From the Stochastic Embedding Sufficiency Theorem to a Superspace Diffusion Framework

Abstract

A generalisation of Takens' delay-coordinate embedding theorem to stochastic systems, the Stochastic Embedding Sufficiency Theorem, is an inverse methodology enabling non-parametric recovery of both drift and diffusion fields from scalar time series without prior assumptions about the governing physics. A blind protocol using only time series data is applied to nine domains: classical mechanics, statistical mechanics, nuclear physics, quantum mechanics, chemical kinetics, electromagnetism, relativistic quantum mechanics, quantum harmonic oscillator dynamics, and quantum electrodynamics. Fundamental constants (the Boltzmann constant, the Planck constant, the speed of light, the Fano factor, and the Van Kampen scaling exponent) emerge in both drift and diffusion channels without prior specification. The recovered diffusion coefficients, viewed across domains, constitute an empirical pattern, the $\sigma$-continuum, in which $k_B$, $\hbar$, and $c$ play structurally distinct roles. The Gravitational Diffusion Theorem, derived from the fluctuation-dissipation theorem, massless mode structure of linearised gravity, and gravitational self-coupling via the equivalence principle, determines the gravitational diffusion coefficient as one Planck length per square root of Planck time. Four canonical axioms formalise the framework, within which the noise character, drift, covariance operator, and fluctuation amplitude are uniquely determined by theorem, yielding the superspace diffusion hypothesis: $\mathrm{d}g_{ij} = \mathcal{D}_{ij}[g]\,\mathrm{d}\tau + \ell_P\,\mathrm{d}W_{ij}$ where all coefficients are non-parametric, first-principles consequences of the axioms. An implication of the hypothesis is that coarse-graining of the superspace Fokker-Planck equation via Mori-Zwanzig projection yields predictions for galactic-scale gravitational acceleration testable against kinematic data.