An Algebraic Exponential-Indicator Method for Event Detection in Time Series: Theory, Numerics, and a Consistent Implementation

This paper introduces an algebraic method for event detection in time series using an exponential indicator. The core idea is to recast threshold crossing as a fixed point check of the indicator and to realize it through a consistent numerical pipeline that avoids cancellation and overflow issues. The discrete implementation combines three elements: a three-region evaluation of the indicator depending on distance to the threshold, light signal smoothing to mitigate noise, and a local validation scheme with minimum slope, hysteresis, and a refractory window to suppress duplicate detections. Event times are refined via a secant interpolation on a normalized function, retaining linear time complexity with respect to the number of samples. We report benchmarks on sinusoidal, quadratic, step, exponential, noisy, and asymptotic cases, together with metrics and reproducible code. Results show accurate and consistent detections, and we discuss limitations and practical extensions to multi-threshold and multivariate signals. The main contribution is a simple algebraic formulation with a coherent numerical realization that is useful for dynamical simulation, control, and signal processing.