Carathéodory Extended: Thermodynamics, Geometry, and Variational Structure A Trilogy on SGCV, MC, DLSFH, and Infinity Algebra

This collected volume develops a modern extension of Carathéodory’s geometric program, unifying thermodynamics, coherence theory, and advanced variational structures. Building on the frameworks of Multifaceted Coherence (MC), the Superluminal Graviton Condensate Vacuum (SGCV), the Dodecahedron Linear String Field Hypothesis (DLSFH), and Infinity Algebra, the work reinterprets classical results—including accessibility, integrability, and umbilic geometry—through the lens of coherence-driven geometry and discrete–continuous duality. The trilogy introduces new formalisms for thermodynamic structure in quantum-gravitational backgrounds, coherence-guided diffusion on discrete dodecahedral lattices, and non-smooth variational calculus on infinite tensor hierarchies. A dedicated chapter outlines a coherence-based reformulation of the Carathéodory umbilic problem, connecting classical differential geometry with discrete and quantum geometric approaches. Together, these studies present a unified mathematical and physical framework in which coherence fields, emergent geometry, and graded tensor dynamics form the foundations of thermodynamic and geometric structure.