Conjugating trivial automorphisms of ????(ℕ)/`ch

A trivial automorphism of the Boolean algebra P ( N ) / F i n \mathcal P(\mathbb N) / \mathrm {Fin} is an automorphism induced by the action of some function N → N \mathbb N \rightarrow \mathbb N . The forcing axiom O C A T \mathsf {OCA}_{\mathrm {T}} implies all automorphisms are trivial, and therefore two trivial automorphisms are conjugate if and only if they have the same (modulo finite) orbit structure. We show that the Continuum Hypothesis implies that two trivial automorphisms are conjugate if and only if there are no first-order obstructions to their conjugacy and their indices have the same parity, if and only if the given trivial automorphisms are conjugate in some forcing extension of the universe. To each automorphism α \alpha of P ( N ) / F i n \raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}} we associate the first-order structure A α = ( P ( N ) / F i n , α ) \mathfrak {A}_\alpha =(\raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}},\alpha ) and compute the existential theories of these structures. These results are applied to resolve a question of Braga, Farah, and Vignati and prove that there are coarse, uniformly locally finite, metric spaces X X and Y Y such that the isomorphism between their uniform Roe coronas is independent of Z F C \operatorname {\mathsf {ZFC}} .