Cognitional Mechanics and Quantum Mechanics: Structural Divergence Through Tier Architecture (Third Edition)

Cognitional Mechanics (CM) establishes an axiomatic framework that formalizes the structural mechanisms of intelligence as a self-contained operational system, abstracting intelligence through non-commutative operations, convergence of semantic states, and structurally inaccessible domains, without invoking physical observables or psychological primitives. While CM exhibits a clear structural correspondence with Quantum Mechanics (QM)---most notably in its non-commutative operator structures and bounded transitions---it departs fundamentally in its treatment of discreteness. Within the standard Hilbert-space formalism, non-commutativity and spectral discreteness are mathematically independent properties: position and momentum operators, for instance, are non-commuting yet possess continuous spectra. In QM, however, the physical framework couples non-commutativity to empirical constants and measurement postulates, producing discreteness as a structural feature of the theory. CM identifies this coupling as a projection condition of the physical layer rather than a structural necessity of the algebra, and demonstrates that non-commutativity is sustainable in a continuous setting without discretization. This analysis is situated within an explicit three-tier architecture. Tier-1 (Noology) constitutes the regulative source. Tier-2 (CM) constitutes the executive operational domain, in which M₃(ℂ) is the unique minimal non-commutative substrate. Tier-3 (MUT/GUT) constitutes the projective display layer, in which physical theories including QM appear as projections of Tier-2 dynamics. CM and QM share a common Tier-1 origin without being in direct correspondence: their shared non-commutativity descends from Tier-1, while QM's discreteness is introduced at Tier-3 by projection conditions absent from Tier-2. The operational limit constant c regulates internal operability and convergence as a logical boundary condition, distinguishing it from Planck's constant ℏ, which fixes physical quantization through empirical calibration. CM thus clarifies that its resemblance to QM is structural rather than physical, and provides a vantage point from which the projection origin of QM's characteristic features is examinable.