Axiomatic Categorical Curvature Theory: Curvature, Invariants, and Structural Rigidity

This paper extends Axiomatic Universal Second Variation (AUSV) into its categorical manifestation, Axiomatic Categorical Curvature Theory (ACCT), demonstrating that categorical structures uniquely express invariants from the intrinsic refinement of AUSV's classifier E. Curvature emerges as an obstruction to functorial flattening, inducing a canonical hierarchical filtration and universal categorical objects. No categorical framework can alter invariant content; admissible representations remain closed under refinement and equivalent via canonical natural transformations. ACCT constructs the category Rep(AUSV) of representations, proving it uniquely determined up to equivalence by AUSV's organizational rigidity. Intrinsic AUSV curvature projects to universal factorization properties in ACCT, with E mapping to a unique factorization functor. Refinements yield adjunction-like structures, and invariants correspond to natural equivalences between functors. Thus, ACCT is not a reformulation but the constrained categorical shadow of AUSV, where universality embodies intrinsic curvature and natural transformations embody invariants. The theory establishes representational equivalence as a structural necessity, independent of categorical assumptions.