As we saw before, the information equilibrium condition is invariant under the transformation:
log A → γ log A + a
log B → γ log B + b
And if the coefficient of the log X terms aren't equal, it's equivalent to a change in the information transfer index (and therefore not necessarily consequential in terms of observables).
One interesting thing is that this invariance eliminates most other terms in an effective information equilibrium theory expansion, in particular the constant term.
As for the meaning of the invariance, I re-wrote the transformation suggestively in terms of logarithms. Basically, the invariance is an affine log-linear transformation (affine group). We'd visualize it as rotations and translations of our original variable in log-log space (blue line is log X = log Y, the others are γ log X + a = log Y):