Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the $Σ^0_1$-Completeness of Cubic Diophantine Systems and their Consequences for Equations

Milan Rosko

Abstract

We prove that $\mathrm{H}\mathbb{N}(3,n)=\mathrm{U}$ for some fixed $n$. The core result is a uniform primitive recursive compiler from sentence codes to finite cubic Diophantine systems over $\mathbb{N}$, such that arithmetic theoremhood is equivalent to cubic solvability on the compiler image. The translation is explicit: finite proof traces are arithmetized, local syntactic and inference checks are compiled to quadratic constraints, and one guard gadget $u = 1+v^2, u\cdot E = 0$ provides the unique degree escalation from $2$ to $3$ while preserving solvability. The construction is mechanized in $\textsf{Rocq}$ and certified in the Calculus of Inductive Constructions, including uniform computability, checker--constraint equivalence, and a global degree-$3$ bound for emitted systems. From this certified core, undecidability is obtained by a classical fixed-point Diagonal Argument: any total correct decider for cubic satisfiability computes a mirror cubic instance on which it fails. Consequently cubic satisfiability is $Σ^0_1$-complete, and in particular there is no total correct procedure deciding all degree-$3$ Diophantine instances.

On the $Σ^0_1$-Completeness of Cubic Diophantine Systems and their Consequences for Equations

Abstract

We prove that for some fixed . The core result is a uniform primitive recursive compiler from sentence codes to finite cubic Diophantine systems over , such that arithmetic theoremhood is equivalent to cubic solvability on the compiler image. The translation is explicit: finite proof traces are arithmetized, local syntactic and inference checks are compiled to quadratic constraints, and one guard gadget provides the unique degree escalation from to while preserving solvability. The construction is mechanized in and certified in the Calculus of Inductive Constructions, including uniform computability, checker--constraint equivalence, and a global degree- bound for emitted systems. From this certified core, undecidability is obtained by a classical fixed-point Diagonal Argument: any total correct decider for cubic satisfiability computes a mirror cubic instance on which it fails. Consequently cubic satisfiability is -complete, and in particular there is no total correct procedure deciding all degree- Diophantine instances.

Paper Structure

This paper contains 28 sections, 43 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Register Arithmetic
  3. Main Result
  4. Proof Environment
  5. Executable Artifact
  6. Reduction
  7. Arithmetic Infrastructure
  8. Fibonacci Preliminaries
  9. Zeckendorf Decomposition
  10. Carryless Pairing
  11. The beta-Function
  12. Extraction
  13. The Hilbert--Ackermann Kernel
  14. Proofs as Finite Sequences
  15. Justification Predicate
  16. ...and 13 more sections

Key Result

Theorem 1.4

There exists a uniform computable transformation within $\mathsf{RA}$ such that (i) $\mathcal{S}_\varphi$ is a finite system of Diophantine equations over $\mathbb{N}$ of total degree $\le 3$, (ii) there is a predicate

Figures (1)

  • Figure 1: Execution traces as finite sequences of configurations within a Von Neumann Architecture. Each step requires only Register Arithmetic $\mathsf{RA} = \mathrm{I}\Delta_0 + \mathrm{B}\Sigma_1$.

Theorems & Definitions (73)

  • Definition 1.1: Register Arithmetic
  • Definition 1.2: ${\rm H}\mathbb{N}(d,n)$ and ${\rm H}\mathbb{Z}(d,n)$
  • Definition 1.3: Cubic satisfiability
  • Theorem 1.4: Certified Degree--3 Threshold
  • Theorem 1.5: RA-verifiable compilation
  • Corollary 1.6
  • Proposition 1.7: Internalization
  • Theorem 1.8: No internally certified cubic decider
  • Remark : Extracted API
  • Remark : Trust assumptions
  • ...and 63 more