Research · Papers · The SHA-256 record and exact synthesis · ML-042
A polar rank 8 separator in the JSC-12 K1x even dual coset
JSC-12 K1x even dual coset separator middle2 * Q has polar rank 8, Hamming weight 37, satisfying K10=K11=0 and FALSE_CUBIC=1
Published 2026-08-29
For everyone
Plain summary
ML-042 searches for a separator in the JSC-12 K1x case—a rule that outputs 0 on valid inputs and 1 on a known false case. The search found one candidate with polar rank 8 and weight 37, written middle2 * Q. It evaluates to 0 on the full K10 and K11 graph cases and 1 on the FALSE_CUBIC test. A targeted sweep tested 277,207,495 candidates across a 92-dimensional slice and found nothing at rank 6 or below. Because the full 2^92 slice wasn't exhausted, this doesn't prove lower ranks are impossible; the status stays UNKNOWN. The search didn't produce the simpler 3-affine form the route wanted. The register lists no prior-art position or novelty claim.
Result
In the JSC-12 K1x even dual coset, synthesis yielded a separator candidate of polar rank 8 and Hamming weight 37:
middle2 * Q
Evaluation satisfies K10=K11=0 on the graph cases and FALSE_CUBIC=1 on the false-cubic test. Scoring applied polar rank to Q within the 110-dimensional odd coset after isolating a coordinate factor. A targeted search for rank ≤ 6 over the 92-dimensional middle2-divisible slice evaluated 277,207,495 candidates across radius-4 Hamming balls around three origins and 64 Gray-code 22-dimensional subcubes, returning 0 hits. With the 2^92 slice unexhausted, rank ≤ 6 remains UNKNOWN rather than UNSAT. The 3-affine target was not obtained.
Setting and definitions
The synthesis target is the JSC-12 K1x even dual coset, paired with an associated 110-dimensional odd coset. Factoring out a coordinate isolates Q, yielding candidates of the form middle2 * Q. Candidates are scored by the polar rank of Q, with Hamming weight tracked as a secondary size metric. The middle2-divisible slice spans dimension 92, giving a state space of size 2^92.
Evaluation conditions are defined by K10=K11=0 across the complete K10 and K11 graphs, alongside FALSE_CUBIC=1 on the designated false-cubic test. The route targets a 3-affine representation, which remained unreached.
Method
Factoring a coordinate from the 110-dimensional odd coset yielded Q, which was filtered by polar rank while recording Hamming weight. This produced the rank-8, weight-37 candidate middle2 * Q meeting the K10=K11=0 and FALSE_CUBIC=1 conditions.
The lower-rank exploration probed the 92-dimensional middle2-divisible slice. It executed 277,207,495 evaluations partitioned across radius-4 Hamming balls around three origins and 64 Gray-code 22-dimensional subcubes. No candidate met rank ≤ 6. Search artifacts comprise the Identity-C lowering JSON, run logs, and findings report recorded in the register.
Discussion
ML-042 carries status EXHAUSTED, classification as a method or instrument result, and page verdict CAVEAT. The verified rank-8 witness exists alongside an incomplete search over the lower-rank slice: the zero hit count covers only the 277,207,495 sampled points and does not prove UNSAT for the full 2^92 space.
The run failed to obtain a 3-affine form. Curation notes confirm that ML-042 asserts neither a 3-affine construction nor a lower-bound impossibility claim for rank ≤ 6. The register records no corrections, novelty claims, or prior-art citations.
For everyone — the takeaway
What this means
ML-042 delivers a working rank-8 separator candidate that satisfies the required graph and false-cubic checks. While a targeted hunt across 277 million candidates turned up nothing at rank 6 or below, most of the 92-dimensional space wasn't checked, leaving lower ranks open. The entry establishes a rank-8 witness without certifying that lower ranks are impossible or providing the simpler 3-affine form sought by the route. It claims no prior art or novelty.
Register references
- ML-042
- Named register entry: MF-068
- CONT jsc_k1_idenc_lowering.json
- CONT jsc_k1_idenc_lowering.log
- CONT jsc_k1_idenc_rank6hunt.log
- ZKGOLF-FINDINGS-JSC-CUBIC-IDENC-2026-08-13.md
- Prior art: the register does not record a 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 (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
- 2026-08-13that ID is not used here.
- 2026-08-29Published on this site.