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A Finite-State Proof of the Well-Definedness of a Perturbed Hofstadter Sequence

Marco Mantovanelli

Abstract

We prove that the perturbed Hofstadter-type sequence Q(1)=1, Q(2)=1, and Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n is well-defined for all n>=1, in the sense that all recursive arguments remain positive. This contrasts with the classical Hofstadter Q-sequence, for which global well-definedness remains open. The proof reduces the infinite recursion to a finite combinatorial constraint system. We introduce a symbolic encoding of local configurations, compute the finite set of admissible contexts, and construct a compatibility relation that captures all valid local transitions. We then show that valid assignments split into two global modes, which reduces all potential obstructions to a finite critical core. A complete finite verification excludes these obstructions and establishes global well-definedness. More generally, the argument shows that certain meta-Fibonacci recursions admit a finite-state description whose global consistency can be decided by exhaustive combinatorial analysis.

A Finite-State Proof of the Well-Definedness of a Perturbed Hofstadter Sequence

Abstract

We prove that the perturbed Hofstadter-type sequence Q(1)=1, Q(2)=1, and Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n is well-defined for all n>=1, in the sense that all recursive arguments remain positive. This contrasts with the classical Hofstadter Q-sequence, for which global well-definedness remains open. The proof reduces the infinite recursion to a finite combinatorial constraint system. We introduce a symbolic encoding of local configurations, compute the finite set of admissible contexts, and construct a compatibility relation that captures all valid local transitions. We then show that valid assignments split into two global modes, which reduces all potential obstructions to a finite critical core. A complete finite verification excludes these obstructions and establishes global well-definedness. More generally, the argument shows that certain meta-Fibonacci recursions admit a finite-state description whose global consistency can be decided by exhaustive combinatorial analysis.

Paper Structure

This paper contains 121 sections, 26 theorems, 116 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method of proof
  3. Position within the literature
  4. Scope and limitations.
  5. Reduction to a Finite Symbolic System
  6. From recursion to symbolic data
  7. Local configurations and contexts
  8. Compatibility constraints
  9. Global consistency problem
  10. Local Contexts
  11. Definition of contexts
  12. Admissibility
  13. Finite enumeration
  14. Role of contexts
  15. Example: A concrete context and its interpretation
  16. ...and 106 more sections

Key Result

Theorem 1

The perturbed Hofstadter-type sequence with initial values $Q(1)=Q(2)=1$ is well-defined for all integers $n\ge1$.

Figures (2)

  • Figure 1: The compatibility graph $(\Lambda_{\mathrm{adm}},\Psi)$. The graph has two connected components, corresponding to debt $0$ and debt $2$. Solid arrows indicate the principal chain structure inside the regimes $R_1$, $R_2$, and $R_3$; dashed arrows indicate the additional compatibility edges from the full explicit list in Appendix B.
  • Figure 2: Automaton-style rendering of the compressed successor relation. Ordinary circles denote debt-$0$ states; double circles denote debt-$2$ states.

Theorems & Definitions (61)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Definition 3
  • Proposition 2
  • proof : Sketch of construction
  • Definition 4
  • Proposition 3
  • ...and 51 more