Umbilic Thermodynamics

This paper develops umbilic thermodynamics, a generalized thermodynamic framework based on the conservation of dimensionless, scale-invariant Weyl information across a non-orientable Möbius topology. Within a conformal projective geometric algebra (PGA) formulation, probability-distribution shapes and pseudo-Riemannian metric shapes are interpreted as grade-dual projections of a single underlying structure, linking microscopic phase entropy with macroscopic metric stress-energy. A corresponding Weyl temperature is introduced to characterize the conversion between these informational and geometric manifestations. The framework formulates four principles of umbilic thermodynamics concerning symplectic-capacity equilibrium, global Weyl-information conservation, topological entropy directionality, and the unattainability of global absolute zero. A singular Legendre transformation at the Gromov symplectic boundary provides a local “backdoor” through which phase-space compression and negative informational temperatures can produce localized metric stress-energy, while global Weyl-information conservation prohibits net creation ex nihilo. The resulting generalized umbilic ergodic theorem identifies local “creatio ex nihilo” events as recurrent, measure-preserving transfers between dual phase sheets. Classical thermodynamics and quantum thermodynamics are recovered as limiting projections of the generalized framework, while a statistical formulation is developed in Boltzmannian, Gibbsian, and Einsteinian editions. The paper further proposes falsifiable experimental tests involving entangled quantum systems, cryogenic cavities, superconducting qubits, and non-orientable photonic structures. Thus, umbilic thermodynamics is presented as a unified topodynamical framework connecting information, entropy, quantum phase structure, spacetime curvature, and thermodynamic evolution.