Research · Papers · Direct sums, wedges and the p14 frontier · MF-041
Structural constraints on the slack gate in seven-wedge one-slack p14 orbits
For each of the three named-prefix seven-wedge, one-slack p14 orbits, the unique slack gate must appear within the first three remaining product positions and carry a nonzero cross
Published 2026-08-29
For everyone
Plain summary
This paper studies three specific ways to build a two-copy carry circuit in 14 multiplication steps. Each design fixes seven named product pieces in advance, leaving one flexible position whose formula is not yet chosen. The theorem proves that this flexible position cannot wait. It must appear within the first three product slots after the fixed prefix, and it must combine inputs from both copies. Because of this requirement, each of the three designs branches into four valid partial setups before the shared step, producing twelve candidate search paths in total. The result applies only to these three acyclic seven-piece prefixes with a single flexible step. It does not settle whether any of the twelve paths leads to a full 14-step circuit, nor does it address general 14-step circuits. The register records no prior art.
Result
For each of the three named-prefix seven-wedge, one-slack p14 orbits, the unique slack gate must appear within the first three remaining product positions and carry a nonzero cross-copy component.
Prior to the slack gate, rank-tight direct-product locality forces every useful gate to be copy-local. The exact component shells for masks 7, 15, 23, and 39 admit at most one target-tight local subspace extension per copy. A copy-local slack gate leaves the opposite copy facing an exact-dead rank-tight shell. Each orbit therefore admits exactly four pre-catalyst subspaces, yielding twelve canonical subspace frontiers in total.
The result is a structural theorem for the named family: it fixes the position and cross-copy requirement of the slack gate without determining whether any of the twelve frontiers completes to a full p14 circuit.
Setting and definitions
The setting is the maximal named seven-wedge/one-slack acyclic family for the exact period-two carry problem. A named wedge is one of the seven prescribed product-shaped components in the prefix. The slack gate is the single remaining product position whose form is unconstrained by the prefix. A p14 circuit contains fourteen product positions, leaving seven positions to order after the seven named wedges.
The first three remaining positions are indexed immediately after the named prefix. A gate is mixed when its cross-copy component is nonzero. A pre-catalyst subspace is the signal subspace available prior to this first mixed slack gate. The four such subspaces across each of the three named orbits form the twelve canonical frontiers.
Method
The classification derives from formal certificates establishing frontier structure as well as positional and mixedness constraints on the catalyst, with raw verification data provided in this paper's downloadable evidence pack. The argument evaluates the exact component shells for masks 7, 15, 23, and 39 under rank-tight locality prior to the slack gate.
Setting the slack gate to be copy-local drives the opposite component into an exact-dead rank-tight shell, eliminating all such candidates within the family. The surviving slack gates are constrained to the first three post-prefix positions and require a nonzero cross-copy component, leaving four valid pre-catalyst subspaces per orbit.
An independent replay confirms this structure, with verification receipts and reports included in the downloadable evidence pack. The method certifies structural constraints on the family rather than executing exhaustive searches across the resulting frontiers.
Discussion
The theorem restricts the search space across the maximal named seven-wedge/one-slack acyclic family: any valid p14 completion must place a mixed slack gate in one of the first three post-prefix positions. This leaves twelve canonical pre-catalyst frontiers open for downstream search.
These twelve frontiers remain undecided. The result does not prove the existence of a full p14 circuit, nor does it constrain general p14 constructions outside this family. It does not apply to positive-slack prefixes, alternate wedge counts, absorbed or alternative quadratic directions, or cyclic rows.
The status is PROVED as a structure theorem. There are no recorded corrections, retractions, or addenda. The register records no prior art.
For everyone — the takeaway
What this means
This result maps out the only remaining paths for building a 14-step two-copy carry circuit under three specific starting patterns. After the first seven fixed pieces, you must introduce a shared operation between both copies within the next three steps. Staying local to one copy hits an immediate dead end. This splits each starting pattern into four surviving branches, giving twelve exact search paths to explore. It does not finish building the circuit, and it says nothing about circuits built under different starting assumptions.
Register references
- Entry:
MF-041. - Receipts: Return E
p14_named_wedge_frontier_certificate.json; Return Ep14_one_catalyst_frontiers.json; CONTreturn6_e_replay_receipt.json; CONTreturn6_e_verification_report.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 4 of 4 receipt files bundled (21 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-30
- 2026-08-29Published on this site.
- 2026-08-30On 2026-08-30, the p14 interpretation was subsumed by the broader graded and named-shell analyses at MF-123 and MF-124. Although it is not a standalone p14 verdict, its three-orbit slack-gate restriction remains valid.