Research · Papers · What rank-one constraints can express · MF-060
An impossibility theorem for the normalized two-witness selector for AND₄
Every normalized 2-witness selector separates 30 wrong witnesses over x ≠ 1111, and no selector separates both wrong witnesses over x = 1111
Published 2026-08-29
For everyone
Plain summary
This result identifies the exact obstruction to a normalized two-witness description of the four-input AND function. A witness is a candidate hidden explanation for a row, and a selector is the rule that chooses or rejects candidates. Away from the all-ones input 1111, every selector separates all 30 wrong witnesses. At 1111, no selector separates both wrong witnesses at once. The failure family is classified exactly: the 1,792 double failures are the 896 four-variable bent functions and the same functions after XOR with AND₄. Bent means a Walsh-transform condition, where the Walsh transform measures a Boolean rule’s pattern, placing a function at maximal distance from affine rules; affine rules use XORs and a constant. The independent census reports that all 896 bent functions have degree 2, so their algebraic descriptions use products of at most two input bits, and that 512 functions have nonlinearity one, meaning one truth-table change from the nearest affine rule. The result isolates a single input corner as the place where the selector requirement becomes impossible.
Result
In the normalized two-witness selector model, every selector separates all 30 wrong witnesses over x ≠ 1111, and no selector separates both wrong witnesses over x=1111. The 1,792 double failures are exactly the 896 four-variable bent functions together with those functions XOR AND₄. An independent Walsh census records 896 bent functions, all of degree 2, and 512 nonlinearity-one functions.
Thus the required two-witness selector condition is satisfiable throughout the inputs away from 1111 and fails at the all-ones input for the stated double-separation requirement. This is the proved obstruction recorded for AND₄ in the normalized model.
Setting and definitions
The input is x ∈ F₂⁴, with 1111 denoting the all-ones input. AND₄ is the named four-variable AND function. The normalized model has two witnesses and a selector tested against the wrong witnesses at each input. The result uses separate in the register’s selector sense: the selector must distinguish the candidates required by the model. A four-variable bent function is a Boolean function meeting the standard Walsh-bent condition. Nonlinearity one means distance one from the nearest affine function in the truth-table metric. Degree is the Boolean algebraic degree reported by the census.
Method
The evidence combines an independent Walsh census with the AND₄ selector-geometry certificate. The census enumerates the four-variable Boolean functions and identifies the 896 bent functions, their degree, and the 512 nonlinearity-one functions. The selector-geometry certificate records the normalized two-witness tests: all 30 wrong witnesses are separated over x ≠ 1111, while the two wrong witnesses at x=1111 cannot both be separated. The same geometry accounts for the 1,792 double failures as the bent family and its XOR with AND₄. The register names a CONT independent audit and a package certificate for these checks.
Discussion
The theorem has a deliberately narrow scope. It concerns the normalized two-witness selector model and the four-variable function AND₄. Within that setting, the behavior splits cleanly by input region. Every listed wrong witness can be separated away from the all-ones corner, while the corner contains an exact double-separation obstruction. The failure family is classified completely by the 896 bent functions and their XORs with AND₄; the independent Walsh census supplies the matching structural count.
This result establishes impossibility for the stated selector requirement. Its scope does not extend automatically to unnormalized models, larger witness budgets, or other Boolean functions. No correction or withdrawn novelty is recorded for MF-060. The register does not record a prior-art position.
For everyone — the takeaway
What this means
The two-witness choice rule works everywhere except the one input where all four bits are one. At that corner, two wrong candidates cannot both be rejected by any selector in the stated model. The complete failure count shows a pattern rather than scattered bad cases: the failures are exactly a well-defined family of four-variable rules and the functions made by XORing each with AND₄. The independent Walsh census confirms the family’s size and that its bent members use products of at most two input bits. This places MF-060 as a sharp impossibility result for one compact representation, with the obstruction located precisely at the all-ones corner.
Register references
MF-060
Receipts: CONT lens_r2a_independent_audit.json; package certificates/and4_selector_geometry.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 2 of 2 receipt files bundled (3 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.