Thermal stability originates the vanishing of the specific heats at absolute zero

The relationship between the vanishing of the heat capacities as $T\to0^+$ and the thermal stability is examined. The heat capacities vanish as fast as or faster than $T$ as $T\to0^+$ for states at the phase space boundary ($T=0$) to sustain the standard thermal stability criterion $U_{ss}>0$. Conversely, weakly vanishing heat capacities, which signify a loss of curvature in $U(S)$ at $T=0$, are the signature of a critical condition precisely at $T=0$, as exemplified in marginal Fermi liquids. %Therefore, the vanishing of the heat capacities should be viewed not as a new, independent law, but rather as a confirmatory result of the existing framework of thermodynamics. Versión en lengua castellana: mainEs.pdf English language version: main.pdf Embargo expires on 3 December 2026 El embargo finaliza el 3 de diciembre del 2026 Earlier versions contain publicly available preprint versions of the manuscript. Las versiones anteriores contiene los preprint públicos del manuscrito.