μ ≈ 0.5 as a Critical Instability Threshold: From Early Modern Certainty to a Graded Model of Paradox Stabilization

This paper develops a formal graded instability framework for analyzing paradoxical and self-referential structures. Departing from classical binary logic and the principle of non-contradiction, we introduce a continuous truth valuation μ: Ω → [0,1]. An instability functional K(P) = 4μ(P)(1 − μ(P)) is defined, attaining its maximum at μ ≈ 0.5 — interpreted as a regime of maximal internal tension. To regulate this regime, we introduce a refusal operator R(P) that attenuates synthesis-prone configurations and a helical transformation operator H that recursively reorganizes states through bounded inversion and memory. A precise Exit Condition ensures finite convergence when instability and variation fall below thresholds. We prove bounded convergence and non-divergence for a class of contradictory propositions. The model is evaluated on the Liar paradox and extended to hallucination control in large language models. Finally, we discuss how the μ ≈ 0.5 critical point reframes the epistemic limits in Descartes and Kant within a graded post-dialectical framework. 5. Keywords graded logic, paradox resolution, refusal operator, helical transformation, instability threshold, post-dialectical model, fuzzy logic, AI ethics, Liar paradox, hallucination control