Global Fedoryuk--Bargmann connection and orientation

Establish the global Fedoryuk--Bargmann connection for the strong-drive normal form, including a univalent chain of punctured-origin, WKB, Weber, and possible Airy domains with progressive paths, controlled overlap and monodromy errors, identification of the compensated Bargmann germ, and the boundary-contraction factorization needed to select the physical imaginary-axis energy branch.

Background

The independent Fedoryuk treatment produces two local strong-drive energy branches associated with real-axis and imaginary-axis Weber sectors. The branch agreeing with the Bogoliubov calculation is not selected by local analysis alone; the selection depends on how the compensated Bargmann solution is transported globally around the pole, turning points, and sheets.

The proposed hypotheses H1--H3 separate the unresolved tasks: constructing the global domain chain, proving that a scalar boundary contraction detects the inadmissible branch, and establishing the factorization that preserves the Weber gamma-function zero. Without these results, the conditional Fedoryuk quantization formula cannot be identified with the exact spectral condition $\Xi(E)=0$.