Every standard expression counts the same parallel classes

Reverse the factorization supplied by Independent arbitrary-rank factorization and simple-class inputs using Transfer to the root-action Brink–Howlett groupoid. The latter identifies its generators with article elementary labels and preserves the label product. Thus the reversed sequence is composable in the article convention and remains Coxeter-length additive. It is a standard expression by Standard expressions and their lengths, proving existence.

Let $w=v_1\cdots v_m$ be any standard expression with $m>0$, and put $p_0=1$ and $p_k=v_1\cdots v_k$ for $1\le k\le m$. Repeated subadditivity from [F2] gives $\mathrm{\ell }(w)\le \mathrm{\ell }(p_k)+{\sum }_{i=k+1}^m\mathrm{\ell }(v_i)\le {\sum }_{i=1}^m\mathrm{\ell }(v_i)=\mathrm{\ell }(w)$. Both inequalities are therefore equalities, so $\mathrm{\ell }(p_k)={\sum }_{i=1}^k\mathrm{\ell }(v_i)$. This also holds for $k=0$, since the empty word has length zero. Consequently $\mathrm{\ell }(p_{k-1}{v}_k)=\mathrm{\ell }(p_{k-1})+\mathrm{\ell }(v_k)$ for every $1\le k\le m$.

Reverse these arrows into $\mathcal{B}$, preserving their label products by Transfer to the root-action Brink–Howlett groupoid. Apply Compressed inversion count is additive for Coxeter-reduced composites to each composable pair with labels $p_{k-1},v_k$, whose lengths add by step 1.2. Since $N(1)=\mathrm{\varnothing }$ by the positive and negative root convention, $C(p_0)=\mathrm{\varnothing }$; induction gives $\mathrm{\mid }C(w)\mathrm{\mid }={\sum }_{i=1}^m\mathrm{\mid }C(v_i)\mathrm{\mid }$. Each $v_i$ is a Brink–Howlett generator by the opposite identification, and Independent arbitrary-rank factorization and simple-class inputs gives $\mathrm{\mid }C(v_i)\mathrm{\mid }=1$. Hence $\mathrm{\mid }C(w)\mathrm{\mid }=m$. If $m=0$, the standard expression is the identity and both counts are zero. The count therefore depends only on the labelled morphism, with no rank assumption.