An Exact Quantile-Energy Equality for Terminal Halfspaces in Linear-Gaussian Control with a Discrete-Time Companion, KL/Schrodinger Links, and High-Precision Validation

Abstract: We prove an exact equality between the minimal quadratic control energy and the squared normal-quantile gap for terminal halfspaces in linear-Gaussian systems with additive control and quadratic effort $E(u)=\tfrac12!\int u^\top M u\,dt$ where $M=B^\top\Sigma^{-1}B$. For terminal halfspace events, the minimal energy equals the squared normal-quantile gap divided by twice a controllability-to-noise ratio $R_T^2(w)=(w^\topW_c^M w)/(w^\top V_T w)$ and is attained by a matched-filter control. We provide an exact zero-order-hold discrete-time companion via block exponentials, relate the result to minimum-energy control, Gaussian isoperimetry, risk-sensitive/KL control, and Schrodinger bridges, and validate to high precision with Monte Carlo. We state assumptions, singular-$M$ handling, and edge cases. The statement is a compact synthesis and design-ready translator, not a universal principle. Novelty: while the ingredients (Gramians, Cauchy-Schwarz, Gaussian isoperimetry) are classical, to our knowledge the explicit quantile-energy equality with a constructive matched-filter achiever for terminal halfspaces, and its discrete-time companion, are not recorded together in the cited literature.