Research · Papers · Direct sums, wedges and the p14 frontier · MF-037
A wedge-span obstruction for the exact Cartesian period-two carry law
dim(T ∩ W)=4, dim(T/(T ∩ W))=6; every p14 realization requires ≥6 gate functions outside W, ruling out k≥9 gate functions in W
Published 2026-08-29
For everyone
Plain summary
The exact period-two carry calculation requires ten independent nonlinear directions. Twelve certified product functions cover only four of those directions once simple input combinations (affine functions) are set aside. That leaves six target directions completely outside the span of those twelve functions.
In a circuit capped at fourteen multiplication gates, each gate supplies at most one new function. A circuit that places nine or more gates inside the certified span has at most five gates left over, which is not enough to cover the six missing directions. That immediately rules out any circuit layout using nine or more gates from this span. The result was checked by replaying an 8,192-row component rank calculation. It does not rule out circuits using eight or fewer span gates, nor does it decide designs that fold these components into deeper product stages. No prior-art position is stated.
Result
Let T be the ten-dimensional nonlinear output quotient of the exact Cartesian period-two carry law, and let W be the span of its twelve certified standalone wedges, both taken modulo affine functions. Exact rank computation yields
dim(T ∩ W)=4,
and therefore
dim(T/(T ∩ W))=6.
Every p14 realization requires at least six gate functions outside W. Any topology materialising k>=9 gate functions in W retains at most five gates outside W and is impossible. This extends the named eleven-wedge obstruction to the full k>=9 wedge-span class.
Setting and definitions
The target is the exact Cartesian period-two carry law from the p14 entry. Quotienting by affine functions isolates the strictly nonlinear components. T is the resulting ten-dimensional nonlinear output quotient. W is the subspace spanned by the twelve certified standalone wedges in this quotient.
The intersection T ∩ W isolates the target directions available from the standalone wedge span. The quotient space T/(T ∩ W) represents the target directions that W cannot provide. A topology materialises k gate functions in W when k of its internal multiplication outputs lie in W.
Method
The result was established by an exact wedge-span intersection computation, recovered independently from existing 8,192-row component subset ranks. The calculation was corroborated by independent replays. Complete verification scripts and raw execution outputs are provided in this paper's downloadable evidence pack.
Discussion
Because dim(T/(T ∩ W))=6, any p14 realization needs at least six gate functions outside W. Dedicating k>=9 gates to W leaves at most 14 - 9 = 5 gates for the remaining six target dimensions, ruling out the entire k>=9 class. This strictly generalizes the eleven-wedge obstruction.
The scope of this obstruction is exact. It does not constrain topologies with k<=8 gate functions in W, nor does it restrict architectures that absorb wedge components into higher-degree product chains. It establishes a structural lower bound and closes a candidate topology class, but it does not claim general p14 UNSAT or rule out all circuits outside the certified span.
No correction, retraction, or restoration is recorded for MF-037. No prior-art position is stated.
For everyone — the takeaway
What this means
This obstruction rules out a large family of circuit layouts using a simple counting test. If a layout spends nine or more of its fourteen gates on combinations of the twelve certified product patterns, it runs out of gates before it can cover the six remaining target directions. It fails automatically. The search can bypass all such layouts and focus entirely on designs that use eight or fewer gates from the span or combine the components differently. No prior-art position is stated.
Register references
- Entry: MF-037
- Receipts: Return B
period_two_p14_wedge_span_obstruction.py;_results.json; CONTpro_request3_p14_rank_independent.json; CONTpro4b_catalog_audit_replay.json; SHA-256d99c5c078775533c67d7dc1fa843a846c235560e682df8e5b938782885287b78 - 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 3 of 4 receipt files bundled (9 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.