Synthetic Machine Learning

We present a research program for synthetic machine learning grounded in (i) Markov categoriesfor stochastic morphisms, (ii) information geometry extended beyond the Fisher–Raosetting via pseudo-Finsler structures endowed with H-functions, and (iii) noncommutativegeometry through spectral triples and pre-Finsler C∗-modules. The program unifies statisticalestimation, model selection, and stability via categorical morphisms and geometricdivergences. Key contributions are: (1) a categorical formulation of information criteria(IC) estimators as morphisms and mixtures thereof, with consistency/efficiency/stabilitynotions made explicit, organized within a 2-category ICSel of IC selectors and expressed asa coend / left Kan extension over a category of loss functionals; (2) an extension of classicalinformation geometry to (pseudo-)Finsler manifolds with α-sprays and dual-flat structuresinduced by regular divergences, accompanied by an explicit α-Christoffel-symbol notation forH-function pseudo-Finsler manifolds; (3) a bridge to noncommutative information geometryby introducing degenerate pre-Finsler structures on spectral triples, yielding a generalizedstatistical manifold category. We provide toy instances illustrating how weighted IC mixturesoperate on simple regression models, sketch operator-algebraic implications for quantum-inspiredlearning, and derive an α-spray-corrected stochastic gradient update as a directconsequence of the Christoffel formulation.