Operational State Selection in Stochastic Analysis: The Least State for a Calibrated Quadratic Ito Program

Abstract

We study the least stable state determined by a causal response program and its quantitative calibration. For nondominated families of continuous Hilbert-valued semimartingale laws with uniform drift and trace-clock bounds, the calibrated quadratic It\^o program selects a state $\mathcal A_2$ with canonical coordinates $(X,A,Q)$, where $A$ is Hilbert--Schmidt skew area and $Q$ is trace-class covariance. Brownian experiments identify the Hilbert--Schmidt/operator coefficient gauges and hence the dual $\mathcal S_2/\mathcal S_1$ state geometry. A lossless encoder--decoder pair shows that every joint realization factors through $\mathcal A_2$. A single sequence of total Borel causal approximants constructs the common state, while the laws verify its classical semantics. A dimension-free defect--energy estimate gives independence from the finite-variation regularization, and spatial trace tightness yields restart-stable compact capacity cores in infinite dimensions. Each fixed higher signature level is a continuous readout of $\mathcal A_2$. Rough flows inherit corewise continuity, while Euler schemes construct a jointly Borel raw-causal It\^o field outside one parameter-independent polar set. On compact covariance-envelope subclasses, covariance responses select a backward first jet which, under the stated completion and continuation hypotheses, has modelwise Galtchouk--Kunita--Watanabe semantics and agrees with the solver's martingale coordinate.