Abstract

This paper establishes the Isometric Extension of the Mathematical Unified Theory of Cognitional Mechanics (CM-MUT), deriving the exact quantitative correspondence between the geometric modal functor and the algebraic modal functor over the historical category Hist. The central result is that for all admissible operational histories H, the κ-scale L¹ norm and the Frobenius norm are related by the Casimir invariant K=√3 of M₃(ℂ): ‖M_A(H)‖_κ = K·‖M_G(H)‖_F. This isometric relationship is derived from two implementation axioms, A3' (Modal Norm Selection) and A5 (Modal Distance Preservation), which formalize assumptions implicit throughout the CM corpus. The proof proceeds via the automorphism structure Aut(M₃(ℂ))≅PGL(3,ℂ), Axiom 4 (Redundancy Exclusion), and a formal lemma establishing that minimal descriptive complexity in the Cartan subalgebra uniquely forces uniform eigenvalue distribution Var(κ)=0 for all admissible histories. Four corollaries follow from a single algebraic source: the inverse fine structure constant α⁻¹=137.03599…, the Iron Stability Attractor κ(Fe)=6/13, the chemical projection constant Π_chem=145/468, and the ABC quality bound q(a,b,c)<13/7. Prior CM results on the Riemann Hypothesis, P≠NP, and the Halting Problem are unified under this framework. The Navier-Stokes regularity problem, the Hodge Conjecture, the Yang-Mills mass gap, the Poincaré Conjecture, and the Birch-Swinnerton-Dyer Conjecture are shown to be Tier-3 projections of Tier-1 structural necessities. The apparent difficulty of the Millennium Problems is an artifact of their Tier-3 formulation, not of mathematical reality. All derivations proceed with zero free parameters from the axiomatic structure of M₃(ℂ). Published on April 4, 2026 DOI: 10.5281/zenodo.19411822