Cognitional Mechanics (CM) establishes an axiomatic framework that formalizes the structural mechanisms of intelligence as a self-contained operational system. Unlike approaches grounded in physical dynamics or engineering implementations, CM abstracts intelligence through non-commutative operations, convergence of semantic states, and structurally inaccessible domains, without invoking physical observables or psychological primitives. The first edition (DOI: https://doi.org/10.5281/zenodo.17994360) established the structural correspondence between CM and QM; the second edition (DOI: https://doi.org/10.5281/zenodo.18062274) strengthened the distinction while preserving the correspondence. This third edition extends the analysis through two additions: the explicit Tier architecture locating QM within Tier-3 as a projection of Tier-2 CM structure, and a structural account of the non-commutativity--discreteness coupling in QM. While CM exhibits a clear structural correspondence with Quantum Mechanics (QM)---most notably in its reliance on non-commutative operator structures and bounded transitions---it departs fundamentally in its treatment of difference and discreteness. Within the standard Hilbert-space formalism, non-commutativity and spectral discreteness are mathematically independent properties, as established in non-commutative geometry: position and momentum operators, for instance, are non-commuting yet possess continuous spectra. In QM, however, the physical framework couples non-commutativity to fixed empirical constants and measurement postulates, producing discreteness as a structural feature of the theory. CM identifies this coupling as a Tier-3 projection condition rather than a Tier-1 structural necessity, and demonstrates at Tier-2 that non-commutativity is sustainable without discretization. Within this framework, intelligence is mathematically tractable and substrate-independent, analyzable without external observation or measurement. The operational limit constant c does not function as a physical quantum but as a logical boundary condition governing convergence and phase transition. This distinguishes c from Planck's constant ℏ: the former regulates internal operability and accessibility, while the latter fixes physical quantization through empirical calibration. The three-tier architecture formally positions CM as follows. Tier-1 (Noology) constitutes the regulative source, specifying the conditions under which a structure qualifies as real. 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. By defining a metric space of semantic states with distance functions satisfying non-negativity, symmetry, and the triangle inequality, CM ensures formal rigor and internal closure. The variability of semantic differences allows CM to reconcile non-commutative structure with classical continuity, thereby avoiding probabilistic collapse or observer-dependence. Inaccessibility and convergence are determined entirely by internal operational constraints. CM thus clarifies that its resemblance to QM is structural rather than physical, and locates that resemblance within a broader Tier architecture from which the projection origin of QM's characteristic features is examinable. The framework generalizes non-commutative formalisms into a domain where intelligence is modeled as a geometric-operational process, not a quantum phenomenon, while maintaining a strict distinction from physical quantum theory.
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