Valuing Optimization Geometry: Intrinsic Value and Finite-Step Transfer Guarantees

Structured preconditioners can substantially alter local optimization dynamics, but end-to-end optimizer comparisons do not isolate what a fixed geometry is worth at a particular training state. We study this local valuation problem for already-constructed, frozen preconditioning geometries. Under a stochastic quadratic model, optimizing over a global step magnitude yields a scale-invariant intrinsic value and a relative break-even interpretation. We then ask when this model-implied ordering survives a finite update. Under a Lipschitz Hessian, we derive an exact endpoint-wise transfer modulus in a cubic action metric $d_3$, yielding an attainable ranking frontier and a stochastic action-law transport bound. A preregistered five-trajectory CIFAR-10/ResNet-18 evaluation finds substantial state dependence: 14 of 20 states have resolved ratios, eight have resolved realized parity, and seven are formal reversals. A separate sealed prospective study does not establish unshrunk residual copying as a reliable repair. Together, the results separate intrinsic geometry value, finite-step transfer, and downstream correction, while motivating an uncertainty-aware robust decision rule derived after the prospective study and not evaluated by that protocol.