Resonance Distance, Phase Locking and Spectral Protection in Neonatal Cardiorespiratory Coupling: What Rhythm Measures Can and Cannot Show

This paper asks what neonatal cardiorespiratory rhythm measures actually measure under explicit signal models and finite windows, and proves eleven propositions and three lemmas. On measurement validity: a phase is defined only where the envelope exceeds the noise, with error bounded by the noise-to-envelope ratio; heartbeats sample the cardiac modulation signal, so respiratory modulation is aliased with reversed phase whenever heart rate is less than twice respiratory rate, a real possibility in preterm infants; and the phase-locking value of a short window is biased upward, with pairwise phase consistency removing the bias. Intermittent strong locking and weak continuous locking can produce the same value. On identification: linear driving of cardiac modulation by breathing produces 1:1 locking whose strength is an exact increasing function of zero-lag complex correlation, so a phase-locking value in the respiratory band is by default a measure of linear transfer, and only excess locking over a linear null indicates nonlinear coupling. By Dirichlet's theorem every frequency ratio lies close to some resonance, so the order searched must be declared. In the Adler phase model, observed resonance distance is exactly zero throughout the locking region, confounding detuning with coupling; these can be separated from phase increments, but phase observation noise requires instrumental variables. A dimensionless tongue margin—detuning relative to coupling, with a second ratio for noise—replaces the heuristic robustness index, and spectral separation protects only underdamped modes, which yields a within-infant test using periodic-breathing episodes. Mismeasured breathing produces spurious reverse directionality, and searching over many orders requires family-wise error control. The paper also gives operational definitions of four structural exits and proposes a validation design and reporting checklist. No clinical data are used and no clinical claim is made.