Research · Papers · Direct sums, wedges and the p14 frontier · MF-034
Closure of the maximal standalone-wedge family for the period-two carry law
Within the certified maximal standalone-wedge family for the exact 26-input Cartesian period-two carry law, all twelve eleven-wedge variants are impossible.
Published 2026-08-29
For everyone
Plain summary
We have ruled out almost all candidate circuits in a specific family for the exact period-two carry problem. This family builds circuits around twelve special product gates called standalone wedges. Every candidate design that uses eleven of these twelve wedges is provably impossible. The same impossibility applies to fifteen eight-wedge starting configurations and 130 seven-wedge configurations that have no spare room in their linear spans.
Only three seven-wedge configurations remain unresolved. These cases have one degree of freedom and pair up into symmetric orbits under swapping the two problem copies. Exhaustive checks and independent replays confirm this boundary, and unrestricted solver runs on the three open cases timed out. This result closes a large, named branch of the search space. It does not settle whether a 14-product circuit exists in general: designs with six or fewer standalone wedges, or designs that blend wedge pieces into later products, remain open. No prior-art position is stated.
Result
Within the certified maximal standalone-wedge family for the exact 26-input Cartesian period-two carry law, all twelve eleven-wedge variants are impossible. For an omitted low-column wedge, the residual rank is seven; for an omitted high-column wedge, it is six. Both exceed the three remaining product gates available to the variant.
The same family-level closure eliminates all fifteen eight-wedge seeds and all 130 rank-tight seven-wedge seeds. The surviving residue consists of three copy-swap orbits with seven standalone wedges, identified by the bitmask orbits (7,15), (7,23), and (7,39). A sound unrestricted fixed-seed fallback timed out on all three cases, leaving their status UNKNOWN.
This result is a scoped closure of the maximal materialized-wedge family. It does not establish general p14 UNSAT. Circuits with six or fewer standalone wedges, or circuits that absorb wedge directions into chained products, remain open.
Setting and definitions
The target is the exact 26-input Cartesian period-two carry law. A standalone wedge is one of the certified wedge functions realized as an individual product gate. The maximal materialized-wedge family comprises the circuits in which these wedge functions appear explicitly as standalone gates.
For an omitted-wedge selection M, the residual-rank instrument is
r(M)=dim((A+W_M+T)/(A+W_M))
where A is the affine subspace, W_M is the span of the retained wedges, and T is the target space in the quotient. A rank-tight seed is a seven-wedge configuration whose quotient rank matches the available product budget with zero slack. A copy-swap orbit groups configurations related by exchanging the two problem copies. Bitmask labels follow the census recorded in the downloadable evidence pack.
Method
Evaluating the residual-rank instrument on the eleven-wedge configurations yields rank seven for low-column omissions and rank six for high-column omissions, both exceeding the three-product budget. Broader seed scans evaluated the full 64-by-64 component-subset map, applied the separated-product theorem, and checked 13 exact component shells, supplying rank and shell certificates that rule out the fifteen eight-wedge seeds and the 130 rank-tight seven-wedge seeds.
Validation proceeded through multiple independent replays:
- A dedicated replay provided a third complete validation of the eleven-wedge closure alongside previous checks and an independent local reconstruction.
- A whole-span calculation with target/wedge intersection dimension four ruled out every prefix with at least nine wedge-span gate functions.
- A reproduction of the 64-subset component quotient census recovered the three surviving skeletons.
- An exhaustive evaluation of all 4,096 named prefixes eliminated all 1,080 rank-tight cases and refined the three surviving orbits to twelve pre-catalyst subspace frontiers.
- An unrestricted fixed-seed fallback on these three orbits timed out, leaving them unresolved.
The primary receipts, replay logs, and component shell certificates are available in this paper's downloadable evidence pack.
Discussion
The established closure applies strictly to the certified maximal standalone-wedge family. All twelve eleven-wedge variants, fifteen eight-wedge seeds, and 130 rank-tight seven-wedge seeds are closed. The fifteen eight-wedge pairs lie entirely in the closed zero-slack stratum and do not expand the three surviving one-slack orbits (7,15), (7,23), and (7,39). Because a solver timeout is not an impossibility certificate, those three orbits remain UNKNOWN.
Independent verification passes reinforce this boundary: a whole-span obstruction establishes the closure for prefixes with at least nine wedge gates, while an exhaustive search over 4,096 prefixes establishes twelve pre-catalyst frontiers. These certificates leave unconstrained any circuit using six or fewer standalone wedges, any circuit that chains wedge directions across products, and the general p14 search outside the maximal materialized-wedge model. The entry makes no claim of general p14 UNSAT.
The register records no corrections, retractions, or restorations for MF-034. No prior-art position is stated.
For everyone — the takeaway
What this means
This result draws a clear boundary through a major candidate family for period-two carry circuits. All twelve near-complete designs with eleven wedges are ruled out, as are fifteen eight-wedge starts and 130 tight seven-wedge starts. Exactly three seven-wedge families remain open inside this framework because the solver runs timed out rather than finishing.
Future attempts on 14-product carry circuits can either target these three specific seven-wedge families or move outside the explicit wedge framework entirely. Circuits with six or fewer wedges, or circuits that combine wedge logic across chained products, remain completely open. No prior-art position is stated.
Register references
- Entry: MF-034
- Receipts:
pro_request3_p14_rank_independent.json;pro_request3_task1_replay_receipt.json;p14_one_slack_exact_attempt.json; Pro Request 3 return component shell certificates; Return Bperiod_two_p14_wedge_span_obstruction_results.json;pro4b_catalog_audit_replay.json;pro4b_rank_frontier_reconciliation.json;pro4b_copylocal_certificate.json;pro4b_return_c_crossmap.md;return6_d_verification_receipt.json;return6_d_analysis_report.md;return6_e_replay_receipt.json;return6_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 12 of 12 receipt files bundled (34 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.
- 2026-08-30Superseded on 2026-08-30: named topology deaths still stand, but the three-orbit picture was replaced first by an 817-cell scoped census and then by MF-124's 221-cell named residual. Unrestricted p14 remains LIVE/UNKNOWN.