Unified Analytic Finite Representation Theory in Differential, Integral, Difference, and Stochastic Operators: A Bidirectional Closure Framework

This paper establishes a unified analytic finite representation theory that encompasses a vast class of operator equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), exterior differential equations (EDEs), total differential equations (TDEs), Fredholm and Volterra integral equations of the first, second and third kinds, singular integral equations (Cauchy principal value, Abel type), convolution equations (Wiener–Hopf), exterior integral equations (the inverse of exterior derivative), total integral equations, difference–integral (summation–integral) hybrid equations, and stochastic integral equations of Itô type (with Wiener, Poisson jumps, and fractional Brownian motion).We prove bidirectional equivalence, construct all closures with minimality properties, provide explicit combinatorial formulas, convergence estimates, interval arithmetic error bounds, and a complete implementation framework. More than thirty physical examples (quantum scattering, population dynamics, image processing, fracture mechanics, Abel inversion, Wiener–Hopf, exterior calculus, discrete-continuous demography, stochastic volatility, etc.) validate the theory. We also resolve several open problems (correct combinatorial coefficients, Borel summability for Gevrey data, complexity lower bound for backward mapping,neural network approximation of high-dimensional coefficients) and state remaining challenges (quantum homotopy, non-smooth kernels, fractional closures, integral Galois group computation).