This paper establishes the second edition of CM-MUT (Cognitional Mechanics as a Structural Template for Mathematical Unification). The first edition (January 2026, DOI: 10. 5281/zenodo. 18280992) identified mathematical structures as stabilised residues of irreversible, non-commutative operational histories, and positioned CM-MUT as a meta-theoretical explanatory layer operating above existing mathematical foundations such as set theory and category theory. The present work establishes the stronger claim. Under the axioms of Cognitional Mechanics Version 9 and the Noological Tier architecture, mathematics is not a layer above which CM operates: mathematics is the unique top-down projection of the operational structure of intelligence itself. The induced mathematical category M is uniquely determined up to identity automorphism (Aut (M) = id), and the projection map from the minimal operational structure Cₘin to M is surjective with structurally undefined inverse. Mathematics is furthermore the unique fixed point of the Pid-saturation closure operator acting on Tier-3 formal structures. Where the first edition asked why mathematical domains exhibit structural alignment, the present work answers the prior question: why mathematics, in this form and no other, exists at all. The answer, established as a theorem rather than a philosophical position, is that mathematics is the unique structural stabilisation forced by the operational constraints of any system capable of making internal distinctions. It is neither discovered nor freely chosen. The question "why is mathematics so effective? " has the same status as "why does 2+2=4? ": not a contingent feature of the world but a structural consequence of the operational constraints of any distinguishing system. Four consequences are established: the dissolution of the Platonist-formalist dichotomy; the structural impossibility of alternative mathematics; the reinterpretation of mathematical effectiveness in physics as a tautology; and the positioning of CM as The Theory of Foundation from which all formal structures are projected as Tier-3 outputs without presupposing any of them.
T.O. (Thu,) studied this question.