Research · Papers · Direct sums, wedges and the p14 frontier · MF-123
Necessary carrier sequence and connected-leakage bounds for hypothetical p14
0→H_6→E_10→C_4→0; rank Q_2(fg) ≤ 2r+2s+rs+2; first mixed catalyst connected rank ≤ 2
Published 2026-09-04
For everyone
Plain summary
In Boolean circuit complexity, multiplicative complexity measures how many AND gates a function needs when XOR gates are free. Whether certain target functions can be computed with at most 14 multiplications—called a hypothetical p14 circuit—remains open.
This work proves structural requirements that any 14-multiplication realization must satisfy. Any valid circuit must split into linear carrier components of sizes 6, 10, and 4, and its algebraic interaction terms cannot exceed strict rank limits. These are necessary conditions: they prune the candidate search space, but they do not prove or disprove the existence of p14 or p15 circuits.
Result
Let f and g be Boolean functions in an XOR-and-inverter graph (XAG) over GF(2). Any hypothetical 14-multiplication circuit realization (p14) satisfies the short exact carrier sequence:
0 → H_6 → E_10 → C_4 → 0
The connected leakage and Boolean two-jet spaces satisfy three rank bounds:
- Boolean two-jets: rank Q_2(fg) ≤ 2r + 2s + rs + 2.
- First mixed catalyst: connected rank at most 2.
- Iterated connected leakage: rank at most 2 + 2r + 2s + rs.
These constraints are necessary structural conditions on p14 circuit realizations.
Setting and definitions
In the GF(2) XAG cost model, multiplicative complexity MC(f) counts AND gates, treating XOR gates and inverters as zero-cost.
Carrier sequence terms and operators denote:
- H_6: 6-dimensional homogeneous carrier space.
- E_10: 10-dimensional intermediate extension space.
- C_4: 4-dimensional co-carrier quotient space.
- Q_2(fg): Boolean two-jet operator measuring degree-2 algebraic differential interactions between components f and g of rank dimensions r and s.
- Connected rank: linear dimension of the cross-term subspace generated by mixed catalyst products.
Method
The carrier sequence and rank bounds were derived analytically and certified under tier P + FC (formal proof and fully checked computational receipts).
Computational validation of two-jet ceilings, mixed catalyst limits, and census consistency ran under AX162 against two machine receipts:
two_jet_and_mixed_rank_receipt.json(SHA-2568452308257a325266450426a602f69b970e92c4ddd5ad6d23bdefa3480ab4dce): establishes rank bounds on Q_2(fg) and the connected rank of the first mixed catalyst.census_crosscheck_receipt.json(SHA-256a5eab1b5af2f0e77d56141cfaac229e4e661215128a34cdb561886b0aed70883): cross-checks the decomposition census.
Derivations are documented in zkgolf-decomp/reports/SB-P14INJ.md and zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md.
Discussion
Satisfying 0 → H_6 → E_10 → C_4 → 0 and rank Q_2(fg) ≤ 2r + 2s + rs + 2 is necessary, not sufficient. The bounds do not establish existence or non-existence of a valid p14 circuit, and they yield no classification verdict for p14 or p15. They bound the algebraic structure of candidate decompositions.
For everyone — the takeaway
What this means
Any 14-multiplication circuit that computes the target function must have internal parts that fit a 6-10-4 structure and stay within these leakage limits. Automated solvers can use these bounds to discard invalid circuit layouts immediately. The constraints narrow the search space without resolving whether a 14-multiplication implementation exists.
Register references
- Register entry: MF-123
- Reports:
zkgolf-decomp/reports/SB-P14INJ.mdzkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md- Receipts:
zkgolf-decomp/sb-p14inj-scratch/two_jet_and_mixed_rank_receipt.json(SHA-2568452308257a325266450426a602f69b970e92c4ddd5ad6d23bdefa3480ab4dce)zkgolf-decomp/sb-p14inj-scratch/census_crosscheck_receipt.json(SHA-256a5eab1b5af2f0e77d56141cfaac229e4e661215128a34cdb561886b0aed70883)
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 2 of 4 receipt files bundled (28 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-09-04
- 2026-09-04Published on this site.