Heat Geometry from a Universal Scalar Process

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
$\operatorname{RCD}$ 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.