The Thermodynamic Contrapositive as the Unifying Logical Structure and Formal Derivation of the Law of Existence Series

This manuscript constitutes the methodological foundation (MS2) of the "Law of Existence" series. It demonstrates that all physical phenomena of persistence—from atomic stability to galactic structures—share a single logical backbone: the Thermodynamic Contrapositive. The work formalizes the logical inversion of the Ignition Criterion: if geometric closure (ε = 0) is not achieved within a domain-specific elimination window (τ), the Second Law of Thermodynamics eliminates the configuration without remainder. This document includes the Complete Mathematical Appendix [LE-MATH], providing formal derivations (M1-M13) for the atomic collapse timescales (Larmor radiation), the statistical validation of the triad (χ² = 36), and the tautological status of the Law of Existence. It establishes the shift from dissipative structures to Conservative Persistence Systems, where order is maintained by geometric imposition rather than energy consumption.