Symbolic Manifolds and Transform Closure: A Geometric Framework for Operator-Invariant Structure

We introduce a geometric framework in which classical transforms are represented as coordinate charts on a symbolic manifold. The construction defines symbolic curvature (κ), strain (τ), compressibility (σ), and the ratio Γ = κ/τ, which together provide a diagnostic coordinate system for comparing representational stability across chart transitions. Within this setting, transforms such as Fourier, Laplace, wavelet, Jordan, and polynomial projection can be treated as charts connected by transition maps that preserve Γ on specified domains. We also introduce a symmetric positive-definite metric tensor Gab to quantify displacement in the invariant coordinates and to formalize minimal-effort paths (geodesics) under modeling assumptions stated in the text. The resulting framework provides a reproducible screening method for evaluating transform stability, diagnosing closure failure, and comparing transform behavior under a shared set of invariants.