Phenomenology of Scale-Dependent Spacetime Regularity: Cusp Interpretation of Strong-Field Gravitational Regimes

This work presents a mathematically rigorous phenomenological framework for describingstrong-field gravitational regimes as scale-dependent loss of geometric regularity,rather than as point-like singularities in classical spacetime. Building on the structure of metric measure spaces ((M, d, μ)), the paper introducesscale-dependent regularity descriptors, explicit curvature proxy functionals with smoothlimits, and invariant formulations via geodesic dispersion and heat-kernel analysis.It defines a “cusp regime” as the simultaneous collapse of local regularity and apower-law growth of proxy curvature at small scales, and it provides a minimal flowmodel for scale evolution. The work also includes a map between these theoreticaldescriptors and observable gravitational-wave signatures, making the framework fullyfalsifiable with current and future data. No modification to Einstein’s equations is assumed; instead, the approach reframesclassical singularities in operational terms compatible with observational constraintsand broadens the mathematical language available for strong-field gravity. **Keywords:** scale-dependent geometry, metric measure spaces, cusp regime,proxy curvature, gravitational waves, quasinormal modes, spectral regularity,heat kernel, phenomenological model