Research · Papers · What rank-one constraints can express · MF-058
Nonexistence of small low-witness AND4 cells
In the affine-factor, affine-decoder, arbitrary-nonempty-fibre model with arbitrary affine-C: 1.
Published 2026-08-29
For everyone
Plain summary
MF-058 tests whether small arithmetic circuits can compute AND4—the four-input operation that outputs 1 only when every input is 1. The setup allows helper bits (witnesses), multiplication gates (rows), and linear mixing (affine factors and decoders). An exact computer search rules out every design with three witnesses and two rows, every design with two witnesses and three rows across 13 combined classes, and every design with two witnesses and four rows across 22 combined classes. A follow-up result in MF-003 settles the remaining two-witness cases using five or more rows. Seven special three-witness, three-row classes with deficient rank remain UNKNOWN. An AND2 control case is confirmed SAT. The register records no prior art.
Result
In the affine-factor, affine-decoder, arbitrary-nonempty-fibre model with arbitrary affine-C:
- Every
3w2cell is dead across all affine-C ranks. - Every
2w3cell is dead across 13 combined orbits. - Every
2w4cell is dead across 22 combined orbits. - The MF-003 selector-closure restoration closes the two-witness
r≥5odd/proper-mixed remainder. - Seven singular affine-C
3w3orbits remain UNKNOWN.
Setting and definitions
Cells are indexed as w·r by witness count w and row count r (3w2, 2w3, 2w4, 3w3). Factors and decoders are affine combinations over rank-one products. The model admits arbitrary nonempty fibres per public input. Affine-C rank partitions the affine C map into regular and singular strata. Combined orbits define the isomorphism classes checked by the census. Fibre configurations are classified into odd and proper-mixed partitions. Selector-closure refers to the MF-003 algebraic restoration resolving the r≥5 tail.
Method
The exhaustive census executed:
- 8,495,472,248 rank-zero
3w2relation pairs, - 931,645,939
2w3word checks across 13 combined orbits, - 35,909,179,160
2w4word checks across 22 combined orbits.
All surveyed candidate cells are UNSAT (dead). An AND2 instance was checked as an explicitly SAT positive control. Verification artifacts and raw outputs are available in this paper's downloadable evidence pack. The MF-003 selector-closure restoration provides the subsequent closure for two-witness r≥5 cases, while the seven singular 3w3 orbits remain UNKNOWN.
Discussion
The census establishes nonexistence across the full 3w2 rectangle at all affine-C ranks, the 13 2w3 orbits, and the 22 2w4 orbits. In combination with MF-003, all two-witness AND4 cells in the odd/proper-mixed strata are closed through r≥5. These exclusions do not span the seven singular affine-C 3w3 orbits, which remain UNKNOWN. The AND2 SAT instance confirms pipeline correctness without altering the AND4 classification boundary. The register records no prior-art position.
For everyone — the takeaway
What this means
Compact AND4 designs cannot be built with two helper bits and three or four multiplication rows, nor with three helper bits and two rows. Later work showed that two helper bits also fail with five or more rows. The only small design space still open consists of seven rank-deficient cases with three helper bits and three rows. The register records no prior art.
Register references
MF-058
Receipts: CONT lens_l3b_verification_receipt.json; lens_l3b_verification_report.md.
Later closure: MF-003 selector-closure restoration.
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 (8 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.