Algebraic Separation of Biexponential Time Constants Using Cross-Ratio Invariance

The extraction of time constants from biexponential signals is a fundamental problem in physics, engineering, and biomedical science. Traditional approaches rely on nonlinear least squares fitting, which suffers from ill-conditioning, multiple local minima, and sensitivity to initial parameter estimates. This paper presents an algebraic method for extracting biexponential time constants that is completely independent of the amplitude coefficients. The method rests on a simple observation: the log-derivative of a biexponential signal is a Möbius transformation of the exponential time variable, and the cross-ratio—a classical projective invariant—cancels the amplitudes exactly while preserving the rate information. The reason the identity works is physical, not mathematical. When two decay processes are independent, their contributions add linearly, and the amplitude ratio enters the log-derivative as a multiplicative constant that the cross-ratio is designed to cancel. The rates enter through the time dependence and survive. This factorization—amplitudes in one slot, rates in another—is a consequence of physical independence, not of any sophisticated mathematical structure. We show that this amplitude annihilation is not an algebraic coincidence but a necessary consequence of the Riccati–Möbius structure of the underlying differential equation. The log-derivative of any positive two-mode linear system satisfies a Riccati equation, whose symmetry group is the Möbius group PGL(2,ℝ). The cross-ratio is the unique fundamental invariant of this group. The method therefore does not merely happen to work—it is the canonical extraction procedure selected by the symmetry of the physics. This structure holds for any positive two-mode linear system, not only exponentials, though the exponential case is the commercially immediate application. We provide complete algebraic derivations, including explicit demonstration of how the amplitude ratio is annihilated in the cross-ratio computation. For noisy measurements, we introduce a manifold projection technique that fits noisy samples to the Möbius curve, achieving robust extraction with sub-2% error at signal-to-noise ratios of 50 dB. The amplitude invariance property is preserved even in the presence of noise. We also place the method in context by noting connections to Hilbert projective geometry, information geometry, and the Euler–Maclaurin formula for optimal sampling, and we describe how the same algebraic structure extends to other basis function families through reparameterization of the time coordinate. The method has immediate applications in battery diagnostics, medical imaging, spectroscopy, and any domain requiring reliable separation of overlapping exponential decay processes.