Binary smoothing and relative Turan densities of ordered triangle-tails

For every $b\ge1$, let $Q_{2,b}$ be the ordered graph obtained from a transitive ordered triangle by attaching a monotone tail of length $b$ at its rightmost vertex. We prove $\rho_{<}(Q_{2,b})=\frac12$ for all $b\ge1$. Thus the previously isolated case $Q_{2,2}$ is one member of an exact infinite triangle-tail family. The lower bound is the sharp forward-template identity $\lambda(Q_{2,b})=1/2$, and it is realized already inside binary-level hosts: a parity-cut construction gives $Q_{2,b}$-free binary-level graphs with density exactly $1/2$ on every level. The upper bound uses the binary rich-level reduction. Its key input is the intrinsic decomposition of a $Q_{2,b}$-free graph into tail-starting vertices $T_b$ and the complement $R$: there are no forward edges from $R$ to $T_b$, $G[T_b]$ is ordered-triangle-free, and $G[R]$ is monotone-$\vec{P}_{b+1}$-free. We control these pieces by weighted binary $\vec{P}_{b+1}$ smoothing and weighted binary Mantel smoothing, the latter following from a binary ultrametric cut-domination theorem. We also record exact path-blow-up values, giving a reusable template/rich-host calculus for ordered relative densities.