What Is Richer Optimization Geometry Worth? Hessian-Blind Pricing of Structured Preconditioners

Richer preconditioners can reduce the number of optimization steps while costing more to construct and apply. We ask a narrower, candidate-specific question: given a baseline and one frozen richer preconditioner, what compute premium is that candidate locally worth? For a stochastic quadratic model we derive the exact optimal one-step progress $G^\star(P)$, its invariance to scalar rescaling, and the break-even premium $q_\star=G^\star(P_1)/G^\star(P_0)$. We estimate the components of this price without revealing the population Hessian, using independent cross-fitted gradients and stochastic secants. On 1,728 analytically calibrated paired-spectrum instances, predicted prices track constructed-candidate oracle prices (Spearman $0.993$; log-price MAE $0.007$). Across an offline compute-price sweep, gain-only routing systematically over-purchases, while value routing nearly eliminates decision regret on this benchmark. A broader synthetic stress test finds that many randomly structured candidates do not justify their modeled compute cost. We therefore view the method as a local pricing diagnostic for structured preconditioning.