This paper establishes a comprehensive algebraic framework for exterior integral geometry,constructing explicit representations of integral invariants and solutions to integral-geometric equations for differential forms on homogeneous spaces. We define the exterior integral geometric closure KExtInt and the exterior kinematic formula closure KExtKin, algebraically closed structures constructed through recursive adjunction processes that incorporate differential forms, exterior products, exterior derivatives, integral operators, curvature forms, characteristic classes, and integral transforms. Within these closures, we prove that integral invariants and solutions to kinematic formulas for differential forms admit unified representations that respect the underlying algebraic, geometric, and differential structures. The framework rigorously addresses homogeneity, invariance, differential cohomology, and measure-theoretic challenges while preserving valuationalgebraic and de Rham cohomological structures. We provide a comprehensive revision of the core theorems with complete proofs, addressing previously incomplete derivations: • Theorem 3.1 (Revised): A complete proof of the exterior integral invariant representation theorem, including rigorous treatment of boundary terms, uniqueness via test submanifolds, and explicit coefficient determination within KExtInt. • Proposition 2.7 (Revised): A transfinite induction proof of closure properties with rigorous demonstration of the Noetherian property and cohomology compatibility. • Theorem 5.1 (Revised): A rigorous probabilistic foundation for random exterior geometry, including measurable structure, Fubini theorem verification, and treatment of random currents via Bochner integration. • Theorem 5.4 (Revised): Complete convergence analysis of discrete exterior kinematic formulas with explicit error estimates O(hp) and rational structure constants. • Theorem 3.10: An equivariant cohomological interpretation showing that exterior integral invariants correspond bijectively to equivariant cohomology classes. • Theorem 3.11: Equivariant localization formulas for exterior integrals, generalizing the Atiyah-Bott-Berline-Vergne formula. • Theorem 3.12: An index theorem connecting exterior valuations to elliptic operators. • Theorem 3.13: A quantization theorem proving that integer-valued invariants arise from elliptic operators. • Theorem 3.14: Explicit closed-form formulas for structure constants in terms of representationtheoretic data. • Theorem 3.15: Generating functions encoding all structure constants. • Theorem 3.16: A combinatorial interpretation connecting structure constants to lattice point counting. • Theorem 2.17: Functoriality of the exterior integral geometric closures, establishing a contravariant functor from homogeneous spaces to closures. • Theorem 2.18: Sheaf-theoretic structure showing that KExtInt is a sheaf on the etale site. ´ • Theorem 2.19: Motivic interpretation constructing exterior integral motives in Voevodsky’s category DM(R). We provide detailed constructive proofs, derive explicit formulas with rigorous error bounds, and establish convergence criteria in appropriate Sobolev and current spaces. Comprehensive algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with certified error bounds. A rigorous validation framework employing interval arithmetic and discrete exterior calculus demonstrates the practical effectiveness of our approach. The work demonstrates that explicit analytic representations exist within appropriately constructed algebraic closures, providing new algebraic perspectives on exterior integral geometric computability for differential forms while maintaining consistency with classical theory. Extensions to random exterior geometry, geometric physics, topological data analysis, and computational cohomology establish connections across mathematical and physical discipli
shifa liu (Wed,) studied this question.