Stability as Memory Mass: Scalar Recovery as the Leading Projection of the Persistence Kernel

Scalar recovery rates are routinely reported as measures of stability and resilience across ecology, physiology, engineering, and machine learning. This paper establishes that such scalars are not primitive system properties but observations of a deeper structure: the zeroth moment of a causal persistence kernel governing how past states constrain present dynamics. We decompose system dynamics into instantaneous forcing (Leo) and lagged retained constraint (Lo), modelled as a causal convolution operator. Via a Taylor expansion of the convolution, we prove that the observable recovery rate k equals the persistence mass m₀ - the integral of the kernel over all delays - whenever a dimensionless ratio called the memory Knudsen number (ε = τ_depth / T_r) is small. When ε is small, scalar measurement is valid. When ε approaches order unity, higher-order kernel geometry governs the dynamics and scalar estimates become model-dependent approximations rather than structural invariants.