In gamma-ray astronomy and cosmic-ray physics, the continuous approximation of inverse Compton scattering (ICS) is widely adopted to model the evolution of electron energy. However, when the initial electron energy approaches $\sim 100\mathrm{TeV}$, the discrete nature of ICS becomes prominent, and the energy of evolved electrons should be considered as a broad distribution rather than a deterministic value. By simulating the evolution paths of individual electrons under ICS, we capture this discrete nature and demonstrate that when the electron injection spectrum exhibits a high-energy cutoff, the correct discrete treatment yields a higher cutoff energy in the evolved spectrum compared to the continuous approximation. Applying the discrete ICS treatment to interpret the gamma-ray spectrum of the Geminga pulsar halo measured by the High Altitude Water Cherenkov Gamma-Ray Observatory (HAWC), we find that the inferred cutoff energy of the injection spectrum is correspondingly lower than that derived using the continuous approximation at a 95% confidence level. This suggests that the systematic bias introduced by the approximation has exceeded the measurement precision. We also expect the application of the discrete ICS correction in the peta electron volt (PeV) regime using the ultrahigh-energy gamma-ray source 1LHAASO $\mathrm{J}1954+2836\mathrm{u}$ as a case study, pointing out that adopting the continuous approximation may considerably overestimate the electron acceleration capability of the source.

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• Cosmic ray acceleration
• Cosmic ray sources
• Gamma ray astronomy