Research · Papers · Direct sums, wedges and the p14 frontier · ML-043
Closure of all 68 p7-eligible directed edges under the rank-tight affine-code model
Zero-catalyst rank-tight affine-code model: 68 p7-eligible edges are closed: 32 rank-tight p7-dead, 28 p6-dead, 8 rank-tight p5-dead
Published 2026-08-29
For everyone
Plain summary
This paper tests a specific way to reduce the number of multiplications needed for a computation. The approach uses affine codes—formulas built from linear combinations plus a constant—under a strict rank constraint. The project register tracks 68 candidate transitions, called p7-eligible directed edges, aimed at a seven-product budget. Without an extra helper input (the zero-catalyst setting), every one of these 68 routes is ruled out. They split into three groups: 32 edges at q-rank 7 fail at p7, 28 edges at q-rank 6 fail at p6, and 8 edges at q-rank 5 fail at p5. Together, these groups account for all 68 candidate edges. The result was established by an exact replay of the search frontier alongside independent verification receipts. One-catalyst variants remain open, and the register notes no prior art.
Result
Under the zero-catalyst rank-tight affine-code model, all 68 p7-eligible directed edges are closed:
- 32 q-rank-7 edges are rank-tight p7-dead,
- 28 q-rank-6 edges are p6-dead, and
- 8 q-rank-5 edges are rank-tight p5-dead.
These three classes exhaust the 68 edges. One-catalyst extensions remain UNKNOWN.
Setting and definitions
An edge is a directed transition in the frontier search object. The set p7-eligible contains the 68 candidate edges evaluated here. The label q-rank-k denotes the rank partition assigned by the register. Rank-tight describes the affine-code model evaluated directly at its tight rank boundary. The zero-catalyst condition specifies that no auxiliary catalyst component is supplied. The remaining p7 search object is the 7-dimensional extension G = W ⊕ ⟨c⟩, governed by the flag theorem on G.
Method
The closure was established through exact frontier replay and independent verification covering all 68 p7-eligible directed edges partitioned by q-rank:
- 32 edges at q-rank 7 verified as rank-tight p7-dead,
- 28 edges at q-rank 6 verified as p6-dead, and
- 8 edges at q-rank 5 verified as rank-tight p5-dead.
The register specifies the active p7 search target as the 7-dimensional extension G = W ⊕ ⟨c⟩ with the flag theorem on G, rather than a 385-selector-bit formulation. The downstream search priority is 29→0, 29→5, 13→0, 24→0, followed by the score-feasible core. One-catalyst 29→0 was not executed in this pass; the register classifies a timeout on that target as UNKNOWN.
The verification receipts and raw replay logs are provided in this paper's downloadable evidence pack.
Discussion
The closure is strictly bounded by the zero-catalyst rank-tight affine-code model. It eliminates all 68 p7-eligible directed edges within this domain while leaving one-catalyst extensions UNKNOWN. The 7-dimensional extension G = W ⊕ ⟨c⟩ remains the active p7 search object under the flag theorem on G.
The recorded q-rank partitions correspond to register cross-references MF-070, MF-071, and MF-072, which locate the exact obstruction rank for each group (p7, p6, and p5, respectively). The priority sequence (29→0, 29→5, 13→0, 24→0, then score-feasible core) identifies future search targets without asserting outcomes; 29→0 was not evaluated here, and any timeout there remains UNKNOWN. Curation notes record no errata and no prior-art claims. Page verdict: SHOW.
For everyone — the takeaway
What this means
This paper completely rules out 68 candidate routes for cutting multiplication counts with tight linear formulas when no helper inputs are used. The exhaustive check leaves no survivors among the zero-catalyst candidates across all three rank categories. Finding a faster construction now requires changing the rules—either by moving into a seven-dimensional extension space or by providing a catalyst input.
Register references
ML-043
Receipts: lens_r2c_frontier_verification_report.md; lens_r2c_frontier_verification_receipt.json; zkgolf-r2-frontier-c-breakthrough/replay/frontier_exact_full_replay.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 3 of 3 receipt files bundled (88 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.