A Dimensionless Retention Criterion for Induced Order in Water-Material Systems

Claims that a driven medium retains an induced ordered state are usually argued qualitatively and therefore cannot be settled. We replace the qualitative question by a dimensionless one. Let τorder be the slowest relaxation time of the system, obtained as an eigenvalue of the measured local generator rather than as a fitted decay, and let τdrive be the characteristic time of the driving. We study the ratio Π = τorder/τdrive and show, for driving at fixed energy per pulse, that the induced amplitude along a soft mode scales as √(E τdrive) g(Π) with g(Π) = Π(1 − e−1/Π), a monotone function saturating at unity. Two exact consequences follow: the point Π = 1 is not a convention but the value at which the induced amplitude reaches 1 − 1/e of its ceiling, and the fraction of order surviving one drive period after the drive is switched off equals e−1/Π. We state the resulting retention window 1 < Π < Πmax, prove its lower edge, and label its upper edge explicitly as a hypothesis with a stated falsification criterion. The criterion converts the disputed question "does the medium remember?" into the measurable one "how long does the induced state live relative to the driving, and what sets the upper edge of the window?" No experimental data are reported. Preprint.