Research · Papers · The SHA-256 record and exact synthesis · ML-040
Piecewise cubic separation of JSC-12 cases in the flip-support subspace
Piecewise pair (k0=0 principal, k0=1 companion) separates K10/K11; no square-free cubic meeting {middle1, middle2, final0} rejects witness
Published 2026-08-29
For everyone
Plain summary
A single cubic rule made from allowed flip coordinates cannot accept all four valid K cases while rejecting a specific failing JSC-12 witness. Splitting the rule in two—using public K to pick between a primary cubic and a companion—does separate the cases tested. This negative result applies only to the named witness within the specified flip-support subspace, not to all cubic or JSC designs. No prior art is on file. Page verdict: CAVEAT.
Result
No square-free cubic in the flip-support subspace meeting {middle1, middle2, final0} vanishes on all four honest K graphs while rejecting the named witness. A piecewise pair—the k0=0 principal cubic alongside a k0=1 companion—separates the named cases on K10/K11.
Setting and definitions
Coordinates and labels include current1, middle1, middle2, final0, K, and k0. A polynomial is square-free if no monomial contains repeated variables. Public K can be substituted per placement; the global alternative spans 41 coordinates.
Method
Exact span logging over the flip-support cubic space shows the named K00 false point sits outside the K00 flip-support cubic span, yet enters the all-K span at rank 2,115 / 2,224. Full certification logs and raw output artifacts are in this paper's downloadable evidence pack.
Discussion
The single global flip-support cubic approach is closed for this witness. Piecewise cubic separation is PROVED; piecewise CSL-2 remains CONJECTURE. False language, masks, replay, recount, and Lean verification remain UNKNOWN. The conditional JSC-12+5 arithmetic count of 21,681 is not a certified score. No corrections. No prior-art position recorded. Page verdict: CAVEAT.
For everyone — the takeaway
What this means
A single compact formula can't tell the four good cases apart from this specific bad case. Splitting the job across two formulas chosen by the public case label solves the problem for these test points. That doesn't guarantee security against untested bad cases or establish a competition score, but it shows that future designs need piecewise rules rather than a single global equation.
Register references
- ML-040.
- Receipts: CONT
jsc_flip_support_h3_allk.log,jsc_k1_companion_cubic.json,jsc_k1_idenc_lowering.json;ZKGOLF-FINDINGS-JSC-CUBIC-H3-2026-08-13.md;ZKGOLF-FINDINGS-JSC-CUBIC-ALLK-2026-08-13.md;ZKGOLF-FINDINGS-JSC-CUBIC-IDENC-2026-08-13.md. - 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 6 of 6 receipt files bundled (13 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.