THE TENSORIAL HEART: FROM ENERGY TRANSITION TO TOPOLOGICAL CARDIOLOGY The Logarithmic Spiral, the Triad of Elastic Derivatives, and the Overcoming of Kinematic Empiricism

ABSTRACT Biomedical engineering and contemporary clinical cardiology find themselves at a critical crossroads today. While the brilliant anatomical discovery of the helical myocardial band has definitively shelved the nineteenth-century hydraulic model (the heart as a simple "pump"), the interpretation of this complex architecture remains confined to mere kinematic empiricism. Modern medicine observes and measures the movement of twisting and untwisting, but ignores its deep causal genesis. This article aims to fill this theoretical void by applying the formal rigor of Tensorial Mechanics 1, making the theory accessible to anyone through a detailed exploration of physical phenomena. We will demonstrate that the heart is not a closed system comparable to a mechanical pendulum, but an open system dynamically forced by external energy (electrons), which constantly drives the organ away from its "normal state" of rest. By replacing linear kinematics with a fractal architecture based on three levels of the Tensorial Elastic Derivative (Ξ₁, Ξ₂, Ξ₃), we will reveal how the myocardial band winds to generate four perfect three-dimensional logarithmic spirals. This is the only geometry capable of simultaneously coupling orthogonal spaces, volumes, and times to obey the supreme constraint of spacetime incompressibility (ΔV = 0). We will explore the vital importance of the Tensorial Radius (RT) and its pathological hyper-extension. Through a continuous, narrative comparison between modern cardiology and Topological Cardiology, we will provide the essential mathematical key to distinguish athletic hypertrophy from clinical hypertrophy, to design zero-friction Total Artificial Hearts (TAH), and to understand HFpEF as the pure collapse of the elastic domain (Toxic Geometry).