From Axioms to Lorentz Invariance in the Vacuum IR Regime - A Regime-Scoped Resonance Theory Derivation

This work develops a rigorous derivation of Lorentzian propagation structure in the defect-free vacuum infrared (IR) regime starting from a minimal set of structural axioms governing wave propagation. Rather than assuming spacetime geometry or Lorentz symmetry a priori, the analysis begins with a general operator description of vacuum propagation together with axioms expressing locality, wave ontology, and vacuum stability. Within an explicitly declared vacuum IR regime, the principal symbol of the propagation operator is analyzed under conditions of constant coefficients, spatial isotropy, quadratic temporal principal part, nondegeneracy, and strict hyperbolicity. These constraints uniquely collapse the admissible characteristic geometry to a Lorentzian light cone. In this sense, Lorentz invariance appears not as a fundamental postulate but as the unique propagation structure compatible with the certified vacuum IR conditions. A distinctive feature of the work is the introduction of a regime-ledger certification framework, which explicitly enumerates every structural dependency entering the derivation and classifies each as an ontic axiom, regime clause, derived consequence, or model-based postulate. This ledger serves as a consistency device that makes the logical structure of the argument transparent and audit-able: each step in the Lorentz classification can be traced to an explicit entry in the ledger, and any modification of assumptions propagates predictably through the derivation. The resulting framework yields a reproducible “regime certificate” for Lorentzian propagation in the vacuum IR regime and provides a template for analyzing how departures from the certified regime may deform the propagation cone. The approach therefore aims to clarify the minimal structural conditions under which Lorentz kinematics arises and to provide a systematic methodology for tracking assumptions in complex physical derivations.