Simple-system geometric axioms

For each simple-system chart ${\mathrm{\Gamma }}_i\subset {\mathbb{R}}^{n_i+1}$, $n_i\ge 1$, let $x=(u,v)$, with one energy coordinate $u$ and $n_i$ work coordinates $v$. The chart is a nonempty open convex domain. The simple-compressible physical case is the fixed-composition slice and embedding of Thermal joins and thermal equivalence, with $n_i=1$, $u=U\mathrm{/}(1\mathrm{J})$ and $v=V\mathrm{/}(1{\mathrm{m}}^3)$. A declared multi-work chart or joined chart instead supplies its own equilibrium endpoint domain and diagonal unit map: $U=u(1\mathrm{J})$ and $W_a=v_a{w}_a$ for fixed positive work-coordinate units $w_a$. The numerical domain is identified with these physical endpoints by that supplied map, and accessibility is pulled back from the admitted endpoint relation. These identifications preserve the internal-energy meaning and fixed composition; extra work coordinates do not follow from the simple-compressible state definition. Adopt these assumptions for every state $x$ and for every thermal join treated as a simple system:

A7 (convex-combination accessibility). For $0<t<1$, the separated scaled pair $(tx,(1-t)y)$ is accessible to the chart state $tx+(1-t)y$, whenever these states are in the declared domains.

For each $x$, define its forward sector $A_x=\{y\in {\mathrm{\Gamma }}_i:x\prec y\}$. Adopt:

S1 (strict successor). Some $y\in {\mathrm{\Gamma }}_i$ satisfies $x\prec \prec y$.

S2 (unique finite-slope support). At $x=(u_x,v_x)$, $A_x$ has exactly one supporting hyperplane through $x$, and it can be written

$$u-u_x+P_i(x)\cdot (v-v_x)=0$$

with finite $P_i(x)\in {\mathbb{R}}^{n_i}$. The map $P_i:{\mathrm{\Gamma }}_i\to {\mathbb{R}}^{n_i}$ is locally Lipschitz: every point has a neighborhood on which one finite Lipschitz constant bounds $\mathrm{\parallel }{P}_i(x)-P_i(y)\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }x-y\mathrm{\parallel }$. A supporting hyperplane means $A_x$ lies in one of its closed half-spaces and meets it at $x$.

S3 (connected relative boundary). The boundary ${\mathrm{\partial }}_{{\mathrm{\Gamma }}_i}{A}_x$, taken in the topology relative to ${\mathrm{\Gamma }}_i$, is arcwise connected.

These are explicit adopted conditions. The slope $P_i$ is a geometric support parameter here; no equation identifying it with measured pressure is asserted. In particular, S1–S3 and A7 do not assert comparison hypothesis for products.