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.

• The paper demonstrates that the perturbed Hofstadter sequence is globally well-defined by reducing an infinite, nonlocal recursion to a finite-state constraint system.
• It employs detailed symbolic state encoding and a compatibility graph to model local contexts, ensuring consistent propagation of recursive assignments.
• The computational analysis establishes two global modes and eliminates potential obstructions through critical core verification, robustly validating the sequence.

Finite-State Proof for Well-Definedness of a Perturbed Hofstadter Sequence

Introduction

The study addresses the well-definedness of a perturbed Hofstadter-type sequence, specifically the recursion

$$Q(1) = 1, \quad Q(2) = 1, \qquad Q(n) = Q(n - Q(n-1)) + Q(n - Q(n-2)) + (-1)^n,$$

where the non-classical perturbation—the $(-1)^n$ term—induces a structural asymmetry which is not present in the classical Hofstadter $Q$-sequence. While the original Hofstadter $Q$-sequence remains remarkably elusive in terms of ensuring global well-definedness, the perturbed version becomes tractable due to its altered combinatorial dynamics.

This work establishes, via a reduction to a finite-state constraint satisfaction problem, that the sequence $Q(n)$ is well-defined for all $n \geq 1$. The implication is that all arguments of the meta-recursive rule always remain strictly positive, precluding undefined behaviors at any step of the recursion—a property which remains open for the classical sequence.

Context: Meta-Fibonacci Recursions and Perturbed Hofstadter Dynamics

Meta-Fibonacci and Hofstadter-type recursions are notable for their dependence of recursive arguments on prior computed values, producing complex non-local dependencies. The classical $Q$-sequence, with its sparse local structure and global unpredictability ([Hofstadter 1979]), is prototypical of such behavior. The proven undecidability of generalized forms of nested recursions ([Celaya & Ruskey 2012 Undecidable]) underscores the profound difficulty in determining even basic properties, such as well-definedness.

Within this context, the constructed perturbed sequence (OEIS A394051; [Mantovanelli2026]) introduces an alternating perturbation, structurally differentiating it from its classical counterpart. Prior computational and symbolic evidence suggested a high degree of regularity in this perturbed model, including persistent self-similarities and widespread local rigidity, motivating a direct, combinatorial proof approach.

Finite-State Reduction: Symbolic Contexts and Compatibility Graph

The central theoretical advance is the explicit reduction of the infinite, nonlocal recursion to a finite combinatorial constraint system. The analysis introduces the following core elements:

• Symbolic State Encoding: Local configurations within a finite neighborhood of $n$ are abstracted into a finite set of symbolic states, $S = \{S_0, \dotsc, S_7\}$, augmented by a bounded "debt" parameter accounting for the $(-1)^n$ alternation.
• Context Construction: Admissible contexts are carefully defined 5-tuples $(-1)^n$0, where $(-1)^n$1 is a finite symbolic pattern, $(-1)^n$2 distinguishes recursion regimes, $(-1)^n$3 locates the window within $(-1)^n$4, $(-1)^n$5 is the local parity, and $(-1)^n$6 indicates debt. Contexts encapsulate all the required local information for recursion propagation.
• Finite Admissibility: The set of admissible contexts $(-1)^n$7 is explicitly computed and shown to be finite of size 28, with each representing an equivalence class of local configurations.
• Compatibility Graph $(-1)^n$8: A directed graph is defined on $(-1)^n$9, with edges denoting local admissibility of consecutive context occurrences given the recursion. This structure is the locus for finite-state propagation of all compatibility constraints imposed by the recursion.

This construction is in the tradition of finite constraint systems as discussed in symbolic dynamics ([Lind & Marcus 1995]; [Hell & Nešetřil 2004]), establishing an effective computational and combinatorial framework for what is, in general, a non-local, infinite recursive process.

Two-Mode Decomposition and Support Structure

A key result is that all globally consistent assignments of symbolic values to contexts reduce to exactly two global modes (A and B), determined by the assignment at a single distinguished "root" context. This rigidity is established by a series of implication chains: the value at the root propagates throughout the entire compatibility graph, uniquely constraining all further assignments (except on a set exhibiting local flexibility).

Associated with this are support sets $Q$0 and $Q$1 for each context, recording the set of symbolic assignments possible at that context in modes A and B, respectively. These are determined by recursive propagation along the compatibility graph and are nonempty (with varying degrees of local multiplicity).

The debt-$Q$2 contexts exhibit full rigidity: each admits only the symbolic value $Q$3, severely restricting possible local behavior.

Obstruction Analysis and Critical Core Reduction

To demonstrate global well-definedness, the possibility of local obstructions—subsets of contexts that cannot be extended to a globally consistent assignment—must be eliminated.

The analysis proceeds by structural decomposition:

• Insensitive Contexts: Contexts where the two modes admit the same support set are shown, via explicit enumeration, to always be locally extendable and thus cannot be part of any minimal obstruction.
• Sensitive Contexts $Q$4: Potential obstructions are restricted to contexts where the two modes differ (i.e., the support sets are not equal).
• Critical Core $Q$5: Further structural reductions, leveraging the compatibility graph and explicit context propagation, confine any possible minimal obstructions to a set of exactly four contexts.
• Acyclicity and Uniform Assignment: The induced core compatibility graph is acyclic and allows a uniform assignment ($Q$6 in Mode~A) over all subsets of the critical core; all compatibility relations are satisfied in this mode.

The ultimate verification is computational: all $Q$7 nonempty subsets of the critical core are exhaustively checked for extendability, and explicit assignments (always in Mode~A) are exhibited, confirming that no obstructions exist.

Computational Certification and Reproducibility

The full symbolic model, compatibility graph, support structure, and final critical-core verification are realized in a deterministic computational pipeline, available as an open-source repository ([MantovanelliCode2026]). The implementation provides a certificate for all claims that depend on explicit finite combinatorics, including:

• Generation and extraction of symbolic traces,
• Construction of admissible contexts and compatibility graphs,
• Propagation and computation of support sets,
• Full enumeration of all possible (partial) assignments on the critical core.

The pipeline ensures complete reproducibility and provides a robust foundation for future formal verification efforts using proof assistants.

Implications and Future Directions

This work demonstrates that even for highly nonlocal, self-referential recursions, suitable structural perturbations can induce finite-state describability, enabling exhaustive combinatorial verification of global properties. In particular, the alternating forcing in $Q$8 fundamentally alters the recursion’s dynamical propagation, making well-definedness accessible to a finite-state analysis which is provably unavailable for the classical $Q$9-sequence.

The methodology has implications beyond the present sequence. Similar finite-state reductions may facilitate proofs of well-definedness for other meta-Fibonacci recursions, especially where perturbations or underlying regularities induce rigid, low-complexity local symbolic models. Furthermore, the explicit computational and combinatorial machinery developed here provides a template for the formal certification of global properties in symbolic dynamics and recursive combinatorics.

Conclusion

The analysis achieves an explicit, finite-state proof that the perturbed Hofstadter sequence with an added $Q$0 term is globally well-defined for all $Q$1 (2603.29622). This is accomplished through a complete reduction to a finite compatibility system on symbolic contexts, the classification of all global extensions via a two-mode structure, identification and verification of a critical core, and exhaustive, machine-certified enumeration of all possible local obstructions. The approach circumvents the traditional undecidability and nonlocality barriers inherent to meta-Fibonacci recursions, opening up new avenues for the systematic, combinatorial analysis of recursive sequences whose global behavior is governed by local symbolic constraints.