Research · Papers · Direct sums, wedges and the p14 frontier · MF-028
A twelve-wedge prefix obstruction for the period-two carry law in p14 circuits
Fixing gates 0–11 of a p14 circuit to 12 input-only quadratic wedge generators cannot realize the 26-input Cartesian period-two carry law
Published 2026-08-29
For everyone
Plain summary
MF-028 rules out a specific 14-gate Boolean circuit layout for an exact 26-input carry rule over its full input domain. The layout assigns gates 0 through 11 to twelve input-only quadratic wedges, where each wedge is a product of two inputs. That leaves only gates 12 and 13 to complete the circuit. Counting the maximum degree at each step shows that gates 0 through 11 reach degree at most two, gate 12 reaches at most four, and only gate 13 can reach degree six. Because the two high carry outputs require two independent degree-six components, a single gate cannot supply both. This prefix design cannot work. Whether any 14-gate circuit exists without this prefix remains UNKNOWN, and the register records no prior art.
Result
Fixing gates 0–11 of a p14 circuit to twelve verified input-only quadratic wedge generators cannot realize the 26-input Cartesian period-two carry law. The prefix signals have degree at most two, gate 12 has degree at most four, and only gate 13 can reach degree six. The resulting degree-six rank is at most one, which fails to span the two independent, disjoint degree-six symbols required by the copy-local high carry outputs. General p14 existence remains UNKNOWN.
Setting and definitions
The target function is the 26-input Cartesian period-two carry law evaluated over the full product domain of its 26 inputs. A p14 circuit consists of gates 0 through 13. Gates 0–11 are input-only quadratic wedge generators, each multiplying two primary input expressions without prior gate outputs. Monomial degree counts input factors. The degree-six component consists of all degree-six monomials, and its rank counts independent linear directions in that subspace. The target carry law contains two copy-local high carry outputs with disjoint, linearly independent degree-six symbols.
Method
The obstruction was verified via a period-two prefix rank screen (with full artifacts provided in this paper's downloadable evidence pack). The screen evaluates degree growth across gates 0–11, gate 12, and gate 13, bounding the available degree-six rank against the target output requirements. An independent recovery confirms the two-symbol degree-six defect on the fixed-twelve prefix and analyzes eleven-wedge tail profiles where at most one gate reaches degree six. This recovery serves as a scoped supporting receipt in the evidence pack rather than an exhaustive proof for all eleven-wedge topologies.
Discussion
MF-028 is strictly a topology-scoped obstruction. It eliminates the family that locks gates 0–11 to the twelve input-only wedges and leaves gates 12–13 to finish the target. Any successful p14 realization must allocate at least three chained gates or absorb wedge products into chained stages. The supporting analysis extends the rank screen to selected eleven-wedge tail profiles, but unexamined eleven-wedge configurations and general p14 UNSAT remain open. General p14 existence is UNKNOWN. Corrections: none. The register records no prior-art attribution.
For everyone — the takeaway
What this means
This particular 14-gate blueprint cannot compute the 26-input carry law. Dedicating the first twelve gates to simple two-input products leaves the last two gates unable to build the two separate six-input combinations the carry outputs need. Future designs must start chaining products earlier or combine inputs differently. The result closes this one construction while leaving other 14-gate layouts open.
Register references
- MF-028
- CONT
period_two_p14_prefix_and_rank_screen.json - Full-table source digest:
dedc04e3…8727 - Additional receipt:
ZKGOLF-PRO-STATUS-RETURN-2-2026-08-12.md - 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 2 of 2 receipt files bundled (5 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.