Mass relations in heavy hadrons from Jensen-like inequalities

We demonstrate that mass inequalities for hadrons with one or more heavy quarks arise primarily from the concavity of binding energies in the quark model, reflecting short-range Coulombic interactions and long-range confinement. Empirical two-body bindings $B_{i\bar{j}}$ are extracted from spin-averaged meson masses, ensuring model independence and direct use of experimental data. Fitting these as functions of reduced mass ${\mu }_{ij}$ reveals a critical confinement scale of 1.34~fm where bindings turn positive. The concave $B(1\mathrm{/}\mu )$ justifies Jensen-like inequalities under flavor permutation, reproducing relations like $m_{x\bar{y}}>\frac{1}{2}(m_{x\bar{x}}+m_{y\bar{y}})$ and baryon analogs, including $m_{xyz}>\frac{1}{3}(m_{xxx}+m_{yyy}+m_{zzz})$. Hadron mass decomposition validates $\mathrm{\Delta }{M}_{\text{EXP}}\approx \mathrm{\Delta }B+\mathrm{\Delta }C$ with $\sigma \sim 2.07$~MeV for mesons and baryons. Promoting inequalities to equalities, we predict masses for unobserved heavy baryons (e.g., $M({\mathrm{\Omega }}_b^{\ast })=6076.6$MeV, $M({\mathrm{\Xi }}_{cc}^{\ast })=3703.6$MeV and $M({\mathrm{\Omega }}_{cc}^{\ast })=3802.4$MeV) and identify favored quark-exchange scattering channels.