Thermodynamics of Coherence: Quantum Ultrametric Rigidity and the Informational Origin of Geometry

Thermodynamics of Coherence (ToC) completes the conceptual arc initiated by Hamiltonian–Lagrangian Rigidity (HLR), developed through Quantum Ultrametric Rigidity (QUR) and Geometry of Coherence (GoC). It establishes the thermodynamic foundation of spacetime coherence, where information, geometry, and rigidity converge into a single statistical law. The universe is described as an informational ensemble governed by the invariant S · ρrig ≃ MP⁴, which unites entropy, rigidity, and Planck-scale finiteness into the first law of informational thermodynamics. Gradients of coherence generate curvature, while the conservation of the product S * ρrig preserves the total informational content of the cosmos. Geometry thus emerges as the visible equilibrium between order and entropy — a self-regulating fabric finite in information yet infinite in extension. At the microscopic level, Quantum Ultrametric Rigidity provides the hierarchical architecture of coherence; at the macroscopic level, it manifests as isotropy and curvature through the informational field equations derived herein. The interplay between χ (coherence) and θ (torsional memory) extends statistical mechanics to topological phases, revealing that spacetime is not a container of information but the very process by which information becomes geometry. This paper closes the Rigidity Program’s foundational cycle (HLR → QUR → GoC → ToC), defining the Thermodynamics of Coherence as the informational bridge between quantum discreteness and geometric continuity — the physics of meaning itself.