From the Stochastic Embedding Sufficiency Theorem to a Superspace Diffusion Framework

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, receiving only raw time series and sampling interval, is applied identically 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 $σ$-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: $\mathrm{d}g_{ij} = \mathcal{D}{ij}[g],\mathrm{d}τ+ \ell_P,\mathrm{d}W{ij}$ where all coefficients are non-parametric, first-principles consequences of the axioms. Coarse-graining of the superspace Fokker-Planck equation via Mori-Zwanzig projection yields predictions for galactic-scale gravitational acceleration testable against kinematic data.

💡 Research Summary

The paper introduces a universal, data‑driven methodology for recovering both the drift (μ) and diffusion (σ) fields of stochastic dynamical systems from a single scalar time series, without any prior knowledge of the underlying physics. The core theoretical contribution is the Stochastic Embedding Sufficiency Theorem (SEST), a probabilistic extension of Takens’ delay‑coordinate embedding theorem. While Takens guarantees a diffeomorphic reconstruction of deterministic attractors, SEST relaxes this to a measure‑theoretic injectivity condition: distinct initial states must generate distinct probability distributions over delay vectors after a finite lag τ. This condition is satisfied when Hörmander’s bracket‑generating condition holds, ensuring smooth, strictly positive transition densities, Malliavin’s non‑degeneracy of the stochastic flow, Varadhan–Léandre’s separation of transition densities, and Frostman’s measure‑geometric control of collision sets. Stone’s results on non‑parametric consistency then provide convergence rates for k‑nearest‑neighbor estimators of the conditional moments.

Building on SEST, the authors construct a three‑stage pipeline: (1) Delay embedding using Cao’s E1 statistic to select embedding dimension d and lag τ, followed by SVD‑based dimensionality reduction; (2) Local geometry estimation on the correlation manifold via KD‑tree nearest‑neighbor queries, from which Kramers–Moyal first and second conditional moments yield non‑parametric estimates μ̂(Y) and σ̂²(Y); (3) Validation through autonomous free‑run forecasting with the Euler–Maruyama scheme, measuring the fraction of true trajectories that fall within the 95 % confidence envelope of an ensemble of simulated paths.

The pipeline is applied blindly—receiving only raw time series and the sampling interval—to synthetic data from nine disparate physical domains: classical mechanics, statistical mechanics, nuclear physics, quantum mechanics, chemical kinetics, electromagnetism, relativistic quantum mechanics, quantum harmonic oscillator dynamics, and quantum electrodynamics. In every case the recovered fields match the ground‑truth equations within statistical error, and the estimated diffusion coefficients reveal a systematic pattern dubbed the “σ‑continuum.” Across the continuum, fundamental constants emerge automatically: the Boltzmann constant (k_B) in thermal regimes, Planck’s constant (ℏ) in quantum regimes, the speed of light (c) in relativistic regimes, the Van Kampen scaling exponent in system‑size expansions, and the Fano factor in counting statistics. This demonstrates that the stochastic structure of physical laws can be inferred directly from data.

A second major contribution is the Gravitational Diffusion Theorem. By combining the fluctuation–dissipation theorem, the massless mode structure of linearized gravity, and self‑coupling via the equivalence principle, the authors derive a unique gravitational diffusion coefficient D_g equal to one Planck length per square root of one Planck time (ℓ_P / √τ_P), with a dimensionless prefactor α = 1 fixed by three independent arguments. This result is embedded within a broader axiomatic framework consisting of four canonical axioms: (i) Wheeler’s superspace as the configuration space of 3‑metrics, (ii) the fluctuation amplitude dictated by SEST, (iii) classical correspondence with general relativity, and (iv) an epistemic probability interpretation of stochasticity. Within this framework the superspace diffusion hypothesis is formulated as

 dg_{ij} = 𝔇_{ij}