Finite-protocol unattainability on a declared zero-temperature boundary

For a declared material constitution $c$, let $D_{c,+}\subseteq D_c$ be its designated equilibrium states with finite positive absolute temperature, and require $D_{c,+}\mathrm{\ne }\mathrm{\varnothing }$ for a nonvacuous application. Let $D_0$ be a separately declared nonempty set of zero-temperature boundary preparations, disjoint from $D_{c,+}$, and let ${\bar{D}}_c:=D_{c,+}\bigsqcup D_0,{\bar{T}}_c:{\bar{D}}_c\to [0,\mathrm{\infty })$ extend the thermodynamic temperature from Entropic temperature, pressure, and molar chemical potentials by ${\bar{T}}_c{\mathrm{\mid }}_{D_{c,+}}=T_c$ and ${\bar{T}}_c(y)=0$ for $y\in D_0$.

Let ${\mathcal{U}}_c$ be the model-declared set of cooling controls, and let $\mathcal{C}\subseteq {\bar{D}}_c\times {\bar{D}}_c\times {\mathcal{U}}_c\times (0,\mathrm{\infty })$ be the model's full declared family of admitted cooling operation records. Each record $\gamma =(x^{-},x^{+},u,\tau )$ has initial-endpoint map $i(\gamma )=x^{-}$, final-endpoint map $f(\gamma )=x^{+}$, control $u\in {\mathcal{U}}_c$, and finite positive duration $\tau (\gamma )=\tau$ seconds. Its positive-temperature portions obey the process conventions of Thermodynamic process records and transfer conventions, while the model separately declares how a boundary endpoint in $D_0$ is represented. Define two records to be compatible exactly when $f(\gamma )=i({\gamma }^{\mathrm{\prime }})$. A finite protocol is a tuple $({\gamma }_1,\dots ,{\gamma }_m)$ for $m\in \mathbb{N}$, $m\ge 1$ (The natural numbers $\mathbb{N}$ (von Neumann)), with each successive pair compatible; its initial and final endpoints are $i({\gamma }_1)$ and $f({\gamma }_m)$, and its duration is ${\sum }_{j=1}^m\tau ({\gamma }_j)$. Let ${\mathcal{P}}_{\mathcal{C}}$ be the set of all such finite protocols and let ${\mathcal{P}}_{\mathcal{C}}^{+}:=\{p\in {\mathcal{P}}_{\mathcal{C}}:i(p)\in D_{c,+}\}$. For each $p=({\gamma }_1,\dots ,{\gamma }_m)$, define $i(p):=i({\gamma }_1)$, $f(p):=f({\gamma }_m)$, and $\tau (p):={\sum }_{j=1}^m\tau ({\gamma }_j)<\mathrm{\infty }$. For a nonvacuous application, require ${\mathcal{P}}_{\mathcal{C}}^{+}\mathrm{\ne }\mathrm{\varnothing }$ and declare $\mathcal{C}$ as the full operation family rather than defining it to exclude boundary endpoints. Define ${\mathrm{Reach}}_{\mathcal{C}}:=\{(i(p),f(p)):p\in {\mathcal{P}}_{\mathcal{C}}\}\subseteq {\bar{D}}_c\times {\bar{D}}_c$.

The adopted operational unattainability postulate is ${\mathrm{Reach}}_{\mathcal{C}}\cap (D_{c,+}\times D_0)=\mathrm{\varnothing }\mathrm{.}$ This is a separate physical assumption from the Nernst heat-theorem postulate; neither is deduced here from the first or second law. If $D_{c,+}$, $D_0$, or ${\mathcal{P}}_{\mathcal{C}}^{+}$ is empty, or if no boundary endpoint is available in the declared operation-record space because the model defines every operation to stay in $D_{c,+}$, the assertion is absent or vacuous and is not evidence for unattainability. The postulate does not remove the need to state the actual state space, controls, and admitted operations.

Masanes and Oppenheim, Introduction, formulate unattainability in terms of finitely many operations in finite time. The reachability notation above is this framework's explicit formulation of that operational principle. Their quantum cooling bounds require additional dynamical, bath, and work-resource assumptions, listed in their Fundamental Assumptions; those bounds are not a derivation of the postulate for the arbitrary declared operation family here. Tong §4.5 and Arovas §2.6.2 provide context for the distinct entropy form of the third law, not this operational exclusion.