Algebraic Separation of Biexponential Time Constants Using Cross-Ratio Invariance and Hilbert Projective Geometry

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 a novel algebraic method for extracting biexponential time constants that is completely independent of the amplitude coefficients. The method exploits the Möbius (linear fractional) structure of the log-derivative function and uses the cross-ratio, a projective invariant, to directly extract the rate difference parameter. A subsequent quadratic equation yields the remaining parameters through closed-form algebra. 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 establish the deep mathematical foundations underlying this method. The biexponential signal is shown to generate a linear flow in R² under the diagonal rate matrix, which projects to a one-parameter subgroup action on the projective line P¹(R). The symmetry group PGL(2,R) acts by Möbius transformations, and the cross-ratio emerges as the canonical PGL(2,R) invariant. Amplitude independence is therefore not algebraic coincidence but structural necessity forced by projective symmetry. Furthermore, we prove that biexponential trajectories are constant-speed geodesics in the Hilbert projective metric on the positive cone, with speed equal to the rate difference |Δλ|. The canonical projective coordinate φ = log(x₁/x₂) simultaneously appears as the Hilbert distance coordinate, the information-geometric natural parameter (log-odds of the Bernoulli family), and the modal log-ratio. In information geometry, this means biexponential trajectories are e-geodesics—straight lines in natural parameter space. This three-way equivalence reveals a unifying principle with deeper connections to entropy-maximizing principles. We introduce the Clock Principle: exponential, power-law, stretched exponential, and logistic decay laws are all manifestations of the same geodesic motion—linear motion in the canonical coordinate φ—under different time parameterizations. Laws are clocks, not curves. Power-law behavior emerges when the system's natural clock is scale-invariant (logarithmic time); stretched exponentials arise from fractional internal clocks; logistic growth is exponential projective motion viewed through a compactifying chart. We provide complete mathematical derivations including explicit algebraic demonstration of how the amplitude ratio r² and denominator products (ruᵢ + 1)(ruⱼ + 1) are annihilated in cross-ratio computation. We present experimental validation across multiple noise levels and amplitude ratios, detailed analysis of why alternative approaches (overdetermined cross-ratios, Riccati phase-plane fitting) fail, and practical implementation guidelines. The method has immediate applications in battery diagnostics, medical imaging, spectroscopy, and any domain requiring reliable separation of overlapping exponential decay processes.