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Description
Blazars are among the most powerful gamma-ray emitters, displaying rapid variability and extreme spectral properties. High synchrotron-peaked blazars (HSPs) and extreme high synchrotron-peaked blazars (EHSPs), with synchrotron peaks exceeding $10^{15}$ Hz and $10^{17}$ Hz, respectively, are crucial for understanding the full range of blazar phenomena and testing models of jet physics. Yet, their understanding remains challenging. This work aims to systematically identify and characterize the most extreme γ-ray blazars using data from the Large Area Telescope (LAT) on board the $\textit{Fermi Gamma-ray Space Telescope}$. The focus is on spectral hardening, where the γ-ray spectrum becomes harder at higher energies, particularly during flaring episodes. This phenomenon is characterized by a flux that decreases with energy up to a break in the GeV range, after which the spectrum hardens as the flux begins to rise. While previous studies have reported spectral hardening in a few individual sources, this work presents the first dedicated, systematic analysis of this effect. We examine 138 blazars with high synchrotron peak frequencies from the 4FGL-DR2 catalog, detecting flaring periods using two methods based on Bayesian Block Analysis. We identify two flaring episodes with indications of spectral hardening, in 4FGL J0238.4-3116 and PKS 2155-304, the latter detected independently by both methods but referring to the same period. These events are consistent with expectations from statistical fluctuations, suggesting that spectral hardening is a rare occurrence (< 0.1 %). These results provide the first population-level constraint on the frequency of such events. The scarcity of events reinforces the notion that the dominant blazar emission mechanism is well described by smoothly varying powerlaw spectra across the $\textit{Fermi}$-LAT range, with sharp spectral hardenings representing rare deviations likely tied to exceptional jet conditions or transient physical processes. Although these flares show notable spectral changes, their statistical significance remains modest and motivates future multi-wavelength studies to assess whether these rare flares reflect genuinely distinct physical processes within blazar jets.