Heat Geometry from a Universal Scalar Process

Abstract

We prove that the law of one smooth scalar process in a finite Wiener chaos completely determines finite families of symmetric tensors on arbitrary real separable Hilbert spaces, up to simultaneous orthogonal equivalence. Gaussian graph characters recover all contractions, while an intrinsic trace class Gram operator reduces the problem to finite dimensional invariant theory. Applied at a single fixed positive time to the canonical quadratic and quartic heat packet, this principle reconstructs the heat generator and forces every reconstructed unitary to be spatial. Consequently, one universal scalar process determines closed Riemannian manifolds, compact $\mathrm{RCD}$ spaces of finite dimension, and Euclidean bundles with metric connection and self adjoint potential. The heat packet functor is fully faithful, and its tensor automorphisms are precisely the geometric ones.