Research · Papers · Symmetry, state encodings and search gauges · MF-004
Functional census and degree bounds for the two-row cyclic model
|S₂| = 575,968, |{Φ(S) : S ∈ S₂}| = 32,768, and ∀S ∈ S₂, deg(Φ(S)) ≤ 3
Published 2026-08-29
For everyone
Plain summary
MF-004 provides a complete census of the two-row cyclic model. It analyzes 575,968 systems—each defined by a choice of the model's two cyclic rows—and groups them by their input-output behavior. Only 32,768 distinct Boolean functions appear across the entire family. Every realized function has degree at most 3, so its standard polynomial description over bits never contains a term with four or more inputs multiplied together. The model produces no quartic behavior. An exhaustive replay confirms the exact count for this named family. The register does not state whether this result is new relative to prior work.
Result
Let S₂ denote the set of systems in the two-row cyclic model, and let Φ(S) be the Boolean function realized by S. The exhaustive census establishes:
|S₂| = 575,968, |{Φ(S) : S ∈ S₂}| = 32,768.
For every S ∈ S₂,
deg(Φ(S)) ≤ 3.
Quartic functions are structurally absent from the model. The census bounds the degree of every realized function from above; it does not assert that every Boolean function of degree at most 3 is realized.
Setting and definitions
The two-row cyclic model is a finite family of systems. Two systems are functionally equivalent when their truth tables match on all binary inputs.
Degree refers to algebraic degree in the multilinear polynomial representation over F₂. Degree at most 3 restricts monomials to products of at most three variables; a quartic term is a degree-four monomial. The register omits row equations, variable names, and explicit cyclic parametrizations, so this analysis treats the model strictly as named in MF-004.
Method
An exhaustive census enumerated all 575,968 systems in the model, evaluated the Boolean function for each, and deduplicated the resulting truth tables to isolate distinct functions. Computing the algebraic normal form for every realized function yielded the uniform degree bound of 3.
An independent replay verified the counts and degree bounds. The raw output datasets and verification records are available in this paper's downloadable evidence pack.
Discussion
The census separates syntactic descriptions from semantic behaviors. The 575,968 system specifications collapse into 32,768 distinct functions under truth-table evaluation, showing substantial functional degeneracy across the cyclic row choices.
The degree bound holds across the entire realized set: no system in the family generates quartic terms. This structural ceiling is specific to the two-row cyclic model and does not constrain other cyclic families, models with more rows, or unrestricted Boolean spaces. The register records no prior-art citations or scope modifications for the entry.
For everyone — the takeaway
What this means
The model defines 575,968 configurations, but they collapse into just 32,768 distinct rules once you look at what they compute. Many different setups produce identical outputs. Every rule can be written using interactions between at most three input bits; no rule needs a four-bit product. Searching within this two-row cyclic setup can never generate quartic rules, though other models with more rows might.
Register references
- MF-004
- DL cyclic_2row_function_census_results.json
- CONT cyclic_2row_census_recheck.json (digest fbf919ad…)
- The register does not record this.
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 1 of 2 receipt files bundled (1 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.