The Axiom of Composition: From Galois Connections to the Regularity of Navier-Stokes Equations

We present a mathematical framework unifying the composition of forms across scales, from quantum phenomena to neural networks and EEG analysis. Building on the Fractal Density Activation Axiom (FDAA) ontology, we formalize the Axiom of Composition through Galois connections between complete lattices of scale profiles. The theory establishes that: (1) forms emerge through thresholded morphological operations on multi-scale density functionals, (2) backpropagation in neural networks implements a Galois adjunction preserving these forms across layers, and (3) EEG envelope dynamics obey the same compositional inequalities as hadronic molecules. This provides the first mathematical realization of Aristotle’s synolon (form-matter composite) and Kant’s synthetic a priori through computable lattice operations.