Research · Papers · Direct sums, wedges and the p14 frontier · MF-032
Copy-locality of target-bearing products at the rank-tight boundary over GF(2)
For affine combinations L and R of separated signals, if L*R is separated and target-bearing, then L*R is copy-local
Published 2026-08-29
For everyone
Plain summary
MF-032 analyzes two separate copies of a binary computation with inputs x and y. A separated signal splits into a contribution from x XORed with a contribution from y. In a rank-tight circuit—one using only as many multiplications as the independent output directions it must produce—any multiplication that produces a new separated output direction must be copy-local. Its useful output comes entirely from x or entirely from y.
This behavior stops once a single extra mixed multiplication is allowed. With one mixed helper product, a subsequent multiplication can combine both copies, cancel the cross terms, and expose a fresh separated output. The proof was independently verified, and all 65,536 abstract core cases were checked exhaustively. MF-084 restricts extensions beyond GF(2) to signal spaces closed under squaring; characteristic two alone does not suffice. The register records no prior art.
Result
Let the signal space and target quotient consist of separated Boolean functions f(x)+g(y) across disjoint input copies. For affine combinations L and R of separated signals, if L*R is separated and target-bearing, then L*R is copy-local.
Modulo constants, the mixed component is a tensor d + c tensor b. Over GF(2), equality of nonzero rank-one tensors forces the nonconstant factor cores to coincide. The zero cases force both factors into a single copy or reduce one factor to a constant. In a rank-tight completion where every intermediate stays in the separated target quotient, induction forces every target-bearing product to split copy-locally.
This structure fails when given one mixed slack product. The minimal boundary witness on 2+1 active bits is
m=(x1+x2)y, g=(x1+y)(x2+y), with g+m=x1*x2+y.
This eight-row identity cylinder-lifts to the full p14 domain. MF-032 therefore does not span the one-slack frontier.
Setting and definitions
Separated signals on disjoint copies take the form f(x)+g(y). The current signal space is the active linear span, and the target quotient is the target space modulo evaluated signals. An affine combination includes a constant term, and a product multiplies two such combinations. A product is target-bearing if it spans a new direction in the target quotient.
A product is copy-local when its nonconstant contribution belongs entirely to one copy. The mixed component contains the cross-copy terms. A rank-one tensor denotes a pure tensor product; the mixed component a tensor d + c tensor b has tensor rank at most two. A completion is rank-tight when the count of new products equals the dimension of the target quotient, leaving zero slack.
The base setting is GF(2). Field-general extensions require the square-closure condition established in MF-084.
Method
The result builds on an independent proof, an exhaustive check of all 65,536 abstract core quadruples, and multiple replay receipts. The central case split evaluates the mixed tensor a tensor d + c tensor b. Because nonzero rank-one tensors over GF(2) have unique factorizations, nonconstant factor cores must match; the vanishing cases yield copy-local or constant factors. Induction across rank-tight steps yields copy-locality for every target-bearing product.
The primary proof records and verification certificates are provided in the downloadable evidence pack. Analysis of this rank-tight lemma maps fifteen eight-wedge pairs to dead component shells, with full certificate output available in the evidence pack.
An independent restatement of the pure-tensor lemma exhausts 16,384 factor pairs in its falsifier and recovers the rank-tight direct-sum corollary, with verification replayed across its named-prefix census (raw verification and replay logs are available in the downloadable evidence pack). Both serve as convergence checks rather than slack extensions.
The boundary construction m=(x1+x2)y, g=(x1+y)(x2+y), g+m=x1*x2+y is certified by an eight-row truth table, minimal on 2+1 active bits, and cylinder-lifts to p14.
Discussion
The induction in MF-032 requires every intermediate product to remain inside the separated target quotient. An available mixed slack product breaks this condition: the eight-row identity shows that a single mixed product allows a subsequent cross-copy product to cancel the mixed terms and expose a new separated direction.
MF-032 does not resolve the live one-slack frontier or establish general p14 impossibility. The fifteen eight-wedge shell eliminations and the independent verification and replay checks confirm convergence on the rank-tight boundary without addressing the slack regime.
MF-084 supplies the necessary condition for broader fields: the component signal spaces U and V must be square-closed (u in U implies u² in U, and v in V implies v² in V). Under square closure, the theorem holds over arbitrary fields. Characteristic two alone is insufficient; for GF(2ᵐ)-valued signals, Frobenius-square closure must be verified directly.
The register records no prior art, and no claim of novelty is asserted.
For everyone — the takeaway
What this means
When running two independent binary calculations side by side without any spare multiplication steps, you cannot mix the two inputs in a useful multiplication and still produce a new separated output. Every multiplication that matters must stick to one copy.
If you have even one extra multiplication to burn, this restriction disappears. You can use the extra multiplication to mix the inputs, then cancel the cross terms in a later step to uncover a fresh separated output. MF-032 pinpoints this exact line: direct-sum behavior holds with zero slack, but the one-slack case remains open. Outside GF(2), the same rule requires the signal spaces to be closed under squaring.
Register references
MF-032 Receipts: CONT verify_separated_product_theorem.py; separated_product_theorem_independent.json; separated_product_theorem_independent_report.md; CONT pro4b_copylocal_certificate.json; CONT return6_d_verification_receipt.json; CONT return6_e_replay_receipt.json. 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 6 of 6 receipt files bundled (11 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.