Research · Papers · Direct sums, wedges and the p14 frontier · MF-044
Exact scans of published p8 factors for three maximal mixed p14 skeletons
Zero 14-target spans across 1,920, 1,920, and 38,400 direct-fusion assignments over GF(2) on (7,15), (7,23), and (7,39) skeletons
Published 2026-08-29
For everyone
Plain summary
This paper closes one route to building a 14-target logic circuit with binary signals. It tests whether two published eight-gate components (p8 factors) can combine through direct fusions to hit all 14 targets. We evaluated three cases: (7,15), (7,23), and (7,39). Complete scans checked 1,920, 1,920, and 38,400 assignments; none covered all 14 targets. A relaxed test also checked all possible seven-gate choices from pools of 49, 49, and 60 candidate gates, ignoring prerequisite rules and execution order, and still found no solution. A p14 circuit in these cases must use components outside the published p8 vocabulary or new catalyst-fed factor pairs. General p14 synthesis remains open. The register records no prior-art position.
Result
Exact scans across the (7,15), (7,23), and (7,39) skeletons exhaust 1,920, 1,920, and 38,400 direct-fusion assignments over GF(2) with zero 14-target spans. Relaxed exact candidate universes of size 49, 49, and 60 also contain no seven-gate span. Any p14 realization in these mixed-slack orbits requires catalyst-fed factor pairs or factors outside the published p8 vocabulary.
Setting and definitions
Scans operate over GF(2). A span generates all fourteen target linear functions. The relaxed candidate universe pools all missing p8 gates and oriented fusions while dropping product ordering and prerequisite gate constraints.
Method
An initial scan enumerated all direct-fusion assignments for each skeleton. A second search evaluated every seven-gate combination within the relaxed candidate universes. An independent NumPy replay script rechecked all evaluation paths against the serialized receipt datasets.
Discussion
These exhaustive searches close the published-p8 seed route for the (7,15), (7,23), and (7,39) orbits. The negative result does not resolve whether these orbits admit other constructions, nor does it constrain general p14 synthesis beyond this seed class. No correction or prior-art position is recorded.
For everyone — the takeaway
What this means
Circuit designers using the published eight-gate components can rule out these three configurations entirely. The failure persists even when dropping order and prerequisite rules. Making a 14-target circuit in these cases requires new catalyst-fed factor pairs or parts outside the published list. While the overall p14 question stays open, this result narrows the search space and shows where new components are needed.
Register references
- Entry: MF-044.
- Receipt artifacts: CONT RETURN6-P8-MIXED-SEED-RESULT.md (SHA-256 f76f3f73013fc5636e4406af6b100fbe770eda2b4d7c5f4c3f66e64434d923da); return6_p8_direct_fusion_scan.json (SHA-256 5cb802421d4d789715d833e1391392069634930a7a9717b20c1d91f314c6e838); return6_p8_factor_universe_scan.json (SHA-256 4699f3d66fba8323ae19eae571b15d88f7a0588607067ff95ea37c6c97414279); independent NumPy return6_p8_seed_numpy_recheck.json (SHA-256 0d0791ecd3fc2a41f1bdbde9b8dd247bae1ad1c95415966f415a18b48f3b6b81).
- Prior-art works: 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 4 of 4 receipt files bundled (11 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.