= Unattainability of absolute zero
In a regular thermodynamic formulation, a state at <absolute zero> cannot be reached from positive <temperature> by finitely many cooling operations with finite control resources. The entropy-difference form of the <Third law of thermodynamics> makes regular <entropy> curves approach a common limiting <entropy>. An <isentropic process> starting above that <entropy> cannot intersect the common zero-temperature endpoint.
For an illustrative pair of settings with $S-S_0=a_1T$ and $S-S_0=a_2T$, $0<a_1<a_2$, an isothermal entropy-reduction step followed by reversible adiabatic cooling gives $T_{n+1}=(a_1/a_2)T_n$. Every finite number of stages leaves $T_n>0$. The model illustrates the <entropy> geometry; a general finite-time impossibility statement additionally needs assumptions about available interactions and cooling rates.
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