Axiomatic Emergence V; The Physical Origin of Noncommutative Geometry

This paper starts from two fundamental principles, information conservation andfinite-step computability, to argue for the natural emergence of the core mathematical structure of noncommutative geometry—the spectral triple. Research showsthat the spectral triple, consisting of a noncommutative C∗ algebra, a Hilbertspace, and a Dirac operator, is not an ad hoc mathematical tool but the inevitablemathematical framework for consistently describing physical systems constrainedby information conservation and simulable via finite computational steps. Thisderivation indicates that information conservation requires the algebra to possess anontrivial cyclic structure, leading to its noncommutativity; whereas computability demands that geometric information be extracted within finite resources, whichuniquely locks the spectral dimension of the Dirac operator and the real structurewith KO-dimension 6. This spectral triple with KO-dimension 6 not only uniquelyreproduces the gauge symmetry, Higgs mechanism, and gravitational coupling ofthe Standard Model but also naturally derives the structure of three generationsof chiral fermions from the dimensions of its algebraic representations. Furthermore, this theoretical framework makes two distinctive and operationally testablenew physical predictions: first, the existence of a long-lived new particle with decaylength on the kilometer scale in the tens of GeV energy range; second, the existenceof metric quantum fluctuations induced by spacetime noncommutativity, featuringa characteristic frequency spectrum in the kHz band. This paper further proposesexperimental schemes to quantitatively test these two predicted phenomena usinglarge hadron collider remote detectors (e.g., MATHUSLA) and solid-state quantum sensing platforms based on diamond nitrogen-vacancy color centers, therebyproviding a clear falsification path for this geometric unified theory.