Research · Papers · Direct sums, wedges and the p14 frontier · MF-112
Status of the FCNS reduction chain for uniform lower bounds
Exact exclusion is Γ_{m, 2m-3} = ∅; local-response theorem proved, but summation across sites and SBD remain unproved (OPEN).
Published 2026-09-04
For everyone
Plain summary
Proving how many multiplications a calculation strictly requires is a central challenge in computer science. The FCNS framework attempts to prove these minimum counts across whole families of formulas at once. This register entry clarifies that FCNS is an active research program, not a finished proof. While individual computational steps have been analyzed and verified, two essential global steps remain unproved: combining responses across multiple sites and proving a structural property called SBD. The targeted condition Γ_{m, 2m-3} = ∅ checks out on small test cases, but general bounds for all parameters remain open.
Result
The uniform exclusion statement Γ_{m, 2m-3} = ∅ remains OPEN.
The reduction chain status stands as follows:
- FCNS and (TMI_m) are sufficient strengthenings for the target lower bound; logical equivalence is not established.
- The local-response theorem is proved over the complete algebra of old variables.
- The summation and no-screening step across sites remains unproved.
- The structural property SBD remains unproved.
- The initial numerical canary is Γ_{5,7}, corresponding to the (TL)_4 cell.
- The first stronger Rees cell is (TMI_4).
Evidence tier: OPEN. Proved and formally checked status (P + FC) is restricted to local statements; computational screens are fully replayed (FR).
Setting and definitions
- Target numerical exclusion: The configuration space emptiness condition Γ_{m, 2m-3} = ∅.
- FCNS reduction chain: A sequence of sufficient conditions for deriving uniform lower bounds under bilinear and tensor rank cost models.
- (TMI_m): Parameterized family of sufficient strengthenings indexed by m.
- (TL)_4 cell: Structural cell corresponding to numerical canary Γ_{5,7}.
- Rees cells: Structural components in the algebraic reduction hierarchy, with (TMI_4) as the first stronger Rees cell.
- Local-response theorem: Proved algebraic theorem governing responses over the complete algebra of old variables.
- Summation/no-screening: Unproved reduction step aggregating responses across independent computational sites.
- SBD: Unproved structural property required to complete the global reduction.
Method
Local algebraic statements carry analytical proofs and formal machine checks (P + FC). Concrete screens and canary evaluations across specific cell configurations carry full computational replay (FR).
Analysis, reduction pathways, and algebraic derivations:
- zkgolf-decomp/reports/CERT-SYNTH.md
- zkgolf-decomp/reports/CERT-TENSOR.md
- zkgolf-decomp/reports/CERT-SYZ.md
- zkgolf-decomp/reports/CERT-COALG.md
- zkgolf-decomp/reports/FCNS-A-TMI.md
- zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
Computational validation and module verification data:
- zkgolf-decomp/cert-syz-scratch/module-data.receipt.txt (SHA-256 68f78bfaa2b3a1c31dc0d514fbcdf3be28f3548bd7d52c94bb47e73666e162aa)
- zkgolf-decomp/cert-coalg-scratch/exact_xag.receipt.json (SHA-256 0702ccae456b9865d27cbc3f46a628a828219b6c5dd03035b681f866811baad1)
- zkgolf-decomp/cert-tensor-scratch/prefix-screen.receipt.json (SHA-256 a7ef24de17806ae977182df864f71d19a0a0cef5b87f7cf2743caf7ff47ca0b6)
Discussion
FCNS is an open reduction chain, not a completed proof. The target property Γ_{m, 2m-3} = ∅ cannot be claimed as a theorem because the bridge from local properties to global exclusions remains incomplete.
Although the local-response theorem holds over the complete algebra of old variables, the reduction lacks a summation or no-screening argument across sites, and the SBD condition remains unproved. FCNS and (TMI_m) operate strictly as one-way sufficient strengthenings rather than exact characterizations.
The canary Γ_{5,7} ((TL)_4 cell) and the stronger Rees cell (TMI_4) define the initial verification boundary.
For everyone — the takeaway
What this means
Proving that an algebraic task demands a minimum number of multiplications is difficult. The FCNS framework maps out a clear path to proving these limits across whole families of functions, but the bridge isn't finished. We have confirmed the math for isolated, individual operations, but linking them across an entire computation still requires proving the summation step and the SBD property. Marking FCNS as open isolates the exact mathematical gaps left to bridge.
Register references
- Register entry: MF-112
- Reports:
- zkgolf-decomp/reports/CERT-SYNTH.md
- zkgolf-decomp/reports/CERT-TENSOR.md
- zkgolf-decomp/reports/CERT-SYZ.md
- zkgolf-decomp/reports/CERT-COALG.md
- zkgolf-decomp/reports/FCNS-A-TMI.md
- zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
- Receipt artifacts:
- zkgolf-decomp/cert-syz-scratch/module-data.receipt.txt (SHA-256 68f78bfaa2b3a1c31dc0d514fbcdf3be28f3548bd7d52c94bb47e73666e162aa)
- zkgolf-decomp/cert-coalg-scratch/exact_xag.receipt.json (SHA-256 0702ccae456b9865d27cbc3f46a628a828219b6c5dd03035b681f866811baad1)
- zkgolf-decomp/cert-tensor-scratch/prefix-screen.receipt.json (SHA-256 a7ef24de17806ae977182df864f71d19a0a0cef5b87f7cf2743caf7ff47ca0b6)
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 6 of 9 receipt files bundled (55 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.