This paper introduces Axiomatic Curvature–Variational Theory (ACVT), a unified geometric framework rooted in Axiomatic Second–Variation Geometry (ASVG), wherein computational complexity, algorithmic information, and thermodynamic entropy emerge as invariants of curvature in finite realizations under second variations. Computation is reframed as a variational process stabilized by minimal curvature density in finite-bias diagrams, recovering the hierarchy P ⊊ NP ⊊ PSPACE as an intrinsic stratification of curvature modes: P via trace curvature (κₜr), NP via trace plus traceless-coherent (κₜr + κₜc), and PSPACE via full skew addition (κₜr + κₜc + κₛk). These components dictate both computational cost and informational compressibility, with Kolmogorov complexity equating to minimal curvature density, randomness to irreducibility, and entropy to degeneracy in realizations. Mutual information reflects curvature density reduction through joint stabilization, yielding thermodynamic correlations structurally. Algorithmic, informational, and statistical mechanics are thus curvature projections of a single variational structure, invariant under models, encodings, or proofs. Separations like P ≠ NP arise from non-embeddability, establishing ACVT as the origin of computational and informational existence, with hierarchies closed under composition and representation-independent.
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