Research · Papers · What rank-one constraints can express · ML-029
Exact-UNSAT of Phase B in the natural P10 missing-plane model
Phase B is exact-UNSAT in the natural 30-coordinate P10 model; product image rank 423, quotient rank 393, rejection family empty
Published 2026-08-29
For everyone
Plain summary
This result rules out a candidate construction in the 30-coordinate P10 model. Phase B searched for a single product-shaped equation across every nonzero missing-plane class, but none exists. The tested product space spans 423 independent directions, and its reduced quotient spans 393. Two solver certificates verify the result. Because no candidate equations exist, the rejection family is empty and Phase C is skipped. Two arbitrary equations whose C-coordinate parts lie outside the missing plane remain open.
Result
Phase B is exact-UNSAT in the natural 30-coordinate P10 model. No genuine rank-one equation exists in any nonzero missing-plane class. The product image has rank 423 and the quotient has rank 393. The rejection family is empty, precluding Phase C. Two arbitrary equations with C-sides outside the missing plane remain UNKNOWN.
Setting and definitions
P10 denotes the natural 30-coordinate model. A missing-plane class is a Phase-B quotient class; the result applies to all nonzero classes. A genuine rank-one equation is the single-product constraint targeted by the rejection family. The C-side denotes the component assigned to coordinate C.
Method
Phase B was solved to exact-UNSAT. Dual solver certificates and rank reconciliation established product-image rank 423 and quotient rank 393, with zero genuine rows in every nonzero missing-plane class. The corresponding certificate and reconciliation receipts are available in this paper's downloadable evidence pack.
Discussion
ML-029 eliminates the Phase-B rejection family in the natural model. Two arbitrary equations with C-sides outside the missing plane remain UNKNOWN. Corrections: NONE. The register records no prior-art position. The result makes no broader claim about P10.
For everyone — the takeaway
What this means
One rejection route has no candidates left. The search looked for a single product-shaped rule in each nonzero slice of the missing plane and found none. The tested product space and its reduced space span 423 and 393 independent directions, confirmed by two solver certificates. With no candidates to inspect, Phase C is not reached. This closes the 30-coordinate route at Phase B. Two arbitrary equations with C-parts outside the missing plane remain untested. The register records no prior-art position.
Register references
ML-029
Receipt artifacts: CONT dp10_phase_ab_certificate.json; dp10_rank_reconciliation.json
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 0 of 0 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 2026-08-29
- 2026-08-29Published on this site.