Algebraic Separation of Biexponential Time Constants Using a Fractional Linear Transformation, the Wronskian Bridge, and a Complex-Zero 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 (NLLS) fitting,which suffers from ill-conditioning, multiple local minima, and sensitivity to initial parameter estimates,and which fails in the close-rate regime even when initialized at the exact answer. This paper presentsan algebraic framework that removes the amplitude coefficients from the rate problem before any fittingbegins. The log-derivative of a biexponential, Λ(t) = S′(t)/S(t), is a fractional linear transformation(FLT, also known as a M¨obius transformation) of the hidden variable u(t) = e−Δλt , where Δλ = λ1−λ2is the rate difference, and the amplitude ratio enters only through the coefficients of that transformation.The cross-ratio, the unique fundamental invariant of the group PGL(2,R), cancels the amplitude ratioexactly while preserving the rate information.The reason the identity works is physical, not merely formal. When two decay processes are independent,their contributions add linearly, and the amplitude ratio enters the log-derivative as a multiplicativeconstant that the cross-ratio cancels; the rates enter through the time dependence and survive. We showthat this amplitude annihilation is not an algebraic coincidence but a necessary consequence of the Riccatistructure of the log-derivative, whose symmetry group is PGL(2,R) and whose unique fundamentalinvariant is the cross-ratio. This holds for any positive two-mode linear system, not only exponentials.The cross-ratio is the noiseless theorem; it is not the estimator. For noisy data the practical estimatorfits the full FLT curve to many log-derivative samples and reads the rate sum λ1+λ2 from a Wronskianbridge identity that never differentiates the data twice. In the clinically demanding close-rate regime(τ2/τ1 = 1.67), on a single dense echo train, this estimator outperforms oracle-initialized NLLS by oneto three orders of magnitude across the full range of amplitude fractions, with slow-rate RMSE below15 ms near the benchmark amplitude fraction. The recovery is amplitude-dependent: its accuracy peaksin a band of amplitude fractions and degrades away from it as a structural bias rather than as noise.We explain this with an exact pool-crossover identity, Λ(t∗) = −(λ1 +λ2)/2 at t∗ = ln(c1/c2)/Δλ,which localizes the rate information at the crossover and shows that a fixed window samples it onlywithin a band. Continuing the signal into complex time recasts this crossover as the real projection ofa complex zero z0 = t∗ ±iπ/Δλ, whose imaginary lift π/Δλ is fixed by the rate separation alone: atfixed separation, amplitude slides the projection left and right, while a change in separation sets thelift and rescales the projection, giving a two-coordinate recoverability map that we chart by simulationrather than collapse into a single distance. The same identity yields a complementary, amplitude-robustestimator whose standard deviation is nearly uniform across amplitude and whose worst-case error islower, which is useful when the amplitude fraction is unknown, as in imaging. We state the scope plainly:this is a quantified, mechanistically explained single-contrast operating envelope, not amplitude-generalrecovery. We place the method in context through connections to projective and information geometry,and all core algebraic identities are machine-verified in Lean 4.