Research · Papers · Adders, counters and the heap law · MF-114
Exact restriction loss conservation law for multiplicative complexity
MC(f) - MC(f|_R) = k_R + e_R with lifted potential k_R + ψ_R 1-Lipschitz; J_2 spectrum has four (1,0) and six (0,1).
Published 2026-09-04
For everyone
Plain summary
Multiplicative complexity counts the minimum number of AND gates needed to build a logic circuit when XOR gates and inverters are free. A standard way to analyze a circuit is restriction: fixing some input wires to constants like 0 or 1. Fixing inputs simplifies the function, which lowers the required AND count.
This result proves an exact conservation law for that drop in complexity. The total decrease splits into two exact parts: cost directly eliminated by the restriction, and residual inefficiency left in the remaining circuit. An accompanying potential function changes by at most one unit per step. On two-input functions, every one of the ten possible restriction steps falls into one of two exact profiles: either one eliminated gate with zero residual inefficiency, or zero eliminated gates with one unit of residual inefficiency.
Result
Let f be a Boolean function and f|_R its restriction under operator R. In the XOR-free multiplicative complexity model over GF(2), restriction loss satisfies the exact conservation law:
MC(f) - MC(f|_R) = k_R + e_R
where:
- k_R is the killed cost,
- e_R is the residual inefficiency,
- the lifted potential k_R + ψ_R is 1-Lipschitz under single-step restriction moves.
Across the space of J_2 restriction transitions, the spectrum of pairs (k_R, e_R) comprises four instances of (1,0) and six instances of (0,1), exhausting all ten non-trivial J_2 restriction configurations.
Setting and definitions
The model is the standard GF(2) XOR-and-inverter graph (XAG), where linear gates (XOR and NOT) have zero cost and multiplicative complexity MC(f) measures the minimum number of non-linear AND gates required to evaluate f.
For an input restriction R:
- MC(f|_R) is the multiplicative complexity of the restricted subfunction.
- Killed cost k_R measures the reduction in non-linear circuit rank eliminated directly by fixing inputs under R.
- Residual inefficiency e_R measures non-linear operations that become redundant or sub-optimally placed post-restriction.
- ψ_R is the residual structural potential of the restricted circuit.
- J_2 denotes the space of two-input Boolean functions and its restriction transitions.
Method
The conservation law and its bounds were proved via dual and coalgebraic verification pipelines combined with exhaustive state-space certificate extraction:
- Linear dual bounds: CERT-DUAL.md establishes the split between killed cost k_R and residual inefficiency e_R, certified in dual-certificate.receipt.json (SHA-256 01fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4).
- Coalgebraic potential tracking: CERT-COALG.md constructs a finite coalgebraic transition system establishing the 1-Lipschitz bound on k_R + ψ_R, certified in finite_certificate.receipt.json (SHA-256 c4a2cab5c1ce818d53affaf9b0acd50bcace59eaeb7a9b29a255457d09796db6).
- Categorical formulation: COUNCIL-LAWVERE.md details the Lawvere categorical construction associated with lawvere_five_cent.receipt.json (the report records no digest).
- Program state synthesis: Summarized in STATE-OF-PROGRAM-V2.md.
- Finite replay on J_2: Exhaustive enumeration over all ten non-trivial two-input restriction configurations verified four cases with (k_R, e_R) = (1,0) and six cases with (k_R, e_R) = (0,1).
Evidence tier: P + FC.
Discussion
The theorem converts the standard inequality on complexity reduction under restriction into an exact balance equation parameterized by killed cost k_R and residual inefficiency e_R.
Scope and limits:
- The result applies to Boolean functions in the GF(2) XAG (XOR-free) cost model.
- The 1-Lipschitz condition on k_R + ψ_R restricts step-wise potential variance to unit changes per single restriction.
- The (k_R, e_R) spectrum is fully classified on J_2 (ten cases: four (1,0), six (0,1)).
- The register records no prior-art position for this theorem.
- The register records no extension to non-Boolean base fields or non-XAG cost models.
For everyone — the takeaway
What this means
Multiplicative complexity is difficult to track because fixing an input can trigger chain-reaction simplifications across a circuit. This theorem shows that saved multiplication gates don't vanish unpredictably. Each saved gate is accounted for as either an eliminated operation or leftover structural slack, while the circuit's potential steps down smoothly by at most one unit at a time. That gives an exact balance sheet for inductive circuit complexity proofs.
Register references
- Entry ID: MF-114
- Report: zkgolf-decomp/reports/CERT-DUAL.md
- Report: zkgolf-decomp/reports/CERT-COALG.md
- Report: zkgolf-decomp/reports/COUNCIL-LAWVERE.md
- Report: zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
- Receipt: zkgolf-decomp/cert-dual-scratch/dual-certificate.receipt.json (SHA-256 01fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4)
- Receipt: zkgolf-decomp/cert-coalg-scratch/finite_certificate.receipt.json (SHA-256 c4a2cab5c1ce818d53affaf9b0acd50bcace59eaeb7a9b29a255457d09796db6)
- Receipt: zkgolf-decomp/council-lawvere-scratch/lawvere_five_cent.receipt.json (the report records no digest)
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 7 receipt files bundled (42 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.