Restricted Dynamic Geometric Complexity: Certificates for Structured Preconditioning

Optimization geometrodynamics views optimizer state as evolving geometry.

Its full positive-definite quadratic benchmark gives the least affine-invariant deformation needed to reduce condition number when arbitrary metrics are allowed.

This paper records that benchmark in the present notation and develops restricted dynamic geometric complexity: an intrinsic certificate distance for reaching a target condition-number class when the metric is restricted to a specified family.

The main proved results are monotonicity and submanifold-distance principles, diagonal and block reachability as linear matrix inequality feasibility problems, an exact two-dimensional diagonal complexity formula, and affine-invariant Kronecker projection theorems with normal equations, computable mismatch certificates, Armijo solver convergence, auxiliary self-conditioned K-target bounds, and Hessian-relative candidate certificates through an exact Kronecker Loewner-sandwich reachability condition, including a Kronecker expression threshold and a fixed-basis exact subproblem.

Low-rank spectral models, curvature-proxy inflation, stochastic restricted complexity, discrete geometric length, and expression--estimation--flow--discretization accounting are presented as diagnostic interfaces rather than full optimizer characterizations.

The resulting language turns structural preconditioner questions into geometric distance, reachability, and certificate problems.

The repository includes deterministic toy and synthetic workflows that check diagonal expression gaps, block primal/dual certificates, Kronecker spectral width, and Hessian-relative Kronecker candidate certificates on small quadratic instances, together with low-rank spectral monotonicity.