Fixed-horizon extremes of spectrally negative Lévy processes with Poisson resetting

Unrefereed research candidate. A spectrally negative Lévy process (no upward jumps) is reset to a home position at the times of a Poisson clock and started in its long-run state. Given that its maximum over a fixed window is very high, where does it end? Up to an error smaller than any exponential order, either no reset has happened and it ends near the peak, or a reset has happened and it ends at an ordinary level. The paper proves an explicit total-variation error bound for this two-profile decomposition, uniform over all endpoint sets, gives the conditional law of the whole path (with an independent exponential overshoot), quantitative overshoot and terminal-threshold statements, exactness for bounded-variation inputs, and an extension to random reset positions. For Brownian motion it supplies the constant left unspecified in Remark 3 of Dębicki, Hashorva and Michna (2026) and proves that their two bounds are strict. The leading constant itself is a standard first-cycle computation. The archive contains the manuscript (PDF and LaTeX), Python check scripts (closed forms, quadrature, reflection brackets, an exact-sampling Monte Carlo of the original process), committed outputs and figures, a replay script, semantic negative controls and SHA-256 sums. Assurance: written proofs; producer-side replay from a clean copy; internal AI-assisted review, AI-assisted proof audits and a verification re-review. This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.