The Conformal Thermodynamic Cycle: Eradicating Singularities, the Information Loss Paradox, and Cosmic Inflation via Holographic Protocol Handovers

This paper addresses the pathological infinities of the early universe by applying the thermodynamic boundary constraints of the Tri-Scale topology to the primordial cosmological timeline. By discarding the assumption of unbounded thermal capacity and treating the initial expansion as a mathematically bounded state, we provide a deterministic framework that resolves the classical initial-value singularity of the Big Bang. Key Breakthroughs: Non-Lipschitz Initial Conditions: Replaces the singular $t=0$ boundary condition with a non-Lipschitz fractional scaling law (homologous to Norton's Dome), allowing for spontaneous, deterministic symmetry breaking without requiring infinite temperature or infinite density. Bounded Thermal Capacity: Utilizes the $N=4$ particle spectrum truncation established in previous work to strictly cap the available thermodynamic degrees of freedom, formally preventing the formation of an infinite-entropy thermal singularity. Picard-Lindelöf Resolution: Demonstrates how the failure of the Lipschitz continuity condition at the absolute geometric minimum ($L_P$) allows a continuous scale factor to emerge deterministically from a finite quiescent state, satisfying local energy conservation without requiring ad-hoc inflationary potentials. This formulation demonstrates that the universe did not originate from an unphysical mathematical point of infinite density, but rather emerged via a continuous, boundary-regulated thermodynamic phase transition.