Signed comparison for forward-sector relations

Let $u,v$ be finite-support real coefficient maps on the tagged state labels of Tagged scalable compounds with equal total coefficient. For a typed state $x$, compound notation uses the represented map $\mathsf{J}(x)$ from Forward-sector hypotheses for a simple state domain; its raw unit point mass is $e_{(\mathrm{\Gamma },x)}$. Signed maps still use the ordinary total of their raw coefficients, exactly the bookkeeping mass of the represented nonnegative maps. Put $d=v-u$, $d^{+}(x)=\max (d(x),0)$ and $d^{-}(x)=\max (-d(x),0)$. Define $u{⪯}_{s}v$ to mean $d^{-}\prec d^{+}$ under the compound relation of Forward-sector hypotheses for a simple state domain. Write $u{\sim }_{s}v$ when both $u{⪯}_{s}v$ and $v{⪯}_{s}u$. These are formal signed comparison maps, not physical negative amounts. The positive and negative parts are finite-support compounds of equal mass. If $u{\sim }_{s}v$, then $cu{\sim }_{s}cv$ for every real $c$: positive $c$ uses scaling, $c=0$ is reflexivity, and negative $c$ swaps the positive and negative parts while equivalence supplies both comparison directions. This Definition introduces the signed notation; ordinary-order compatibility and the signed-order laws are proved separately.