Adithiya Dinesh UC Madrid and IPARCOS, Alberto Dominguez UC Madrid and IPARCOS, V. Paliya, J. L. Contreras, S. Buson, M. Ajello

Blazars are a subclass of active galactic nuclei (AGN) that emit non-thermal radiation through relativistic jets, characterized by rapid flux and polarization variability. Extreme high synchrotron-peaked blazars (EHSPs), with synchrotron peaks exceeding 10$^{17}$ Hz, are essential for understanding the full range of blazar phenomena and testing jet physics models. However, the number of known extreme blazars is small, so this class of objects remains poorly studied. This work aims to systematically identify and characterize the most extreme $\gamma$-ray blazars using data from the Large Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope. The focus is on spectral hardening, where the $\gamma$-ray spectrum becomes harder at higher energies, particularly during flaring episodes. This represents the first dedicated analysis of spectral hardening across a population of EHSPs, as previous studies explored it only in individual sources. We analyze 138 blazars selected from the 4FGL-DR2 catalog with high synchrotron peak frequencies and well-sampled light curves. Flaring periods are automatically identified, and each flare is analyzed, with the significance of spectral hardening assessed through a test statistic based on the likelihood ratio of two spectral models. 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 constrain its frequency and support a smoothly varying power-law blazar emission model, motivating future multi-wavelength studies to clarify whether these rare flares reflect distinct physical processes within blazar jets.