  Recent findings from DESI BAO combined with Planck CMB data have set an upper
limit on the total neutrino mass of $\sum m_\nu < 0.072\, \text{eV}$ (95%
confidence level), ruling out the inverted hierarchy. Indeed, methods that rely
on the background expansion of the Universe tend to suggest negative neutrino
masses. In this work, we contribute to the quest for accurately constraining
neutrino mass using cosmological probes. By conducting a full-shape analysis on
data from BOSS, eBOSS, and synthetic power spectra, we showed that projection
effects can significantly influence constraints on neutrino mass, rendering
these measurements largely unreliable. Our results highlight the need for
better techniques to measure the neutrino mass accurately. Based on the
large-scale structure suppression, we identified a critical blind spot in the
full-shape analysis. By splitting the galaxy power spectrum into broadband and
wiggles, we noticed that information on neutrino mass is primarily extracted
from the suppressed wiggles rather than broadband suppression. This opens the
possibility of developing alternative methods based only on the wiggles of the
power spectrum that can be more robust than those heavily reliant on background
evolution.
