On a connected symplectic manifold (M, ω) equipped with a family of surjective Poisson maps (coarse-grainings), we define a star product to be coarse-graining stable if it pushes forward to an equivalent star product under every coarse-graining. We prove that coarse-graining stability forces the characteristic class in the Fedosov–Kontsevich classification to be trivial, so that the star product is equivalent to the standard Moyal product with a single global deformation parameter (Theorem 5.1). The proof uses the Hochschild–Kostant–Rosenberg isomorphism to show that a non-degenerate Poisson bivector cannot be a Hochschild coboundary, which forces the local deformation parameters on overlapping Darboux charts to agree. Combined with the Stone–von Neumann theorem and Gerstenhaber rigidity (H² = 0), this yields: on any finite-dimensional Darboux chart, the unique irreducible separable quantization of a non-degenerate symplectic structure is the standard Weyl algebra q, p = icδ with a universal constant c (Theorem 6.1). The value of c is not determined by the formalism. Applications to coarse-graining in statistical mechanics, renormalization, and foundational programs in quantum gravity are discussed.
Marcus Schmieke (Fri,) studied this question.