture proposes a built-in rank-two bilinear gate that fits both pieces into one row. The current verifier allows only one rank-one product per row, so this proposed gate is not verifier-legal and carries no zkgolf score claim. The register specifies a test: run an exact mixed-tensor count across the three one-slack wedge orbit families under the new rule. If that count is zero, the proposed gate fails on this barrier. The register records no prior art.
Result
MF-035 is a conjecture. The mixed component of a separated-factor product,
a tensor d + c tensor b,
has tensor rank at most two. A proof system with a native rank-two bilinear custom gate could represent this cross-copy bridge in one row, replacing the current verifier's limit of one rank-one product per row.
This primitive falls outside the current verifier and carries no zkgolf score claim. The entry does not assert gate existence, an explicit one-row representation, or verifier legality.
Setting and definitions
Factors derive from separated signals on disjoint copies; their mixed component couples both copies. Tensor rank at most two indicates representation by at most two elementary products: a tensor d + c tensor b.
A native rank-two bilinear custom gate admits this rank-two expression in a single row. The active verifier permits only one rank-one product per row. The p14 bridge denotes the cross-copy construction under evaluation. The three one-slack wedge orbits form the target test family; coordinate representations are unrecorded.
Method
The derivation supplying the rank-two bound and the one-row proposal, along with the records for the one-slack setting, are in this paper's downloadable evidence pack.
The designated falsifier is an exact mixed-tensor census across the three one-slack wedge orbits under native rank-two operations. An empty census falsifies the primitive against this obstruction. The census is proposed rather than executed. No positive construction, solver certificate, exhaustive replay, or verifier execution is recorded.
Discussion
Scope is restricted to encoding a rank-two bilinear mixed component in one custom gate row. The primitive is not verifier-legal in zkgolf and asserts no score claim.
The three one-slack wedge orbits provide the falsification benchmark. An empty census refutes the primitive for this obstruction; a non-empty census demonstrates representability under the proposed gate model without resolving verifier admission or benchmark scoring. The conjecture does not address alternative p14 routes or general custom-gate capabilities beyond this component.
No corrections are recorded. The register records no prior art.
For everyone — the takeaway
What this means
Today's checker handles one elementary product per row. MF-035 explores adding a single instruction that can handle two at once. The cross-copy part of the bridge needs at most two elementary pieces, so a rank-two gate could fit it into one row. This is a proposal, not a working proof. The gate is not part of the current zkgolf checker, and there is no score claim. Testing it requires checking the three one-slack orbit patterns with the new rule allowed. If none work, this gate cannot clear the barrier. If any work, the gate is viable, though getting it into the official checker remains a separate step. The register records no prior art.
Register references
MF-035 Receipts: Pro Request 3 theory return §9; p14_one_slack_mixed_tensor_manifest.json. 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 1 of 1 receipt files bundled (2 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.