Infinite-Dimensional Levy Area: Probability-Selected Critical Geometry and Sharp Spectral Selection

Abstract

We develop a critical second-order geometry for infinite-dimensional Lévy area. A faithful exterior representation selects a canonical scale of graded state spaces, while stochastic integration selects its universal critical member. Modulo zero energy, the operational kernel space is isometric to the range of Itô integration in $L^2$, and the same topology quantitatively controls the full second-order lift, yielding stability of projection, Galerkin, semigroup, coefficient, and parameter approximations. The exterior representation also gives an exact covariance calculus. For revealed predictable brackets, the conditional exterior covariance satisfies a fermionic Doléans law, whose degree-two component identifies the precise atomic defect in Lévy-area covariance. The universal area regularity is sharp weak trace class, with logarithmic Ky Fan growth and no finite Lorentz refinement. Retaining the full bracket spectrum yields finer sharp Schatten classes; for polynomial semigroup profiles the critical exponent is the reciprocal of the generator-growth exponent plus twice the noise-decay exponent.

16 pages