Hierarchical Self-Referential Reconstruction Cosmology D: Joint Distribution of Cosmic Mathematical Abundance and Information-Computation Constraints and Their Observable Signals

This paper, within a unified theoretical framework, explores the intrinsic relationship between the ”mathematical abundance” of a universe—i.e., the extent towhich its physical laws can be described by formal mathematical structures—and its”information-computation constraints”—namely, information conservation and thecomputability of physical processes. The logical boundaries of mathematical abundance are delineated by fundamental theorems in the foundations of mathematics,while the information-computation constraints are classified into four discrete levels. By constructing a potential landscape model based on information geometryand statistical manifolds, this paper derives the probability density function describing their joint distribution. The analysis shows that universes possessing highmathematical abundance and strong logical constraints are extremely rare in themeasure-theoretic sense, while low-abundance, weakly-constrained universes dominate. This model further predicts that such logical structures leave observableimprints in the early universe, manifested as specific non-Gaussianity in the cosmicmicrowave background radiation, distortions in the shape of the primordial gravitational wave power spectrum, and temperature steps in the 21-cm backgroundradiation. These predictions provide a clear path for testing the mathematical andlogical foundations of the universe using next-generation astronomical observations.