Motion-Based Structural Mathematics v4.0: Collapse and Persistence in Directional Systems

This paper presents a structural mathematical framework for modeling system persistence and collapse using directional motion thresholds. System persistence is defined by the accumulation of directional motion (Σ∆m), while collapse occurs when either motion ceases (∆m = 0) or acceleration exceeds a structural tolerance (∆∆m ≥ Ct). Time is not treated as a governing dimension, and no thermodynamic or probabilistic assumptions are required. The framework introduces a compression threshold (Ct) that represents the maximum rate of change a system can absorb before failure. Collapse is modeled as a binary trigger (Ke = 1), activated under structural stress. This approach provides a general-purpose method for analyzing system integrity across physical, computational, and transition-based environments, without relying on temporal or statistical constructs.