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 qCDIR / Cbase ≲ (Bₖ / Nfull) * (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 Cₜau, Sₜau, and DeltaTⁿ (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 Lᵢmg). The result is a principled recipe: keep quotients gentle (small Lᵢmg), stay in the interior (1/4 coefficient, barrier), certify splitting (Cₜau, Sₜau, DeltaTⁿ), and shrink boundary activity (Bₖ). 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.
Takahashi K. (Tue,) studied this question.