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

arXiv: https://arxiv.org/abs/2606.22039 · HTML: https://mlsanity.com/paper/2606.22039/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 46 references that are in the map.
- First posted on arXiv 20 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [Tree-Structured Parzen Estimator: Understanding Its Algorithm Components and Their Roles for Better Empirical Performance](https://mlsanity.com/paper/2304.11127/) (2023)
- [Rethinking Multimodality: Optimizing Multimodal Deep Learning for Biomedical Signal Classification](https://mlsanity.com/paper/2508.00963/) (2025)
- [Explainable Deep Neural Network for Multimodal ECG Signals: Intermediate vs Late Fusion](https://mlsanity.com/paper/2508.11666/) (2025)
- [Cardiac Stability Theory: An Axiomatically Grounded Framework for Continuous Cardiac Health Monitoring via Smartphone Photoplethysmography](https://mlsanity.com/paper/2604.23876/) (2026)
- [Attractor-Vascular Coupling Theory: Formal Grounding and Empirical Validation for AAMI-Standard Cuffless Blood Pressure Estimation from Smartphone Photoplethysmography](https://mlsanity.com/paper/2605.10871/) (2026)
- [A Nonlinear Complexity Index for Wearable PPG Cardiovascular Stability: Multiscale Validation, Systematic Evaluation Correction, and Bayesian Parameter Optimization](https://mlsanity.com/paper/2605.18802/) (2026)

## Abstract (arXiv)

The cardiovascular system evolves along a bounded trajectory in physiological state space that converges to a compact geometric object: the cardiac attractor. A wearable photoplethysmograph (PPG) or electrocardiograph (ECG) observes a one-dimensional projection of this attractor; by Takens' embedding theorem, delay coordinates reconstruct its full geometry. Three decades of nonlinear cardiac dynamics have extracted Lyapunov exponents, recurrence statistics, and sample entropy from reconstructed attractors, yet no principled account exists of which attractor properties capture which cardiovascular quantities, or why, leaving feature selection as a search problem and negative results uninterpretable. We introduce Attractor Domain Theory (ADT), which proves that the reconstructed attractor's information partitions into three mutually non-redundant domains: the Geometry Domain G (delay embedding; native capability: artifact rejection), the Ergodic Domain S (asymptotic statistical invariants; native capability: stability estimation), and the Variational Domain V (finite-time Lyapunov exponent field; native capability: hemodynamic inference). We prove a Domain Sufficiency Theorem (the Parseval analog for attractor information) and establish that three domains are necessary and sufficient. Geometry Domain validation via the SCSI framework across 176,742 PPG segments from four datasets yields AUC = 0.757 [0.686-0.828] and NPV = 0.966 after correcting three systematic evaluation artifacts (+0.179 net inflation). Ablation confirms C_NL as the dominant Geometry Domain component (Delta AUC = -0.413) and intra-domain redundancy across five components.