Tagged scalable compounds

Let $X$ be a set of state labels. A tagged scalable compound over $X$ is a function $c:X\to {\mathbb{R}}_{\ge 0}$ whose support $\mathrm{supp}(c):=\{x\in X:c(x)\mathrm{\ne }0\}$ is finite. The coefficient is a dimensionless formal amount; state labels retain their tags, so equal coefficients on different labels do not identify those labels. Compounds are equal exactly when their coefficient functions are equal.

For compounds $c,d$ and $t\ge 0$, define composition and nonnegative scaling pointwise by $(c+d)(x):=c(x)+d(x)$ and $(tc)(x):=tc(x)$. The zero function is the empty compound; in particular $0c=0$. Comparison axioms involving scaling still require a strictly positive scalar unless explicitly stated otherwise. The mass of $c$ is the finite sum $m(c):={\sum }_{x\in \mathrm{supp}(c)}c(x)$. It is dimensionless bookkeeping mass, not thermodynamic mass or energy. The compound is positive when $m(c)>0$.

For every state label $x\in X$, the unit-state compound $e_x$ is the coefficient function with $e_x(y)=1$ if $y=x$ and $e_x(y)=0$ otherwise. Its support is $\{x\}$, so it is a tagged compound. In compound comparisons, a state label $x$ denotes $e_x$. Every compound has the canonical finite decomposition $c={\sum }_{x\in \mathrm{supp}(c)}c(x)e_x$, with an empty sum when $c=0$; coefficients on this support are strictly positive. The mass-one property $m(e_x)=1$ and the equivalence $m(c)=0⟺c=0$ are proved in Compound mass is well-defined and additive ↗.

The formula for mass is independent of the chosen finite set containing the support, and composition and scaling obey $m(c+d)=m(c)+m(d)$ and $m(tc)=tm(c)$; Compound mass is well-defined and additive ↗ proves these assertions.

No topology, continuity, or physical interpretation is part of this definition. The finite-support restriction is essential: no infinite series or convergence convention is implicit.