Eq. (2): thermally regularized vacuum density $\Lambda_0=\int_0^\infty \frac{\omega^3\,d\omega}{e^{\omega/m}-1}\propto m^4$

Let $m>0$ be a cutoff (characteristic temperature). Then the Bose–Einstein regularized vacuum energy integral is

This is the exact form of Carneiro's estimate $\Lambda_0\approx m^4$ (Eq. (2)): a thermal distribution of vacuum modes at temperature $m$ regularizes the divergent flat-space vacuum density $\int_0^\infty\omega^3\,d\omega$ and gives a density proportional to $m^4$.

Formalization Note The integral is the Lebesgue integral over $(0,\infty)$; the paper writes $\approx m^4$, and the theorem states the exact constant $\pi^4/15$.