Gaussian Operator Algebras: Observability, Reconstruction, and Microscopic Limits

Abstract

We reconstruct Gaussian operator algebras from finite source-labelled observations, recovering coefficients, ordinary multiplication, and completion in independently specified Schatten norms. Gaussian duality and ordered contractions give a reconstruction theorem including the trace-class endpoint, while entire Mehler responses describe the resulting topology. In the Hilbert coefficient geometry, symmetric Rademacher systems realize the same algebra as a fluctuation limit: microscopic products converge at inverse site-number rate in the original norm scale, and finite decoding commutes with this limit. Relative Gram residuals determine target-dependent inverse costs. For bounded tuples, positive word observations close into completely positive maps, with contextual Schwarz defects controlling realization and multiplication. Positive mass and source innovations characterize compactness. Spectral invariants, Wick operators on closed manifolds, and Gaussian networks provide applications with explicit noise and tail bounds.