S-Modal: Recursive Free Will, Threshold-Based Necessity

This paper introduces a radical new approach to modal logic that replaces static Kripkean accessibility relations with recursive, threshold-based agent intervention. We define necessity and possibility through Synchronization Dynamics, allowing free-willed agents to force state transitions once their intervention strength exceeds a critical collapse threshold. By bridging recursive free will, emergent quantum parallels, and even a reimagined arithmetic of synchronization (see Appendix~Beta), S-Modal lays the groundwork for a transformative paradigm in logic, decision theory, and computation.