Research · Papers · Symmetry, state encodings and search gauges · MF-113
Shortest-path and LP-dual formulation of multiplicative complexity via semantic gate states
MC equals shortest-path distance on the complete semantic gate-state graph; its LP dual is the 1-Lipschitz potential problem
Published 2026-09-04
For everyone
Plain summary
Multiplicative complexity counts the minimum number of nonlinear AND gates needed to compute a Boolean function when XOR operations are free. This work proves that multiplicative complexity equals the shortest-path distance between circuit states in a complete semantic gate-state graph. Because shortest-path problems have linear programming duals, proving lower bounds translates directly into assigning numbers, called 1-Lipschitz potentials, to each state such that no valid step increases the potential by more than 1. For a lower-bound certificate to hold unconditionally, it must check every possible transition edge in the graph; skipping edges invalidates the lower bound.
Result
In the GF(2) XOR-free XAG cost model, the multiplicative complexity MC of any Boolean function equals the shortest-path distance on the complete semantic gate-state graph. The linear programming dual of this shortest-path formulation is the 1-Lipschitz potential problem. Edge completeness is necessary for quotient certificates: verifying potentials on a selected edge subset does not produce an unrestricted lower bound on MC.
Setting and definitions
The model is the GF(2) XOR-free XAG, where affine operations cost zero and multiplicative complexity MC counts the minimum number of AND gates needed to evaluate a target Boolean function or map over GF(2). Reachable computational states under affine spans and nonlinear gate transitions define the complete semantic gate-state graph. A potential function on this graph is 1-Lipschitz if the potential difference across every directed semantic gate transition edge is at most 1.
Method
Formulating circuit synthesis across semantic gate states as a network shortest-path problem yields the 1-Lipschitz potential problem via linear programming duality. Implementations and dual certificates are verified under tier P + FC on the filed small cells.
Artifacts and reports filed in the register:
- Technical report:
zkgolf-decomp/reports/CERT-DUAL.md - Status report:
zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md - Primary certificate receipt:
zkgolf-decomp/cert-dual-scratch/dual-certificate.receipt.jsonwith SHA-256 hash01fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4 - Linear programming dual model file:
zkgolf-decomp/cert-dual-scratch/j3-normal-form-dual.lpwith SHA-256 hash6c4d382908e80ca879c4b3f58b8ec7004b32cbe1f4c60b473a7093099aba15fe
Discussion
Mapping multiplicative complexity to shortest paths links circuit minimization directly to LP duality and network flow. The key operational requirement is edge completeness: dual values obtained over restricted or heuristically pruned edge sets do not guarantee valid unrestricted lower bounds on MC. Sound certificate checking requires exhausting the complete semantic transition graph. Filed small-cell certificates satisfy tier P + FC.
For everyone — the takeaway
What this means
Finding the most efficient nonlinear circuit is mathematically identical to finding the shortest path across a map of semantic gate states. Linear programming duality lets us prove a circuit cannot be compressed further by building a valid landscape of potentials across states. Crucially, this framework proves that lower-bound certificates cannot skip any valid state transitions; every transition must be checked for the bound to hold unconditionally.
Register references
- Register Entry: MF-113
- Report:
zkgolf-decomp/reports/CERT-DUAL.md - Report:
zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md - Receipt:
zkgolf-decomp/cert-dual-scratch/dual-certificate.receipt.json(SHA-25601fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4) - Receipt:
zkgolf-decomp/cert-dual-scratch/j3-normal-form-dual.lp(SHA-2566c4d382908e80ca879c4b3f58b8ec7004b32cbe1f4c60b473a7093099aba15fe)
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 3 receipt files bundled (27 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.