When Can a Transport Index Be Trusted? Projection, Signal Quality and Measurement Invariance in Neonatal Cardiorespiratory Monitoring

This paper asks when a transport index computed from a neonatal monitor reflects the physiology it is meant to describe, and proves two theorems, fourteen propositions, one corollary and three lemmas. The monitor is modelled as an observation map from physiological state to displayed signal, and the paper determines which quantities that map preserves. The central result is that the thinness ratio—transport over a horizon divided by distance to an endpoint-specific failure region—is invariant for every state and every failure point exactly when the observation map is a similarity, which for a single channel means an increasing or decreasing affine map. The ratio is therefore an interval-scale quantity, and the coordinate in which distance is measured is part of its definition. By contrast, reach, hitting times, time spent in the failure region, and the number and durations of excursions are invariant under every failure-preserving map, including every strictly increasing recalibration of a channel; these event quantities are the right anchor for validation. A bi-Lipschitz map with distortion changes the ratio by at most that factor, and under the canonical oxygen dissociation curve the ratio computed in saturation understates the ratio in oxygen tension for falls toward a hypoxemic threshold, substantially at high baseline tensions. Integer display makes a reading of high saturation consistent with a wide range of tensions, so an endpoint defined by high oxygen tension is not visible there. A per-channel maximum ratio is invariant under separate affine recalibration of each channel, while a Euclidean ratio is not. A certificate computed in a projection is either valid for hidden failure but possibly vacuous, or informative only about failure the monitor can see. Smoothing delays a ramp by exactly the kernel's mean lag and erases events shorter than an explicit duration; averaging cannot enlarge the fluctuation of a stationary signal; and sampling, signal hold and common imputation rules cannot increase supremum transport, so all err toward false reassurance. For a fixed mean and variance, the time below a threshold and the depth of the excursion have sharp upper bounds but do not determine event structure. When lower signal quality is a garbling of higher quality it cannot reduce expected posterior uncertainty, and unbounded artifact bias forces abstention. The paper proposes an output grammar and a measurement passport, and a two-monitor test. All results are checked by simulation of canonical models; no clinical data are used.