Hall wall crossing preserves the completed spectrum generators

For each of $Z_{-}$ and $Z_{+}$ the hypotheses of [F1] hold. The decreasing-phase Hall factors in [F1] are the $S_Z(\varphi )$ of [F2], and its continuous integrations are the maps used in [F2]. Therefore the two Euler-integrated products are equal and the two skew-integrated products are equal in their respective completions. By [F2] these are exactly $P_{Z_{-}}^E=P_{Z_{+}}^E$ and $P_{Z_{-}}^{\mathrm{s}\mathrm{k}}=P_{Z_{+}}^{\mathrm{s}\mathrm{k}}$.

Let $P$ be either generator and write $P=1+G$. Every term of $G$ has positive total dimension, so for each total-dimension bound $D$ only the finitely many powers $G^n$ with $n\le D$ contribute to ${\sum }_{n\ge 0}(-G{)}^n$ in degrees at most $D$. The finite geometric identity in each such truncation gives both $P{\sum }_{n\ge 0}(-G{)}^n=1$ and ${\sum }_{n\ge 0}(-G{)}^{n}P=1$ degree by degree. Hence $P$ is a unit. Multiplication is continuous in the stated completion, so $a\mapsto Pa{P}^{-1}$ is a continuous automorphism with inverse $a\mapsto P^{-1}aP$. The equal generators from step 1.1 therefore induce equal automorphisms.