A Minimal Framework for Non-Commutative Quantum Mechanics

T.O.

Cognitional Mechanics (CM) is a non‑commutative, deterministic computational framework that treats order‑sensitive operator composition as a standalone computational resource, independent of quantum physics, probabilistic interpretation, or measurement theory. While non‑commutativity has historically been monopolized by quantum mechanics, CM restores it to its natural mathematical domain and demonstrates that it can be executed directly on classical hardware. This work presents the first complete operationalization of CM as an executable computational model. It defines semantic states as vectors embedded in an abstract manifold, introduces a minimal set of non‑commutative operators, and formalizes the Logical Leap, a deterministic convergence regulator that replaces probabilistic collapse. The paper provides a reproducible reference implementation in Python and shows how CM maps naturally onto GPU/TPU batch‑parallel execution without requiring quantum simulation or physical quantization. By establishing CM as a practical, hardware‑compatible computational system, this work positions non‑commutative computation as a new domain of computer science. It offers a foundation for deterministic semantic computation, structured exploration, and hybrid symbolic‑algebraic reasoning, and clarifies the structural distinction between CM and quantum mechanics. The framework is intended as a starting point for further theoretical development and engineering‑level applications of non‑commutative computation.

Andrew C. Millard

T.O.

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.

Larisa Jonke, Stjepan Meljanac

T.O.

Non-commutativity, first formalized in the 1920s within quantum mechanics, has historically been associated with physical quantization, measurement postulates, and Planck-scale constants. This historical association has led to the long-standing misconception that non-commutative structures are inherently tied to quantum physical systems, limiting their consideration as general-purpose computational resources. Cognitional Mechanics (CM) repositions non-commutativity as a fundamental computational property, independent of physical constants or probabilistic measurement. By treating operator sequences as order-dependent yet deterministic, CM establishes a substrate-independent framework for computation and semantic reasoning. This work traces the historical trajectory from Heisenberg's matrix mechanics to contemporary quantum computation and demonstrates how CM operationalizes non-commutative computation as a fully executable model. CM defines semantic states as vectors in an abstract manifold, introduces a minimal set of non-commutative operators, and formalizes deterministic convergence control mechanisms that replace probabilistic collapse. By providing a reproducible implementation on classical hardware and showing compatibility with GPU/TPU parallel execution, this work positions non-commutative computation as a distinct domain within computer science. CM offers a foundation for deterministic semantic computation, structured exploration, and hybrid symbolic–algebraic reasoning, clarifying the structural distinction between computational non-commutativity and quantum mechanics.

Jonas F. G. Santos, F. S. Luiz, O. S. Duarte +1 more