Additive separated systems and tagged bulk scaling

Let $A$ and $B$ be two named subsystems with disjoint physical regions as in Thermodynamic systems and permitted exchanges, combined as a compound system $A\bigsqcup B$ inside a declared outer boundary. Fix their material constitutions $c_A,c_B$, each with a finite ordered list of distinct species labels as in Equilibrium state domains. Fix an ordering $({\sigma }_1,\dots ,{\sigma }_r)$ of the labels occurring in either list, each occurring once, and use this as the species list of the compound constitution $c_{A\bigsqcup B}$. For $s\in \{A,B,A\bigsqcup B\}$, write $D_s:=D_{c_s}$ and use $U_s,V_s$ for its dimensioned coordinate maps. For every $1\le j\le r$, define the aligned amount map $a_{s,j}:D_s\to [0,\mathrm{\infty })\mathrm{m}\mathrm{o}\mathrm{l}$ by $a_{s,j}(x):=n_{c_s,k}(x)$ if ${\sigma }_j$ is the $k$-th species in $c_s$'s list, and $a_{s,j}(x):=0\mathrm{m}\mathrm{o}\mathrm{l}$ if it is absent. Distinctness makes $k$ unique. The absent-species value is a bookkeeping map on the existing $D_s$; it does not insert a zero coordinate into that constitution's positive-mole chart or assert a boundary-state extension.

The model's admitted separated pairs form a specified subset ${\mathcal{C}}_{AB}\subseteq D_A\times D_B$ using Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$ and The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$. Adopt a composition function ${\kappa }_{AB}:{\mathcal{C}}_{AB}\to D_{A\bigsqcup B}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain) such that, for every $(x_A,x_B)\in {\mathcal{C}}_{AB}$ and every $1\le j\le r$, its output satisfies $U_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))=U_A(x_A)+U_B(x_B),V_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))=V_A(x_A)+V_B(x_B),a_{A\bigsqcup B,j}({\kappa }_{AB}(x_A,x_B))=a_{A,j}(x_A)+a_{B,j}(x_B)\mathrm{.}$ All quantities have the SI units of SI quantities and unit conventions. Each union species is present in at least one constituent with positive amount, so the summed amount is positive and agrees with the compound's $j$-th coordinate. The existence of an output in the declared $D_{A\bigsqcup B}$ is part of the adopted model, not a consequence of positivity.

Here “extensive coordinates” means precisely the named coordinates declared additive by this postulate. This is an idealized composition rule: cross-interface and long-range interaction energy is omitted from $U_{A\bigsqcup B}$, rather than asserted to vanish in nature. The postulate does not assert that every pair of subsystem states is physically admissible.

The framework's bulk family is a tagged family of material types. For each dimensionless scale $\lambda >0$ the model may designate a tag $c^{(\lambda )}$ with the same ordered species list and SI coordinate conventions as $c$, and a subset $B_{c,\lambda }\subseteq D_c$ of states admitting a scaled copy, with inclusion as in Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$. For $x=(\widehat{U},\widehat{V},{\widehat{n}}_1,\dots ,{\widehat{n}}_r)\in B_{c,\lambda }$, the scaling map is the function ${\delta }_{\lambda }:B_{c,\lambda }\to D_{c^{(\lambda )}}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain) given by ${\delta }_{\lambda }(x)=(\lambda \widehat{U},\lambda \widehat{V},\lambda {\widehat{n}}_1,\dots ,\lambda {\widehat{n}}_r)$. The tag matters: $c^{(\lambda )}$ is not silently identified with $c$. This postulate specifies the coordinate scaling model and energy additivity only; it does not postulate entropy additivity or exact scaling for every finite physical body.

This is an adopted physical composition rule, not a deduction from SI units or from the mathematics of Cartesian products.