Research · Papers · Direct sums, wedges and the p14 frontier · MF-045
The kernel of the Boolean product-residue map on separable signal cuts
ker(μ) = ker(μ_0) ⊕ ker(μ_1)
Published 2026-08-29
For everyone
Plain summary
When two separate Boolean components share a constant 1 signal, combining their product information produces no unexpected zero-output combinations. Every zero output comes entirely from one component or the other. Because of this, catalyst planes that span both components with rank two yield a unique concrete residue. When two choices collide at cross-component rank one, the collision always traces back to a decomposable plane inside a single component's local kernel. Full 8,192-row censuses on masks 7, 15, 23, and 39 confirm this behavior. The quotient structure is mathematically sound, but it remains too small to replace live exact search or resolve the broader p14 construction. The register records no prior art.
Result
Let S = S_0 + S_1 be a two-copy separable signal cut with shared constants, and set V_i = S_i / <1>. Let μ be the Boolean product-residue map on Λ^2(V_0 + V_1), with local product maps μ_0 and μ_1. Under the direct-sum decomposition into local blocks and the cross-tensor part, μ consists of the two local maps and an injective cross-tensor embedding. Therefore
ker(μ) = ker(μ_0) ⊕ ker(μ_1).
Every catalyst plane with cross-tensor rank two yields a unique concrete residue. Rank-one collisions are generated entirely by decomposable local-kernel planes.
Full 8,192-row component censuses for masks 7, 15, 23, and 39, evaluated before and after canonical local extension, find 21/28, 28/35, 28/35, and 28/35 nonzero decomposable kernel planes, respectively. Every collision-active factor direction has a two-dimensional kernel modulo that direction, bounding each rank-one residue fiber to at most size four.
Exact counting gives 12,297,829,216,765,457,700 distinct pre00 catalyst residues and 196,765,269,276,340,245,782 pre11 residues per orbit.
Setting and definitions
The separable signal cut is S = S_0 + S_1, sharing the constant signal 1. The quotient spaces are V_i = S_i / <1>, and Λ^2(V_0 + V_1) encodes pairs of factor directions. The maps μ_0 and μ_1 are the local Boolean product maps, while μ is the full product-residue map.
A catalyst plane is a two-dimensional subspace of factor directions in this encoding. Its cross-tensor rank is the rank of its projection onto V_0 ⊗ V_1. A local-kernel plane lies in ker(μ_0) or ker(μ_1), and a decomposable local-kernel plane is a decomposable element within that kernel. A residue fiber is the preimage of a single concrete residue under μ. A rank-one collision occurs when distinct planes with rank-one cross-tensor parts evaluate to the same residue.
Method
The result was verified via full 8,192-row component censuses for masks 7, 15, 23, and 39, run before and after canonical local extension. The verification script counted nonzero decomposable kernel planes, tracked full local cosets, and computed exact residue counts. The construction passes the Return-5 eight-row regression. Full verification scripts, census outputs.
Discussion
The direct-sum identity ker(μ) = ker(μ_0) ⊕ ker(μ_1) establishes that the cross-tensor embedding introduces no new kernel elements. Consequently, all rank-one collisions originate from decomposable local-kernel planes. The fiber bound of four and the exact orbit counts turn this structure into an explicit quotient for pruning search spaces.
The quotient is sound, but the register notes that it is insufficient to replace live exact search or settle the existence of p14. The construction preserves full local cosets and passes the Return-5 eight-row regression. No errata appear in the curation notes. The register records no external prior-art position for MF-045.
For everyone — the takeaway
What this means
Combining two separate Boolean components doesn't create new duplicate records out of thin air. If two setups produce the exact same product record, the duplicate always stems from an internal cancellation inside one of the individual components. This allows search tools to group candidate planes cleanly and caps the ambiguity for rank-one cross interactions at four planes per record. The classification holds across all four tested masks, before and after canonical extension, and through an independent audit. While it doesn't settle the full p14 problem or bypass large searches, it draws a strict boundary between local ambiguity and cross-component information.
Register references
Entry: MF-045.
Receipt artifacts: return6_p14_decomposable_kernel.py (SHA-256 0e5619a441985113fbde8b17e9e673a5eb53b7f56dfa74c2fc5d13585c3af855); return6_p14_decomposable_kernel.json (SHA-256 f75c08cd77ad36f007d8a974d42685f5b5288b1380a1777eb2e95c095b7a967f); return6_p14_decomposable_kernel_audit.json (SHA-256 b92db8db255056d52fceed4534be01d45508691dcd198767420d4444b448c38d).
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 3 receipt files bundled (27 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.