A finite-step geometric cooling protocol

Fix $T_{\ast }>0$ kelvins, an initial state coordinate $T_{\mathrm{i}\mathrm{n}}\in D_{+}:=(0,T_{\ast }]$, a dimensionless ratio $r\in (0,1)$, and an operation duration ${\tau }_{\ast }>0$ seconds. The model state is the scalar coordinate $T$ with state set $\bar{D}:=D_{+}\bigsqcup D_0$, where $D_0:=\{0\}$; $T$ is assigned units of kelvins. This abstract coordinate is not identified with the entropy-derived thermodynamic temperature $T_c=1\mathrm{/}{S}_{c,U}$ of Entropic temperature, pressure, and molar chemical potentials. Each admitted operation is the partial map $f:D_{+}\to D_{+}$, $f(T):=rT$, with duration ${\tau }_{\ast }$. An admitted finite protocol is a finite composable chain of these operations; write $T_N$ for its endpoint after $N$ updates. Since the operation has codomain $D_{+}$, this restricted model has no operation with endpoint in $D_0$; the proof below is a direct property of its update rule, not evidence for the physical unattainability postulate. It does not assert that a real refrigerator or every thermodynamic model has this update rule.