Gradient-Flow-Based Compute--Performance Trade-offs for Intelligent Systems

This paper develops a mathematically explicit framework for compute–performance trade-offs in intelligent systems whose coarse-grained dynamics can be modeled as Evolution Variational Inequality (EVI) gradient flows on quadratic Wasserstein spaces. Under a “gradient-flow universal intelligence process (UIP)” hypothesis, the work isolates structural mechanisms that constrain how far a given architecture can push performance under finite compute, rather than proposing another empirical scaling law.

The first module studies observation quotients: Borel observation maps from a UIP state space into a lower-dimensional observation space. Under a metric analogue of Riemannian submersions, the paper proves an Image–EVI theorem showing that Wasserstein EVI flows are stable under such quotients. This provides conditions under which coarse observables (embeddings, task metrics, or compressed statistics) still evolve as EVI gradient flows with the same contraction parameter.

The second module introduces a notion of preimage Minkowski dimension for localized sets of epsilon-optimal parameters in a metric parameter space, together with a resolution-dependent prototype complexity. Under an explicit geometric regularity assumption and a separate coverage-based implementability condition, any empirical scaling law of the form error ≲ N^{−α} must satisfy a structural ceiling α ≤ κ / d_pre, where κ is a local Hölder exponent and d_pre is the preimage dimension. This links approximation exponents to the geometry of near-optimal preimages, independently of a specific training algorithm.

The third module models deep residual or iterative architectures as fractal residual spines: nonexpansive limit maps decomposed into 1-Lipschitz residual updates with summable tails. The paper proves linear-in-depth truncation bounds and combines them with standard JKO time-discretization error estimates to obtain joint bounds of the form error ≲ C_disc h^γ + n ∑_{k>L} a_k. These bounds can be read as design rules for time step, iteration count, and effective residual depth in compute-limited AI systems.

Keywords: Wasserstein gradient flows, EVI, compute–performance trade-offs, observation quotients, scaling laws, preimage Minkowski dimension, prototype complexity, residual networks, JKO scheme, optimal transport, intelligent systems.