Research · Papers · Adders, counters and the heap law · ML-056
Restriction conservation law for multiplicative complexity under exact restriction
MC(f) - MC(f|_R) = k_R + e_R under exact restriction, with 1-Lipschitz potential k_R + ψ_R; exhaustive J_2 spectrum is 4×(1,0), 6×(0,1)
Published 2026-09-04
For everyone
Plain summary
Multiplicative complexity measures the minimum number of multiplication (AND) gates needed to evaluate a Boolean function when additions (XOR) are free. Fixing some input variables to constants—called a restriction—simplifies the circuit and lowers its multiplicative complexity.
This result proves an exact accounting rule for that drop. For any exact restriction, the reduction in multiplicative complexity equals the number of multiplication gates directly eliminated plus a residual inefficiency term. A potential function tracking these quantities changes smoothly by at most one unit per step (it is 1-Lipschitz). This identity provides an exact ledger rather than an automated lower-bound engine; proving circuit lower bounds still requires aggregating these restriction losses across full trees.
Result
Let f be a Boolean function evaluated in the GF(2) XOR-free XAG cost model with multiplicative complexity MC(f). Under any exact restriction R, the drop in multiplicative complexity satisfies:
MC(f) - MC(f|_R) = k_R + e_R
where k_R is the number of killed multiplication gates and e_R is the residual inefficiency.
The lifted potential k_R + ψ_R is 1-Lipschitz across restriction transitions. For the J_2 system, the full spectrum of restriction transitions decomposes into 4×(1,0) and 6×(0,1).
Setting and definitions
Multiplicative complexity MC(f) is the minimum number of AND gates required to compute f over the basis {AND, XOR, NOT, 1} in the GF(2) XOR-free XAG cost model.
An exact restriction R assigns Boolean constants to a subset of input coordinates such that the restricted function f|_R is realized by simplifying the original circuit. The term k_R counts multiplication gates whose outputs collapse to affine functions under R, while e_R measures the residual inefficiency in the resulting decomposition. The term ψ_R is the structural residual component of the restriction potential.
Method
Algebraic structural analysis and finite proof certificates establish the conservation theorem and potential bounds (Evidence tier: P + FC).
Dual and coalgebraic certificate frameworks verified the state space and the J_2 spectrum:
- Dual certificate: recorded in
zkgolf-decomp/reports/CERT-DUAL.md, artifactzkgolf-decomp/cert-dual-scratch/dual-certificate.receipt.json(SHA-25601fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4). - Finite coalgebraic certificate: recorded in
zkgolf-decomp/reports/CERT-COALG.md, artifactzkgolf-decomp/cert-coalg-scratch/finite_certificate.receipt.json(SHA-256c4a2cab5c1ce818d53affaf9b0acd50bcace59eaeb7a9b29a255457d09796db6). - Categorical and program reports: recorded in
zkgolf-decomp/reports/COUNCIL-LAWVERE.md,zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md, and artifactzkgolf-decomp/council-lawvere-scratch/lawvere_five_cent.receipt.json(no digest printed in the report).
Discussion
The identity MC(f) - MC(f|_R) = k_R + e_R accounts for all complexity loss under exact restrictions through directly eliminated gates and residual inefficiency, ruling out unobserved non-local losses. The 1-Lipschitz property of k_R + ψ_R bounds step-wise variance.
The identity functions as an exact ledger rather than an automated lower-bound generator. Obtaining non-trivial lower bounds on MC(f) still requires an edge-complete potential or a verified method to sum restriction losses across full trees.
The status of the result is CONFIRMED under the GF(2) XAG cost model. See MF-114 for related structures.
For everyone — the takeaway
What this means
When you fix some inputs of a logic circuit to constants, the circuit gets simpler and uses fewer multiplication gates. This result gives an exact ledger for that reduction: every lost multiplication gate either disappears directly or turns into measurable leftover inefficiency. The potential function tracking this process moves smoothly, shifting by at most one gate per step. While it does not automatically prove that a given function is hard to compute, it guarantees that no complexity is lost without a trace.
Register references
- Register Entry: ML-056
- Cross-reference: MF-114
- Verification Reports:
zkgolf-decomp/reports/CERT-DUAL.mdzkgolf-decomp/reports/CERT-COALG.mdzkgolf-decomp/reports/COUNCIL-LAWVERE.mdzkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md- Verification Receipts:
zkgolf-decomp/cert-dual-scratch/dual-certificate.receipt.json(SHA-25601fe39f50fa66314c0356cda17480b94124c5ef496dca132a08ed7969cb695c4)zkgolf-decomp/cert-coalg-scratch/finite_certificate.receipt.json(SHA-256c4a2cab5c1ce818d53affaf9b0acd50bcace59eaeb7a9b29a255457d09796db6)zkgolf-decomp/council-lawvere-scratch/lawvere_five_cent.receipt.json(no digest printed in the report)
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.