Integral representation of the neutrino mass-squared differences

Determining the absolute neutrino mass scale remains one of the most compelling challenges in particle physics. To constrain theoretical models, establishing precise relations among neutrino masses is essential. We propose a simple integral representation of the neutrino mass-squared differences $\Delta m_{ij}^2$ that provides a complementary perspective on these oscillation parameters. We then demonstrate its utility through several examples. Specifically, assuming stringent cosmological bounds that confine the sum of neutrino masses near the normal ordering floor, we derive an analytical condition for the lightest neutrino mass, $m_1 < \sqrt{61\Delta m^2_{21}}/30$. Using recent data from the JUNO experiment, this yields a competitive upper limit of $m_1<0.0023$ eV (95\% C.L.). We also formulate practical analytical bounds for $m_{2}$ and $m_{3}$ adaptable to future data, and translate the results into allowed ranges for the effective electron and Majorana neutrino masses $m_{\nu_e}$ and $m_{\beta\beta}$. Finally, we show that neutrino mass relations of the Gatto-Sartori-Tonin type emerge directly from the proposed integral representation.