Cardiac Stability Theory: An Axiomatically Grounded Framework for Continuous Cardiac Health Monitoring via Smartphone Photoplethysmography

Abstract: We present Cardiac Stability Theory (CST), an axiomatically grounded framework formally defining cardiovascular health as a stability margin around a cardiac dynamical attractor. From four axioms we derive the Cardiac Stability Index (CSI), a composite scalar in [0,1] integrating the largest Lyapunov exponent, recurrence determinism, and signal entropy via time-delay embedding. The ECG-based model (CSISurrogateV2, CNN-Transformer) achieves $R^2=0.8788$, MAE$=0.0234$ on PTB-XL (21,799 recordings). We extend CSI to smartphone PPG via Complementary Domain Transfer (CDT): CSISurrogateV2 generates pseudo-labels for the BUT PPG dataset (48 recordings, 12 subjects), training TinyCSINet (122,849 parameters), achieving MAE$=0.0557$, $ρ=0.660$ on the held-out test set ($n=1065$ windows) at ${<}30$ ms mobile latency. CDT is validated on BIDMC, Welltory, and RWS-PPG. Paired validation on 5,035 BIDMC windows yields $r=0.454$ ($ρ=0.485$, $p<10^{-295}$), confirming correlated cardiac stability across modalities. CSI is negatively correlated with age (slope $= -0.000225$ CSI/year, PTB-XL), discriminates atrial fibrillation from normal sinus rhythm (AUROC$=0.89$), and is robust under Perturbation Invariance Training (max AUC drop 1.65\%). We derive HeartSpan, a longitudinal stability metric relative to population age norms, enabling continuous non-invasive cardiac monitoring from commodity smartphones for longevity tracking and cardiac risk stratification.

• The paper introduces Cardiac Stability Theory, grounding cardiac health monitoring on axioms and nonlinear dynamics with a novel Cardiac Stability Index.
• It establishes a methodology combining Lyapunov exponents, recurrence analysis, and signal entropy to quantify cardiac health and validate against diverse datasets.
• Practical deep learning models, such as TinyCSINet and CSISurrogateV2, enable efficient cross-modal CSI estimation for real-time, individualized cardiac screening.

Cardiac Stability Theory: Axiomatic Foundations for Smartphone-Based Cardiac Health Monitoring

Introduction

"Cardiac Stability Theory: An Axiomatically Grounded Framework for Continuous Cardiac Health Monitoring via Smartphone Photoplethysmography" (2604.23876) presents a comprehensive, principled approach to defining and quantifying cardiac health in the context of continuous, consumer-device-based monitoring. The study introduces Cardiac Stability Theory (CST), establishing a theoretical and mathematical foundation for the Cardiac Stability Index (CSI), and implements a robust methodology for inferring this index from both ECG and smartphone PPG signals. Through a combination of validated axioms, nonlinear dynamics, and deep learning-based domain transfer, the paper advances both the interpretability and deployability of cardiac monitoring technologies.

Theoretical Framework and Axiomatization

The core contribution is the formalization of cardiac health as dynamical stability, expressed through a stability margin around a bounded cardiac attractor. CST is built on four explicit axioms governing the nature of the cardiovascular system (as a deterministic nonlinear dynamical system), its projection onto observable signals (ECG/PPG as information-preserving mappings), the monotonic relationship between health and attractor stability, and a novel complementary domain universality principle, positing that ECG and PPG are both projections of the same latent attractor.

From these axioms, four theorems are derived: attractor boundedness, stability-complexity duality, strict monotonicity of CSI with respect to attractor health, and a proven negative correlation between CSI and age in adults. These provide mathematical justification for key properties of the proposed index and its longitudinal interpretation.

The Cardiac Stability Index and Computation Pipeline

The Cardiac Stability Index is defined as a weighted composite of the largest Lyapunov exponent ($\lambda_{\max}$), recurrence determinism ($R_{\mathrm{det}}$), and normalized signal entropy ($H$), extracted from Takens-delay embedded representations of cardiac signals:

$$\mathrm{CSI} = w_1 e^{-\lambda_{\max}} + w_2 R_{\mathrm{det}} + w_3 (1 - H)$$

This formulation ensures that CSI is strictly decreasing in both age and disease severity, aligning with empirically observed deformations of the cardiac attractor in pathology.

The signal processing pipeline is built upon established nonlinear dynamical systems methods: optimal delay estimation via average mutual information (AMI), attractor reconstruction, Lyapunov exponent calculation (Rosenstein method), recurrence quantification analysis, and signal entropy estimation. The derived scalar index is calibrated to $[0,1]$ for comparability and implemented efficiently for mobile deployment.

From the CSI, the authors introduce HeartSpan, a derived longitudinal metric expressing an individual's cardiac stability relative to population age norms, enabling population-level risk stratification and detection of longitudinal stability trends.

Deep Learning Models and Complementary Domain Transfer

ECG Model: CSISurrogateV2

The first stage employs a CNN-Transformer hybrid (CSISurrogateV2) trained on the PTB-XL ECG dataset (21,799 recordings). The model predicts CSI from raw single-lead ECG with strong numerical results ($R^2 = 0.8788$, MAE = 0.0234), robust discrimination of normal rhythm versus atrial fibrillation (AUROC = 0.89), and empirically confirms the negative age-CSI relationship (slope $-0.000225$/year).

Cross-Modal PPG Model: TinyCSINet and CDT

A two-stage Complementary Domain Transfer (CDT) approach leverages the CSISurrogateV2 model to generate surrogate CSI labels for PPG segments from the BUT PPG dataset (12 subjects, 48 smartphone camera recordings, with simultaneous high-fidelity ECG). A lightweight CNN-Transformer (TinyCSINet, 122,849 parameters) is then trained to predict CSI directly from PPG. Under rigorous subject-disjoint splits, TinyCSINet achieves MAE = 0.0531 (validation), $R^2 = 0.42$, and Spearman $\rho = 0.684$ (test $\rho = 0.660$), with latency $R_{\mathrm{det}}$0 ms per window on commodity mobile hardware.

The CDT principle eschews traditional domain adaptation, instead asserting—under the formal axioms of CST—that both modalities manifest the same latent dynamics up to affine transformation. This is empirically supported by significant cross-modal correlation in the BIDMC dataset (Pearson $R_{\mathrm{det}}$1, Spearman $R_{\mathrm{det}}$2, $R_{\mathrm{det}}$3) between PPG- and ECG-derived CSI.

Universality, Calibration, and Robustness

The models are extensively validated across heterogeneous datasets: clinical (BIDMC), consumer wearable (Welltory), and large-scale real-world smartphone recordings (RWS-PPG). Raw CSI distributions are context- and device-dependent; however, universal affine calibration aligns them onto a common operational scale (target $R_{\mathrm{det}}$4, $R_{\mathrm{det}}$5), preserving rank invariance ($R_{\mathrm{det}}$6) and facilitating interoperable reporting. The evidence for CDT universality is qualified by persistent label noise and context variability, but the statistical alignment remains consistent with the axiomatic structure.

Perturbation Invariance Training (PIT) is applied to maintain robustness under acquisition artefacts without undermining sensitivity to real physiological changes. Across 40 independent runs, the model's AUROC drops by no more than 1.65% under significant signal perturbations, evidencing effective invariance.

Implications and Future Directions

CST provides a principled alternative to empirical feature engineering or black-box discrimination, offering explainable, dynamical-system-theoretic interpretability at both feature and index levels. By defining cardiac health explicitly as attractor stability and quantifying it axiomatically, CST establishes a substrate for explainable AI advancements (e.g., integration with EAT) and robust, theory-grounded clinical deployment.

Practically, the study demonstrates that cardiac health indices with strong theoretical justification and cross-modal interpretability can be continuously estimated on smartphone hardware at population scale. This enables individualized, longitudinal cardiac risk stratification, early trend detection, and democratization of cardiac screening outside of clinical infrastructure.

Theoretically, CST's universality rests on the assumption of smooth, information-preserving modal projections and the empirical success of affine calibration. Establishing predictive and causal validity relative to clinical outcomes, and generalizing the approach to further biosignals (e.g., from the broader observable projection class), are concrete priorities for future research.

Conclusion

This work establishes Cardiac Stability Theory as an axiomatically grounded framework for quantifying cardiovascular health based on dynamical systems principles and nonlinear time series analysis. The derived Cardiac Stability Index integrates Lyapunov stability, attractor recurrence, and entropy, enabling interpretable, cross-modally valid monitoring from both ECG and PPG. Combined with efficient, lightweight deep learning models and robust cross-modal transfer validated against multiple datasets, the approach provides a pathway to truly continuous, actionable cardiac health assessment on widely available consumer devices. The formalism and methodology are extensible, setting a foundation for further theoretical development, improved clinical integration, and broader physiological monitoring based on attractor stability concepts.