The Internal Language of M3(C) Structure within Cognitional Mechanics: Cyclotomic Polynomials as the Necessary Formula for the Derivation of Physical Constants

This work establishes the Internal Language Theorem for the algebra M3 (C) within the framework of Cognitional Mechanics (CM). CM derives all four Standard Model dimensionless coupling constants with zero free parameters, but the mechanism by which M3 (C) generates the structural constants used in these derivations has remained implicit. In particular, the repeated appearance of the cyclotomic evaluation values Φ1 (3) =2, Φ2 (3) =4, Φ3 (3) =13, and Φ6 (3) =7 has lacked a formal explanation. The present paper demonstrates that these values arise necessarily from the axioms A1–A4 defining the M3 (C) Structure. Using the Cayley–Hamilton theorem, the evaluation point x=n=3 is shown to be uniquely determined by information completeness and redundancy exclusion. The minimal noncommutative realization of A1 yields the circulant matrix C3, whose characteristic polynomial generates Φ3. The Cartan reflection H↦−H produces its dual Φ6. The linear factors Φ1 and Φ2 arise from the terminal Cayley–Hamilton polynomials x3±1. All other cyclotomic polynomials are excluded by degree closure or by the integer-eigenvalue condition derived from A4. The result is a fully self-generated internal language: the set Φ1, Φ2, Φ3, Φ6 is complete, unique, and necessary. This mechanism has no analogue in conventional physics or in Connes’ noncommutative geometry, where M3 (C) functions only as a carrier of particle ontology. In CM, the algebra itself is the origin of physical constants through the self-evaluation formula Φn (n).