Research · Papers · Direct sums, wedges and the p14 frontier · MF-084
The separated-product theorem over any field under square closure
If u ∈ U ⟹ u² ∈ U and v ∈ V ⟹ v² ∈ V, the separated-product theorem holds over any field k
Published 2026-08-29
For everyone
Plain summary
A Boolean product rule splits a combined calculation into separate pieces for each component. MF-084 shows that this rule holds over any field if both component signal spaces are square-closed, meaning squaring any signal keeps it in the same space. Without square closure, multiplying mixed factors can generate new separated terms that escape the subspace. In a polynomial setting, mixed factors yield x^2 - y^2. Over a field where 2 is nonzero, two-bit inputs yield the extra term 2x_1x_2. Characteristic two (where 1 + 1 = 0) is therefore not enough on its own. The result generalises MF-032 and records no external prior-art position. It provides a portability check before applying Boolean proofs to other algebraic settings.
Result
Let k be a field, and let U and V be the component signal spaces. Assume square closure in both components:
u ∈ U ⟹ u^2 ∈ U, v ∈ V ⟹ v^2 ∈ V.
Under this hypothesis, the separated-product theorem holds over any field k. Square closure is the exact missing condition required for the field-general statement.
Characteristic two alone is insufficient. In k[x] ⊗ k[y], take E = ⟨1, x, y⟩, L = x + y, and R = x - y. Then
LR = x^2 - y^2
is separated and new even though neither factor is local. In characteristic two, the same calculation gives
(x + y)^2 = x^2 + y^2.
A second counterexample on X = {0,1}^2 and Y = {0,1} occurs over characteristic ≠ 2. With a = x_1 + x_2, b = y, L = a + b, and R = a - b,
LR = x_1 + x_2 + 2x_1x_2 - y.
The mixed factors expose the new local direction 2x_1x_2. The original theorem therefore rests jointly on Boolean idempotence and the trivial multiplicative group F_2^* = {1}.
Setting and definitions
Let k be a field, with component signal spaces U and V. A space is square-closed if it contains the square of every element it contains. A term is local if it depends on only one component; a term is separated if it is a sum of a term local to U and a term local to V.
The first counterexample operates in the tensor product k[x] ⊗ k[y] with signal space E = ⟨1, x, y⟩. The second uses component domains X = {0,1}^2 and Y = {0,1}. Characteristic two denotes fields where 1 + 1 = 0. Boolean idempotence is the identity u^2 = u for Boolean signals. F_2^* = {1} is the multiplicative group of nonzero elements of F_2.
For GF(2^m)-valued signals, the relevant squaring operation is the Frobenius square. Closure under this map must be verified explicitly for any target signal subspace.
Method
The field-general theorem, the square-closure condition, and both counterexamples are established analytically. The full derivation, supporting notes, verification scripts, and certificate data are available in this paper's downloadable evidence pack.
The entry is recorded as a proved structure theorem generalising MF-032 beyond characteristic two.
Discussion
MF-084 establishes the field-general separated-product theorem under component square closure. Field characteristic two alone does not permit transfer to an arbitrary signal subspace; the counterexamples show mixed factors generating new separated terms when closure fails.
For GF(2^m)-valued signals, Frobenius-square closure is not automatic and must be checked per subspace. The result lifts a characteristic-two circuit lemma into a general statement about commutative signal algebras.
No corrections are recorded for this entry. MF-032 is the internal predecessor, with no external prior art recorded.
For everyone — the takeaway
What this means
Proving a rule for 0-and-1 Boolean signals does not mean it works in other number systems. MF-084 gives the exact requirement: square a signal and check whether it stays in the allowed set. If it falls outside, multiplying mixed pieces can create new separated terms. Counterexamples in polynomials and two-bit inputs show that arithmetic where 1 + 1 = 0 is not enough to guarantee safe transfer. MF-032 is the internal predecessor, and no external prior-art position is recorded.
Attribution and prior art
Prior art: This entry builds on predecessor MF-032; no position on external prior art is stated.
Register references
Entry: MF-084.
Receipt artifacts: 02-direct-sum-tensor/REPORT.md §2 (Theorem 2); ALGEBRA-NOTES.md; check_transfer_algebra.py; transfer_algebra_certificate.json.
Prior art: MF-032, identified by the register as the internal predecessor; the register does not record an external prior-art work.
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 (24 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.