Global closure of the weak-drive multiplicative expansion

Determine whether the weak-drive multiplicative perturbation series can be globally completed into a Bargmann-admissible eigenfunction by controlling its moving zeros, Stokes data, cumulant poles, and growth, or by constructing a zero-carrying canonical-product factor compatible with the exponential factor.

Background

The weak-drive analysis exactly resums selected upper-edge contributions using a ${}_0F_1$ function and extends the construction multiplicatively to a subdiagonal. However, the multiplicative logarithmic representation develops poles at zeros of the resummed factor, while finite exponential truncations can create essential singularities and cannot carry zeros themselves.

The paper explicitly identifies the unresolved issue as the global control of these moving zeros and the associated sectorial growth and Stokes behavior. A successful construction would turn the local perturbative dressing into a globally defined Bargmann eigenstate rather than merely a formally accurate expansion.