Compute-Optimal AI via Image--EVI, Interior Bures--HK Control, and Fractal Dendritic Approximation (DIR)

Compute-Optimal AI via Image–EVI, Interior Bures–HK Control, and Fractal Dendritic Approximation (DIR) presents an audit-ready theoretical framework for reducing training and inference compute in modern AI systems without degrading performance. The work unifies (1) image quotients where Evolution Variational Inequalities (EVI) transfer under length-submetry with epsilon-lifts and fiberwise inf-projection; (2) interior control of a fibered Bures–Hellinger–Kantorovich (HK) entropy–transport geometry, including a unique reaction coefficient 1/4 fixed by convex duality; and (3) Fractal Dendritic Approximation (FDA), which certifies boundary-dominated compute with attenuation q<1. We derive a closed-form DIR compute ratio that combines boundary size, slack, and curvature degradation: C_DIR / C_base ≲ (B_k / N_full) * (1 − eta − rho)^(−1/p) * L_^2 with p = (alphabeta)/(alpha+beta). Here alpha and beta are the empirical scaling exponents for parameters and tokens, L_* is a robust image Lipschitz proxy, and (eta, rho) encode slack from acceptance tests. The paper defines measurable audit monitors C_tau, S_tau, and Delta_T^n (normalized), gives acceptance thresholds tied to Strang splitting and BCH constants, proves EVI transfer and epsilon-complexity monotonicity under pushforwards, and states a sufficient condition for 10x compute reduction under common scaling (e.g., alpha=beta=1). Mapping to practice is provided for distillation/MoE (quotient observation), pruning/structured sparsity/low-rank (FDA), quantization (fiber perturbations with barrier), and paging/ZeRO (locality controlling L_img). The result is a principled recipe: keep quotients gentle (small L_img), stay in the interior (1/4 coefficient, barrier), certify splitting (C_tau, S_tau, Delta_T^n), and shrink boundary activity (B_k). This manuscript is theoretical; measurement protocols and reference implementations of the audit monitors will accompany follow-up work. Keywords: compute reduction, scaling laws, entropy–transport, Hellinger–Kantorovich, Bures metric, Petz monotone metrics, EVI, gradient flows, length-submetry, quotient maps, image Lipschitz, Strang splitting, BCH expansion, pruning, structured sparsity, low rank, distillation, mixture of experts, quantization, auditability.