Geometric--Nongeometric Optimizer Calculus: A Modular Language for Reachable Gradient Methods

Adaptive optimizers mix several mechanisms: a metric or preconditioner maps gradients to descent directions, while estimation, memory, step-size control, constraints, stochasticity, target modification, and discretization determine which directions are available and how they are used.

We introduce geometric--nongeometric optimizer calculus, a modular language for auditing reachable gradient methods under explicit oracle, budget, state, and rule constraints.

The geometric module is a positive cometric family that maps covectors to parameter-space directions; the nongeometric modules are information, memory, control, operator, noise, target, and discretization mechanisms.

The main formal result is a direction-expressivity theorem: away from critical points, full positive-definite geometry expresses exactly the strict descent directions.

We then define restricted direction residuals for admissible metric families, prove exact expressivity conditions for diagonal and block geometries, and separate this direction-level diagnostic from condition-number geometric complexity.

The resulting design problem is a Pareto optimization over module budgets, not a single universal optimizer ordering.

We also lift pointwise residuals to a trajectory-level residual complexity that couples direction mismatch with the variation of the explaining geometry.

We include diagnostic prototypes only as evidence for the language: a high-information full-metric probe solves deterministic quadratic benchmarks to numerical precision, while a practical Muon-style PyTorch candidate gives small-scale evidence that matrix-operator updates can be audited by the calculus.

The paper is a theory and benchmark-language manuscript; it does not claim large-scale optimizer state-of-the-art performance.