Attractor Domain Theory: A Mathematical Framework for Cardiovascular Attractor Analysis with Wearable Photoplethysmography (PPG) Validation

Abstract

The cardiac attractor provides a phase-space representation of cardiovascular dynamics reconstructed from scalar observations under standard delay-embedding conditions. We introduce Attractor Domain Theory (ADT), a theory in which linked mathematical transformations expose distinct properties of reconstructed dynamics: the Geometry Domain is obtained through the delay-reconstruction transformation, while Ergodic and Variational transformations act on that geometry to characterize long-horizon organization and localized finite-time deformation. Thus, ADT specifies a three-representation operator architecture for reconstructed dynamics rather than retrospectively grouping existing nonlinear features; the architecture is proposed as an operational organization, not as a uniqueness or exhaustiveness theorem. A Conditional Domain Decomposition quantifies endpoint-specific incremental information with an explicit residual, and a domain-transformation impossibility theorem establishes that a property cannot be exactly recovered when an operator maps property-distinct inputs to the same representation. Empirical evaluation includes PPG analysis across 176{,}742 segments with 18 patient-level held-out BIDMC records (AUC 0.757 [0.686--0.828]) and an independent ECG analysis of 5{,}000 PTB-XL records (AUC 0.719 [0.686--0.750]). Cross-observation transfer is property-specific: the Lyapunov exponent shows subject-level ECG--PPG agreement ($\rho=0.703$, 95\% CI [0.503, 0.836]), whereas the predicted class-wide separation of invariant and observation-coupled descriptors is not supported ($p=0.54$). These results provide empirical support for attractor-domain analysis across cardiovascular measurements while leaving cross-observation recoverability as a property-specific empirical question.