Chemistry's rules carry exceptions: the octet rule, Hess's Law, detailed balance, orbital symmetry selection rules, all with disclaimers memorised separately. Their cause: a question from a richer structural level posed in the vocabulary of a simpler one, i.e. level incompleteness. This monograph makes the levels explicit, constructing a canonical tower of nine categorical levels from stoichiometry through thermochemistry, equilibrium, kinetics, electron-pushing mechanisms, stereochemistry, potential energy surfaces, and electronic structure to all-particle quantum mechanics. Each level emerges from pairs of reactions distinct yet indistinguishable at the previous level; the minimal extension resolving each ambiguity is provably unique, certified by a non-trivial cokernel in an automorphism exact sequence, and recovers Feinberg's deficiency theorems as homological corollaries. A perpendicular dimension: every ML model for chemistry (yield predictors, neural kinetic networks, equivariant force fields, learned wavefunctions) is a morphism in the Para-enrichment of one tower level, with equivariance and thermodynamic consistency as universal properties. Three incompleteness results (Eyring, Wegscheider, topological output gaps) apply to the current literature. The framework descends to code: an operational functor from a Para-enriched product of the first four levels into the Kleisli category of the probabilistic sub-monad of Haskell IO, instantiated as a simulator of the Briggs-Rauscher oscillating reaction: the first Kleisli semantics of Gillespie's next-reaction method and first Para application outside ML. The passage to all-particle quantum mechanics, Born-Oppenheimer as the classical limit of a continuous field of C*-algebras, remains the deepest open construction; four candidate conjectures including Woolley-Primas have obstructions the framework makes specific.