The Cognitive--Algebraic Isomorphism: A Structural Equivalence Between Multiplicative Group Spectra and the Law of Cognitive Phases (CPL 4.0)

This preprint presents an independent theoretical result establishing a structural equivalence between algebraic group invariants and the cognitive phase geometry defined by the Law of Cognitive Phases (CPL 4.0). The analysis focuses on normalized Euler–totient spectra of multiplicative groups modulo n and demonstrates that their entropy, maximal component, and induced phase classification coincide exactly with the invariant criteria used in CPL 4.0. The work is purely structural and does not introduce new dynamics, operators, or extensions of the CPL framework. Its purpose is to clarify the invariant foundations of CPL 4.0 by exhibiting an explicit algebraic realization, thereby situating cognitive phase classification within a broader mathematical context. The paper is self-contained and independent of subsequent developments of the Law of Cognitive Phases.