This paper establishes a comprehensive algebraic closure framework for integral geometry, extending classical theory from homogeneous spaces to nonlinear, nonsmooth, and fractal geometric spaces. We define the Integral Geometry Algebraic Closure KIntGeo and the Kinematic Formula Algebraic Closure KKin, structures constructed through a recursive adjunction process that incorporates geometric objects, invariant measures, curvature measures, and integral transforms. Within this framework, we prove that integral invariants and kinematic formulas admit unified algebraic representations while preserving the underlying algebraic and geometric structures. The framework is further extended to nonsmooth manifolds, fractal sets, high-dimensional and infinite-dimensional manifolds, establishing corresponding representation theorems and computational algorithms. We develop efficient algorithms with certified error bounds and provide a rigorous validation framework based on interval arithmetic and discrete integral geometry. Additionally, we explore applications of this framework in geometric deep learning, stochastic geometry, and physical geometry (including general relativity and quantum field theory), demonstrating its cross-disciplinary connecting potential. This work demonstrates that within appropriately constructed algebraic closures, integralgeometric objects admit explicit algebraic representations, providing a new algebraic perspective for geometric computation while maintaining consistency with classical theory.
shifa liu (Wed,) studied this question.