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The Solver's Paradox in Formal Problem Spaces

Milan Rosko

TL;DR

This work analyzes why global decision problems over arithmetically represented domains incur reflective structure when quantified universally, showing that arithmetization forces diagonal fixed points internal to the problem space and imposes $\Pi^0_2$-style verification obstructions. It introduces the Solver's Paradox as a minimal model, linking universality, representability, and diagonalization to a unavoidable self-reference that any finitary theory cannot discharge. The main contribution is a unified framework tying class-quantification to arithmetical provability, with a detailed application to uniform complexity questions such as $P=NP$, where the obstruction arises from structural impredicativity rather than algorithmic limits. The results yield the Matryoshka Principle of nested reflective barriers and a constructive reformulation perspective, highlighting fundamental epistemic limits on global reasoning and suggesting principled ways to reframe such global assertions for constructive justification.

Abstract

This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection beyond finitary means, producing Feferman-style obstructions independent of computational technique. We use this mechanism to analyze uniform complexity statements, including $\mathsf{P}$ vs. $\mathsf{NP}$, showing that their difficulty stems from structural impredicativity rather than methodological limitations. The focus is not on deriving separations but on clarifying the logical status of such arithmetized assertions.

The Solver's Paradox in Formal Problem Spaces

TL;DR

This work analyzes why global decision problems over arithmetically represented domains incur reflective structure when quantified universally, showing that arithmetization forces diagonal fixed points internal to the problem space and imposes -style verification obstructions. It introduces the Solver's Paradox as a minimal model, linking universality, representability, and diagonalization to a unavoidable self-reference that any finitary theory cannot discharge. The main contribution is a unified framework tying class-quantification to arithmetical provability, with a detailed application to uniform complexity questions such as , where the obstruction arises from structural impredicativity rather than algorithmic limits. The results yield the Matryoshka Principle of nested reflective barriers and a constructive reformulation perspective, highlighting fundamental epistemic limits on global reasoning and suggesting principled ways to reframe such global assertions for constructive justification.

Abstract

This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection beyond finitary means, producing Feferman-style obstructions independent of computational technique. We use this mechanism to analyze uniform complexity statements, including vs. , showing that their difficulty stems from structural impredicativity rather than methodological limitations. The focus is not on deriving separations but on clarifying the logical status of such arithmetized assertions.

Paper Structure

This paper contains 26 sections, 22 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Solver’s Paradox
  3. Impredicativity Transport in SAT
  4. Total Classifiers and Encoded Problem Spaces
  5. The Self-Misclassification Instance
  6. Fixed Point Theorem
  7. PSSP Isomorphism
  8. Global Assertions over Coded Domains
  9. Verification and Reflection
  10. Application: Uniformity in Class--NClass Separations
  11. Deterministic and Nondeterministic Classes
  12. Uniformity Implies Class-Quantification
  13. Diagonal Fixed Points
  14. Verification Collapse
  15. Structural Limits of Unbounded Problem Specifications
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $\mathsf{SAT}$ denote the class of propositional satisfiability instances, encoded by a primitive–recursive bijection with $\mathbb{N}$. Let $S$ be a total computable function intended to decide $\mathsf{SAT}$, and let Assume: Then there exists a SAT instance $\psi_S$ such that $S$ misclassifies $\psi_S$.

Figures (1)

  • Figure 1: Illustration: An evolving map of complexity classes, illustrating the expanding structure of the Complexity Zoo, a project of complexityzoo.

Theorems & Definitions (39)

  • Theorem 2.1: Universality Forced Impredicativity
  • Lemma 2.2: Internal Diagonalization
  • Example 2.3
  • Remark
  • Definition 2.4: Gödel-Representable Class
  • Definition 2.5: Total Classifier
  • Lemma 2.7: Existence of $P_1$
  • Theorem 2.8: Reflective Obstruction for Total Classifiers
  • Remark
  • Example 2.9: Solver's Paradox
  • ...and 29 more