Research · Papers · The SHA-256 record and exact synthesis · ML-071
Refutation of the Maj Edge Phase -1,024 Candidate
Maj edge phase -1,024 candidate phase defect grows linearly at 32k bits for k=1..4, refuting the candidate with zero solver time.
Published 2026-09-04
For everyone
Plain summary
Designing efficient digital circuits and cryptographic primitives requires evaluating functions with as few nonlinear multiplications as possible. The Maj edge phase -1,024 candidate was proposed as an optimized design for majority-voting logic.
This candidate fails completely. Testing its phase defect—a measure of how many unconstrained bits build up across operational steps—showed that the defect grew by 32 bits at every tested step instead of stabilizing. Each pair of calculation rounds adds an independent 32-bit word that cannot be eliminated, causing the defect to grow without bound. Because of this structural flaw, the candidate is ruled out analytically without spending compute time on search solvers.
Result
The Maj edge phase -1,024 candidate is proved dead. Under the XOR-free GF(2) XAG cost model, the candidate phase defect scales linearly as 32k bits for k ∈ {1, 2, 3, 4}, with measured values of 32, 64, 96, and 128 bits. This unbounded linear growth prevents defect stabilization, refuting the candidate with zero solver time.
Setting and definitions
The setting is multiplicative complexity optimization in the GF(2) XAG (XOR-free) cost model. The target is the Maj edge phase -1,024 candidate. The verification metric is phase-defect growth evaluated across k relaxed disjoint round pairs for k = 1..4.
Method
Phase-defect growth was evaluated analytically across k = 1..4 relaxed disjoint round pairs. Each successive round pair introduces a fresh, unconstrained 32-bit intermediate word. The sequence produced defect sizes of 32 bits for k = 1, 64 bits for k = 2, 96 bits for k = 3, and 128 bits for k = 4. Strict linear scaling at 32k bits refutes circuit closure directly, bypassing SAT solvers and automated search routines.
Measurements are recorded in RECORD-MAJEDGE.md.
Discussion
This result closes the Maj edge phase -1,024 candidate for XOR-free multiplicative complexity synthesis. The register records no prior-art positions, addenda, or corrections. The refutation applies strictly to this candidate configuration and establishes that relaxed disjoint round pairs inject unconstrained degrees of freedom that prevent circuit closure.
For everyone — the takeaway
What this means
This eliminates a dead end in circuit minimization. Because the Maj edge phase -1,024 design leaks 32 unconstrained bits per round pair, researchers should allocate no solver time to it. Any viable design in this setting must bind intermediate words across round pairs to prevent accumulation.
Register references
- Entry: ML-071
- Receipt: RECORD-MAJEDGE.md
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 0 of 1 receipt files bundled (1 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-09-04
- 2026-09-04Published on this site.