Macro Integer Theory: Number-Theoretic Statistical Mechanics and Critical Phenomena --- Universality and Scaling from Number-Theoretic BEC and Hessian Degeneracy ---

his paper aims to connect macro quantities introduced for integer mechanics (typified by Collatz-type maps) to the standard language of statistical mechanics and critical phenomena, and thereby provide a universal coarse-graining theory for number-theoretic objects. The central idea is to make explicit the map from micro descriptions (complete configurations via ABN/CPE) to macro descriptions (degrees of freedom, total carry, symmetry entropy, and related quantities). Starting from counting the number of microstates that realize fixed macro quantities, we build a thermodynamic picture as a construction rather than an assumption. Structure of the paper. The paper consists of three parts. Part I: We systematically organize the correspondence between integer mechanics and statistical mechanics, step by step, from the microcanonical picture to the canonical picture, and further to the (genuine) grand-canonical picture. Part II: We classify generalized Collatz-type maps and organize the macro quantities that should be fed into Part~III (effective degrees of freedom, local interactions, scale separation, etc.). Part III: Based on the Two-scale viewpoint (keeping the degrees of freedom $\mu$ and the interaction distance $R_{\max}$ as independent quantities), we present one-parameter scaling, universality, effective dimension, critical exponents, and a route to class classification.