Telescoping Solutions of Differential Equations: A High-Order Exponential Approximation Framework

We construct high-order approximations to solutions of linear ordinarydifferential equations (ODEs) by replacing the exponentials in their closed-formsolutions with \emph{telescoping exponential approximants}$E_n^{(s)}(c) = (1 + c/n + s\tfrac{\sqrt3}{6}c^2 n^{-2})^{\,n + c(1/2 - s\sqrt3/6)}$,$s=\pm1$, whose successive differences decay as $O(n^{-4})$. We give explicitorder-3 constructions for first-order linear ODEs $y'=ay$, the harmonicoscillator $y''+\omega^2 y=0$, and matrix systems $x'=Ax$; a general theoremshowing that any constant-coefficient linear ODE whose solution is a finitecombination of $t^j e^{\lambda t}$ inherits telescoping sequences of arbitraryorder; and error analysis giving uniform bounds $O(n^{-(k+1)})$ on differencesand $O(n^{-k})$ on errors over compact time intervals. The exact algebraicidentities the framework rests on --- the vanishing of the $n^{-1}$ and $n^{-2}$log-series coefficients, the leading coefficient $g_3$, the finite-differencefactor $-3$, and the Pythagorean base identity --- are kernel-checked in Lean~4(Appendix~\ref{app:lean}); the asymptotic statements themselves are confirmed byhigh-precision computation and are \emph{not} claimed as formalised. Every claimis labelled by evidence tier.