Temporal Density and the Limits of Time Establishing TD_min and TD_max

This paper develops a comprehensive framework treating time as an emergent quantity inferred from the density of physical state transitions, called Temporal Density (TD). It establishes strict lower and upper bounds, TD_min > 0 and TD ≤ TD_max, to ensure continuous, causal, and finite temporal evolution while preventing singularities and divergent energy densities. The framework links TD to proper time (dτ = dt / TD), energy density (TD = k √(ρ / ρ_0) ⇒ ρ ≤ ρ_max), and gravitational effects (a = −∇TD), providing thought experiments, operational constraints, and falsifiability criteria. Applications include modeling black hole interiors, the early universe, and high-energy astrophysical phenomena without invoking unphysical temporal or spatial behavior.