Theorem: The two-way bridge under (TC)

Under Assumption ass:TC, (SB) holds for every retained first-stage pair. In particular, the first-stage commutator vanishes on every retained pair satisfying both actual orderings.

Proof. By Proposition prop:Wexact, every $y\in G$ belongs to ${\mathrm{supp}}_{\mathrm{fun}}{W}_g$. Suppose an observation pair $x\in F,y\in G$ belongs to the closed commutator slab

Then $t-s\le R\le t+s$. Lemma lem:sphereball supplies an initial sphere point $z=(0,\mathbf{z})$ of $x$ with $|\mathbf{z}-\mathbf{y}|\le s$. Theorem thm:sphere places $z$ in ${\mathrm{supp}}_{\mathrm{fun}}{B}_f$, while $z\in J^{-}(y)$. Thus $B_f\gtrsim W_g$ fails.

Consequently a two-way pair has $F\times G$ disjoint from the entire closed support in (eq:imagsupport). Multiplication by $fg$ annihilates $\mathrm{Im}K$, including its direct cone and apex. Equation (eq:commutator) proves vanishing.