Finite-time queue peak laws in stochastic networks show a transition from square-root to logarithmic growth beyond a geometry-dependent threshold. This shift occurs due to self-normalization of fluctuations relative to drift, eliminating capacity geometry from the logarithmic coefficient while preserving it in the threshold. Lower bounds confirm both the logarithmic term and threshold are unavoidable, with local bottleneck geometry enabling sharper thresholds in input-queued switches.