Comparison and stability for compounds

Fix a nonempty set $X$ and the compounds of Tagged scalable compounds. A comparison relation $⪯$ is defined only for pairs $(A,B)$ with $m(A)=m(B)$. Write $A\sim B$ when both $A⪯B$ and $B⪯A$; this is equivalence, not equality of coefficient functions. The following are explicit hypotheses on $⪯$:

1. Preorder: $A⪯A$, and $A⪯B⪯C$ implies $A⪯C$.
2. Composition consistency: $A⪯B$ implies $A+C⪯B+C$ for every compound $C$.
3. Positive scaling: $A⪯B$ implies $tA⪯tB$ for every real $t>0$.
4. Splitting and recombination: for $x\in X$, $t>0$ and $0<s<t$, the one-state compound $t{e}_x$ is equivalent to $s{e}_x+(t-s)e_x$.
5. Stability: for equal-mass $A,B$, fixed positive compounds $Z_0,Z_1$ with $m(Z_0)=m(Z_1)$, and a sequence ${ϵ}_n>0$ decreasing to $0$, if $A+{ϵ}_n{Z}_0⪯B+{ϵ}_n{Z}_1$ for every $n$, then $A⪯B$.
6. Comparison hypothesis (CH): for every equal-mass pair $A,B$, either $A⪯B$ or $B⪯A$ (or both).

These are mathematical assumptions, not conclusions about a physical system. No antisymmetry, continuity, convexity, or strict comparison is included. Strict comparison means $A⪯B$ and not $B⪯A$.

In the stability clause, a sequence ${ϵ}_n$ is a function $\mathbb{N}\to \mathbb{R}$; “decreasing to $0$” means ${ϵ}_m\le {ϵ}_n$ whenever $n\le m$ and ${ϵ}_n\to 0$ in the sense of Limits and Cauchy sequences of reals. The separate requirement ${ϵ}_n>0$ is stated in the clause.