Thermal joins and conditional contact hypotheses

For two simple domains ${\mathrm{\Gamma }}_i\subset {\mathbb{R}}^{n_i+1}$ and ${\mathrm{\Gamma }}_j\subset {\mathbb{R}}^{n_j+1}$, write their coordinates $(u_i,v_i)$ and $(u_j,v_j)$. Their thermal-join domain is

${\mathrm{\Delta }}_{ij}:=\{(u,v_i,v_j):\mathrm{\exists }{u}_i,u_j,(u_i,v_i)\in {\mathrm{\Gamma }}_i,(u_j,v_j)\in {\mathrm{\Gamma }}_j,u=u_i+u_j\}\mathrm{.}$

The joined simple type is a distinct formal tag with amount function ${\nu }_{{\mathrm{\Delta }}_{ij}}={\nu }_{{\mathrm{\Gamma }}_i}+{\nu }_{{\mathrm{\Gamma }}_j}$ and weight $w_{{\mathrm{\Delta }}_{ij}}=w_{{\mathrm{\Gamma }}_i}+w_{{\mathrm{\Gamma }}_j}$, as defined in Forward-sector hypotheses for a simple state domain. Encode the separated pair by ${\mathsf{J}}_{{\mathrm{\Gamma }}_i}(x)+{\mathsf{J}}_{{\mathrm{\Gamma }}_j}(y)$ and a joined coordinate $Z$ by ${\mathsf{J}}_{{\mathrm{\Delta }}_{ij}}(Z)$. Their bookkeeping masses agree, and their aggregate composition functions agree. These encoded maps and their raw tags remain distinct. Positive coordinate-scaled copies use the reversible preparation transport of that definition; nested joins preserve their finite primitive amount function under reassociation and permutation. The empty type contributes the zero map.

Write $J_{ij}(x,y)$ for the join state with the summed energy coordinate and the unchanged work coordinates. A compound pair $(x,y)$ and $J_{ij}(x,y)$ have the same total energy and work coordinates but are distinct tagged objects. Define $x{\sim }_{T}y$ exactly when ${\mathsf{J}}_{{\mathrm{\Gamma }}_i}(x)+{\mathsf{J}}_{{\mathrm{\Gamma }}_j}(y)$ and ${\mathsf{J}}_{{\mathrm{\Delta }}_{ij}}(J_{ij}(x,y))$ are accessible in both directions. Nested joins use the declared reassociation and permutation identifications.

Assume every finite join is itself a simple domain satisfying Forward-sector hypotheses for a simple state domain. The contact hypotheses are mathematical assumptions on the common relation: (T1) $(x,y)\prec J_{ij}(x,y)$; (T2) every join state has at least one reversible split into component states at its stated coordinates, and the equal-state split of any state into positive scaled copies of itself is reversible; (T3) ${\sim }_T$ is transitive; (T4) for every simple state $x$ there are states $x_0,x_1$ in its domain with $x_0\prec \prec x\prec \prec x_1$ and $x_0{\sim }_T{x}_1$; (T5) for every ordered pair of simple systems, every $x$ in the first and every work coordinate $v$ in the second's projection $\rho ({\mathrm{\Gamma }}_j):=\{v:\mathrm{\exists }u,(u,v)\in {\mathrm{\Gamma }}_j\}$, there is a state $y$ with work coordinate $v$ and $x{\sim }_{T}y$.

For the completed calibrator branch, add (C1) join coherence: whenever $x{\sim }_{T}y$, the join state $J_{ij}(x,y)$ is thermally equivalent to each of $x$ and $y$ under the nested-join identifications. The coordinate sum in the definition alone does not assert C1. In later theorem statements, T1–T5 and C1 are explicitly listed as hypotheses; they are not treated as established physical laws.