This paper establishes a rigorous and constructive extension of the Atiyah–Singer index theorem to broad classes of integral operators, including those with non-polynomial symbols, oscillatory kernels, and non-local effects. We introduce a novel filtered differential algebra Kgenint, the generalized index algebraic closure for integral operators, which provides a unified framework for certified representations of such operators, their symbols, and associated regularization procedures. Within this framework, we develop explicit index formulas accompanied by certified,computable error bounds. Our main contributions are: (1) a systematic classification of integral operators based on kernel properties and symbol behavior; (2) the construction of Kgenint with its certified symbolic calculus and error propagation rules; (3) derivation of index formulas for elliptic, weakly elliptic, and non-Fredholm integral operators, including singular integral and Fourier integral operators; (4) certified criteria and algorithms for determining Fredholm properties; (5)development of equivariant index theory for group-invariant operators; (6) detailed constructive algorithms with complexity analysis and guaranteed error control; and (7) validated applications in image inpainting, quantum field theory, and geophysical imaging. This work bridges abstract index theory with computationally realizable, certified numerical methods, providing a mathematically sound foundation for index computations in scientific applications.
shifa liu (Wed,) studied this question.