Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invariant Entropy, and Directional Degrees of Freedom
We give mathematically self-contained formulations, in the complex-time (kime) representation, of three open problems from the foundations of classical mechanics: (I) the extension of the classical entropic uncertainty principle to non-canonical variables and to multiple degrees of freedom; (II) the characterization of coordinate-invariant measures and entropies, i.e., the question of why continuous physical quantities must be paired for an invariant entropy to exist; and (III) the construction of a classical relativistic directional degree of freedom (a classical analogue of a spin-1/2 system). Throughout, the kime phase is interpreted {statistically as a latent circular random variable whose law Φmodels the intrinsic trial-to-trial variability of repeated, identically controlled experiments indexed by the kime magnitude. The mathematical bridge is an exact symplectic identification of the kime cone with the action-angle chart of a one-degree-of-freedom phase space, under which the kime measure is the Liouville measure and the phase law becomes the angular conditional of a Liouville density. Specifically, we (i) prove a sharp entropic uncertainty relation on the kime cylinder whose extremal family is von Mises x Gaussian, together with a sharp circular Fisher-information inequality saturated exactly by von Mises laws; (ii) prove an exact non-canonical uncertainty relation in which the correction term is the geometric mean of the Poisson bracket, clarifying the conjectured role of the expected bracket; (iii) prove aggregate multi-degree-of-freedom bounds via the Williamson normal form and Fischer's inequality, and isolate the per-degree-of-freedom refinement as a precise open problem of symplectic Schur-Horn type; (iv) prove that diffusion of the kime phase produces monotone entropy growth with the equipartitioned (Haar-uniform) phase law.