Recurrence Before Return: Recovery-Relative Clustering of Apnea–Bradycardia–Desaturation Events in Preterm Infants. Mathematical Analysis, Simulation and Study Protocol

This paper rigorously examines what common rhythm measures actually capture in the cardiorespiratory signals of preterm infants, under realistic noise, finite recording windows, and simple generative models. Drawing on an applied reading of a “Trojan” framework that emphasizes protected spectral separation between two dominant modes, the author tests the resonance distance between respiratory and cardiac-modulation frequencies, the phase-locking value, and a directionality index. The work supplies eleven propositions and three lemmas that clarify both measurement validity and what the statistics identify. A phase is trustworthy only where the signal envelope substantially exceeds noise; otherwise the error can be arbitrary. Because heartbeats sample the cardiac signal, respiratory-frequency modulation is replaced by an aliased, phase-reversed component whenever heart rate falls below twice the respiratory rate—an occurrence that cannot be excluded in late-preterm infants. The phase-locking value computed over short windows is inflated by sampling variability; the pairwise phase consistency statistic removes that bias. The same numerical value can arise from brief intervals of strong locking or from continuous weak locking, so occupancy and strength must be reported separately. Under linear driving of heart-rate modulation by breathing, one-to-one locking is guaranteed and its strength is a fixed function of linear coherence; excess locking above a linear surrogate null is therefore required before nonlinear coupling can be claimed. Every frequency ratio lies near some low-order resonance, so “near resonance” is meaningless without a pre-specified maximum order. In the noisy Adler phase model the observed resonance distance is identically zero throughout the locking region and therefore confounds intrinsic detuning with coupling strength; the two quantities can be recovered from phase increments only when instrumental variables correct for phase-observation noise. The ratio of detuning to coupling, together with the ratio of coupling to noise, supplies a dimensionally consistent tongue margin that replaces an earlier heuristic robustness index. A damped linear mode exhibits a resonance peak only when its quality factor is sufficiently high; periodic-breathing episodes then furnish a within-infant test of under-damping. Finally, measurement error in the respiratory channel alone can reverse the apparent direction of influence. Synthetic simulations confirm each analytic result. No clinical data are analyzed and no clinical claim is advanced. The paper closes with operational definitions of four structural exits, a proposed validation protocol, and a reporting checklist that makes the conditions under which the original hypothesis can be tested explicit.