Transition as the gain annulus collapses

Characterize the transition in singular local identifiability and minimax behavior as the strict complex-gain annulus collapses toward a fixed gain magnitude.

Background

The paper shows that the strict gain-annulus assumption is essential: for one analytic dark germ, allowing gain magnitudes in a nondegenerate annulus yields a singular exponent r+5, whereas fixing the gain magnitude yields r+3. The paper does not determine the intermediate transition or the limiting behavior as the upper and lower gain-magnitude bounds collapse together.