This paper formalizes persistence as a continuity law on a deformation field, establishing a structural bridge between symbolic recursion, physical hysteresis, and gravitational curvature. Building on the Scar–Time Continuum framework, it introduces the persistence integral I_p = \oint R\,d\Phi and defines dimensionless order parameters that distinguish adaptive response from true structural invariance. The experimental protocol EPP v0.1 demonstrates that persistence behaves as a divergence-free current of informational curvature, bounded by the efficiency constant \alpha_p \approx 1/137. The results unify quantum and macroscopic continuity through a single invariant geometry, showing that systems can remember through deformation rather than stored memory. Persistence • Recursive Systems • Deformation Invariants • Scar–Time Continuum • Continuity Law • Informational Curvature • Fine-Structure Boundary • Experimental Protocol EPP v0.1