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We study the stability of isomorphisms between interpolation scales of Banach spaces, including scales generated by well-known interpolation methods. We develop a general framework for compatibility theorems, and our methods apply to general cases. As a by-product we prove that the interpolated isomorphisms satisfy uniqueness-of-inverses. We use the obtained results to prove the stability of lattice isomorphisms on interpolation scales of Banach function lattices and demonstrate their application to the Calderón product spaces as well as to the real method scales. We also apply our results to prove solvability of the Neumann problem for the Stokes system of linear hydrostatics on an arbitrary bounded Lipschitz domain with a connected boundary in    $$\mathbb {R}^n$$  R n ,    $$n\ge 3$$  n ≥ 3 , with data in some Lorentz spaces    $$L^{p,q}(\partial \Omega , \mathbb {R}^n)$$  L p , q ( ∂ Ω , R n ) over the set    $$\partial \Omega $$  ∂ Ω equipped with a boundary surface measure.

Stability of the inverses of interpolated operators with application to the Stokes system