DFIM Companion Note 4: Mathematical Formalization of the Soft Breach of the Equivalence Principle—Deriving Frequency-Dependent Horizon Dynamics (Version 1.0)

This companion note provides the rigorous microscopic derivation of the horizon dynamics in the Dual-Field Interface Model (DFIM). It formalizes the "Soft Breach" of the Equivalence Principle proposed in Note 3 by deriving a Modified Dispersion Relation (MDR) for the spacetime weld. Connecting the resonant EM feedback loop (from EM Extension v1.2) to the geodesic equation, we derive an effective dispersion relation for test particles: E2=p2+m2+f(κ,ω) We demonstrate that the interaction term f(κ,ω), modeled as a Lorentzian resonance, creates a regime-dependent horizon: Macroscopic Regime (High ω/E): The interaction term vanishes (f→0), restoring standard General Relativity geodesics and preserving the Equivalence Principle for infalling matter (no Firewall). Wave Regime (Low ω): The interaction dominates near the resonance frequency (ω≈ωres), creating an effective potential barrier that reflects low-frequency perturbations, generating the Gravitational Wave Echoes predicted by the model. This derivation proves that the "Weld" acts as a frequency-dependent filter, unifying the mechanical, geometric, and quantum aspects of the theory without violating Lorentz invariance in the macroscopic limit. Keywords DFIM, Equivalence Principle, Modified Dispersion Relation, Quantum Gravity Phenomenology, Horizon Dynamics, Gravitational Wave Echoes, Lorentz Invariance, Effective Field Theory, Lorentzian Resonance. Related Identifiers Is Supplement To: [DOI of DFIM v3.3] Is Supplement To: [DOI of EM Extension v1.2] Is Supplement To: [DOI of Note 3]