The KAIROS Invariance Theorem: Projective Isolation of Intrinsic Identity in Exponential Decay Systems - A Universal Method for Measuring Thermodynamic Entropy via Geometric Projection

We present a general mathematical method for isolating the intrinsic time-constant (τ) of an exponential decay system, independent of initial magnitude (Scale) or equilibrium state (Offset). By projecting the observed signal onto a log-derivative manifold, we demonstrate that Scale and Offset are geometrically orthogonal to Identity. We formalize this as E = P(v) − U, where Entropic Identity (E) equals the residual deformation of the Projected Observation P(v) from the Universal Taylor Skeleton (U). Critically, we demonstrate through simulation that E is not merely correlated with thermodynamic entropy—E maps isomorphically to thermodynamic entropy, measured in the natural units of exponential decay. Monte Carlo verification (500 trials, Python 3.11) shows a correlation of r = 0.9481 (p < 0.001) between E and configurational entropy S = k_B × ln(1 + CV), where CV is the coefficient of variation of the time-constant distribution. With R² = 0.8989 and an intercept representing the sensor noise floor, this demonstrates a robust physical identity within realistic measurement constraints.