Research · Papers · What rank-one constraints can express · ML-020
An impossibility result for two-witness AND4 in the affine-C model
In the affine-C model, the two-witness AND4 route is unsatisfiable for every finite row count r.
Published 2026-08-29
For everyone
Plain summary
AND4 is a four-input logic gate that outputs true only when all four inputs are true. This entry tests whether AND4 can be built algebraically using two helper values (called witnesses) and any finite number of equation rows, when the right-hand side of each equation is restricted to affine sums of variables and a constant (the affine-C setting). That two-witness route has no solution for any finite row count. Several bounded cases with unrestricted right-hand sides are also closed. Exactly seven specific three-witness, three-row cases remain unknown. This result settles the two-witness route without ruling out other ways to represent AND4. The register records no prior art.
Result
In the affine-C model, the two-witness AND4 route is unsatisfiable for every finite row count r. Exactly seven singular affine-C 3w3 orbits (three witnesses, three rows) remain UNKNOWN.
Setting and definitions
Let w be the number of Boolean witnesses and r the number of rows. The affine-C setting restricts the C side of each relation to an affine expression over variables and a constant. An orbit groups cases identified by the recorded symmetries.
Method
An initial two-witness, two-row search ran 955,369,472 equation-pair iterations with cross-validation. Extended exact censuses closed every affine-C rank at 3w2, all 13 arbitrary-C 2w3 orbits, and all 22 arbitrary-C 2w4 orbits. Selector-closure then closed all remaining 2w, r >= 5 cases, establishing the arbitrary-r two-witness result. For the identity-C 3w3 rectangle, an exact UNSAT certificate (4,038 variables, 15,836 clauses) was verified by DRAT, with solver concordance between Glucose and CryptoMiniSat. Verification receipts and replay artifacts are available in this paper's downloadable evidence pack.
Discussion
The scope expanded in stages from bounded row searches to the general affine-C two-witness impossibility result. Seven singular affine-C 3w3 orbits remain unresolved and are demoted as a scalar AND4 score route only. Broader nonidentity-C and four-witness models remain open. The register records no prior-art position.
For everyone — the takeaway
What this means
Adding more rows cannot rescue the two-witness affine-C construction of AND4. Searches can stop treating that route as live and focus on the seven unresolved three-witness, three-row cases, or on broader models with four witnesses or unrestricted right-hand sides.
Register references
- Entry: ML-020.
- Receipt artifacts: CONT phantom_pair_a2m3_semilattice_results.json; package3_replay_receipt.json; fleet-c-return7/phantom_and4_3w3r_verified_unsat_receipt.json; lens_l3b_verification_receipt.json; lens_l3b_verification_report.md; lens_r2a_catalyst_verification_report.md; lens_r2a_catalyst_verification_receipt.json.
- Prior-art works: 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 4 of 4 receipt files bundled (15 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-13Scope update: Verified in the enlarged exact rectangle, closing every affine-C rank at 3w2, all 13 arbitrary-C 2w3 orbits, and all 22 arbitrary-C 2w4 orbits. Positive-even two-witness fibres are ruled out across all rows and arbitrary-positive fibres through row four, leaving odd/proper-mixed 2w,r>=5 and seven singular affine-C 3w3 orbits unknown.
- 2026-08-13An internal verification confirmed the two-witness case is proved unsatisfiable, fully closing the `2w,r>=5` remainder (MF-003 / MF-059). Seven singular affine-C `3w3` orbits remain unknown and are demoted to a scalar AND₄ score route.
- 2026-08-29Published on this site.