Pinching antenna systems (PASS) have attracted much attention recently owing to their capability to maintain reliable line-of-sight communication in high-frequency bands. By guiding signals through a waveguide and emitting them via a movable pinching antenna, PASS enable dynamic control of signal propagation and spatial adaptability. However, their performance heavily depends on effective resource allocation, including power allocation and antenna positioning, which becomes challenging under user localization uncertainty. Existing studies largely assume perfect user locations, while our prior work considered uniformly distributed localization errors. In this paper, we develop a robust resource allocation framework for multi-user downlink PASS under Gaussian-distributed localization uncertainty. An energy-efficiency (EE) maximization problem is formulated subject to probabilistic outage constraints, and the minimum transmit power is analytically characterized for any given antenna position. For the unbounded Gaussian model, we prove that the resulting one-dimensional antenna-placement objective is convex and derive a globally optimal derivative-bisection algorithm. We further consider a circularly truncated Gaussian model for bounded localization-error regions and develop a deterministic branch-and-bound method that returns an $ε_{\rm B}$-globally optimal antenna position. Finally, we discuss how position-dependent waveguide attenuation modifies the placement objective and may destroy its convexity. Simulation results verify that the proposed branch-and-bound solution closely matches the bounded grid-search benchmark, demonstrate substantial EE gains from optimized pinching-antenna placement over the bounded fixed-antenna architecture, and reveal an increasing distinction between the bounded and unbounded Gaussian models as the location uncertainty grows.

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