A large-radius analytic criterion for the local F0 grand-potential sum

We give a conditional large-radius convergence theorem for the worldsheet and Nekrasov–Shatashvili combination in the local F0 grand potential. The coefficient hypotheses are a finite character expansion with fixed parity and an exponential weighted bound in the total curve degree. On compact intervals of positive real Planck constants, a single Mellin–Barnes block and finite character corrections give normal convergence after grouping all covers of each curve class. A finite-difference identity controls arbitrarily large imaginary translates of this block. Combined with the quantum mirror map, it gives a normally convergent integer periodization on a common right half-plane. We also verify directly from the two-partition formula the known base-degree-one characters and prove normal convergence of that slice, including all its covers, independently of the full-degree hypotheses.