Research · Papers · Adders, counters and the heap law · ML-059
Bounds on strict width-nine complexity E_9
18 ≤ E_9 ≤ 36
Published 2026-09-04
For everyone
Plain summary
Multiplicative complexity measures the minimum number of nonlinear multiplication steps needed to evaluate a logic function when additions cost nothing. For symmetric operations across nine variables, called strict width nine, the exact cost E_9 is unknown. The proven range spans from 18 to 36 multiplications. Solver verification establishes the lower bound of 18, while a symmetric circuit construction sets the upper bound of 36. Neither automated searches nor counting arguments have closed this gap.
Result
The multiplicative complexity of strict width nine, denoted E_9, satisfies:
18 ≤ E_9 ≤ 36
under the GF(2) XOR-and-AND graph (XAG) multiplicative cost model. The exact value of E_9 remains open.
Setting and definitions
Let E_9 denote the multiplicative complexity of the strict width-nine target system over GF(2). The metric counts the minimum number of AND gates (GF(2) multiplications) in a valid XAG representation, with XOR gates (GF(2) linear additions) incurring zero cost.
Method
Two complementary bounds establish the bracket on E_9:
- Lower floor: E_9 ≥ 18 is recorded in
zkgolf-decomp/sb-tax-scratch/n9_one_full_floor_receipt.json(SHA-2565a5237e61432a07345684ff252cb112ad2aeb2606ffda9bcfab3500ca009b25e). - Upper ceiling: E_9 ≤ 36 is established by the clean-room strict-equivariant construction in
zkgolf-decomp/sb-tax-scratch/cleanroom_upper.py(SHA-256c03bb6312c08c9b35633cb3817c69e41ba3da401e3ad5957a594709adef51a00).
Analytical verification and taxonomic evaluations are documented in:
zkgolf-decomp/reports/SB-TAX.mdzkgolf-decomp/reports/EXP-EQ-TAX.mdzkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
The evidence tier for the interval [18, 36] is recorded as P + FC + FR.
Discussion
The exact value of E_9 remains unresolved. The upper endpoint of 36 is an explicit construction rather than a proven minimum, and SAT solver runs and inventory counting arguments have not closed the bracket. Progress requires either a higher certified floor or a lower-cost algebraic construction.
For everyone — the takeaway
What this means
Evaluating symmetric nine-variable Boolean functions takes between 18 and 36 multiplications when additions are free. We know the cost cannot fall below 18 or exceed 36, but finding the exact number requires faster search tools or new mathematical insights. Pinning down E_9 will establish the true efficiency limits for cryptographic primitives and zero-knowledge proof circuits that minimize non-linear operations.
Register references
- Entry: ML-059
- Cross-reference: MF-117
- Reports:
zkgolf-decomp/reports/SB-TAX.mdzkgolf-decomp/reports/EXP-EQ-TAX.mdzkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md- Artifacts:
zkgolf-decomp/sb-tax-scratch/n9_one_full_floor_receipt.json(SHA-2565a5237e61432a07345684ff252cb112ad2aeb2606ffda9bcfab3500ca009b25e)zkgolf-decomp/sb-tax-scratch/cleanroom_upper.py(SHA-256c03bb6312c08c9b35633cb3817c69e41ba3da401e3ad5957a594709adef51a00)
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 3 of 5 receipt files bundled (33 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.