Degenerate thermometers can fail product comparison

Given: A state is a positive dimensionless energy coordinate $U$ as defined in the Statement. For one copy, accessibility is $U\le U^{\mathrm{\prime }}$. Thermal equilibrium of scaled copies ${\lambda }_1{U}_1,{\lambda }_2{U}_2$ holds exactly when their normalized coordinates $U_1=U_2$. For two unit copies starting at $(U_1,U_2)$, put $\bar{U}=(U_1+U_2)\mathrm{/}2$, let

$A=\{(U_1^{\mathrm{\prime }},U_2^{\mathrm{\prime }}):U_1^{\mathrm{\prime }}\ge U_1,U_2^{\mathrm{\prime }}\ge U_2\},B=\{(U_1^{\mathrm{\prime }},U_2^{\mathrm{\prime }}):U_1^{\mathrm{\prime }}\ge \bar{U},U_2^{\mathrm{\prime }}\ge \bar{U}\},$

and define the product forward sector as $\mathrm{conv}(A\cup B)$, where $\mathrm{conv}(C)$ is the set of all finite convex combinations ${\sum }_{k=1}^m{t}_k{z}_k$ with $z_k\in C$, $t_k\ge 0$, and ${\sum }_k{t}_k=1$. $A$ describes componentwise energy increase and $B$ thermal equilibration followed by energy increase, as in Lieb–Yngvason §IV.C. To respect the permutation identification in [F1], a two-copy state here is the class $[u]$ of $u=(U_1,U_2)$ under swapping its coordinates. Define $[u]\prec [v]$ exactly when some representative of $[v]$ belongs to $\mathrm{conv}(A_u\cup B_u)$, where $A_u,B_u$ are the displayed sets for the initial representative $u$. This explicitly declares the permutation-compatible version of the source construction; its independence of the initial representative is checked below.

To realize these relations by formal process records, use $X=\mathrm{\Gamma }$ for one copy and $X={\mathrm{\Gamma }}^2$ for two ordered copies. In the latter case admit endpoint pairs $a,b$ exactly when $[a]\prec [b]$ under the displayed sector relation; in the former admit exactly $a\le b$. For either domain, put $Z^{-}=Z^{+}=Z=X$, take identity endpoint maps, a singleton control set, transfer-record set $X\times X$, and a fixed outer boundary. For each admitted pair use exactly the one-second record $P_{a,b}=([0,1],a,b,z_{a,b},\ast ,(a,b))$, where $z_{a,b}(t)=a$ for $0\le t<1$ and $z_{a,b}(1)=b$. Its transfer record supplies $H_{P_{a,b}}^{\mathrm{i}\mathrm{n}}(t)=H_{P_{a,b}}^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in [0,1]$, with numerical values in joules, so $Q[P_{a,b}]=0$. Define $\mathcal{E}(U)=U(1\mathrm{J})$ for one copy and $\mathcal{E}(U_1,U_2)=(U_1+U_2)(1\mathrm{J})$ for two, and assign $W_{\mathrm{o}\mathrm{n}}[P_{a,b}]=\mathcal{E}(b)-\mathcal{E}(a)$. The same one-copy construction applies to a scaled copy using its actual energy coordinate $\lambda U$. Permuting initial and final representatives transports their endpoint labels and the corresponding path labels; the inverse permutations give record bijections preserving boundary, thermal histories, total energies and admission by the representative invariance checked in step 1.1 below. Quotienting therefore gives exactly the stated class relation while retaining both factors up to permutation. These are formal records without auxiliaries, path regularity, core-class membership or material realization asserted; their zero histories are explicit data, not an inference from zero net heat.

Tagged scaled systems and separated products: product states are tagged scaled copies and product comparison is the declared relation, not coordinate addition.

Thermal-contact axioms T1–T4: T4 requires a thermal-equilibrium class crossing strictly below and above every state.

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: the real natural logarithm is strictly increasing on positive reals, has $\log 1=0$, and satisfies $\log (ab)=\log a+\log b$.

SI quantities and unit conventions: energy is measured in joules, and numerical energy coordinates are divided by their displayed units before they are used as Euclidean coordinates.