Considering The Satisfiability of Cubic Diophantine Equations

Abstract

Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every emitted equation has degree at most 3. Degree-3 terms arise only when a linear selector variable activates a quadratic verification obligation. Earlier versions of this manuscript claimed a reduction from unbounded theoremhood to satisfiability of a fixed bounded-domain cubic polynomial instance. That claim is withdrawn. The error and its source are identified precisely. The bounded construction, the degree bookkeeping, and the Zeckendorf-based carryless encoding stand independently of the withdrawn claim. The note closes by identifying the uniformization gap that separates a family of decidable bounded slices from a single many-one reduction target, and records why closing that gap would require a compression principle not supplied here.