Extreme-Value Criticality and Gain Decomposition at the Integer Quantum Hall Transition

Extreme-value fluctuations at quantum critical points remain poorly understood in the presence of strong correlations and openness. At the integer quantum Hall transition in the open Chalker--Coddington network, we show that the maximal wave-function amplitude separates into a global gain and an intrinsic extreme component, $|\psi|_{\max}=A\,|\tilde{\psi}|_{\max}$. We introduce extreme-moment scaling for $|\psi|_{\max}$ and observe an approximately parabolic exponent function $\tau_{\max}(q)$ over moderate $q$, while $\ln|\psi|_{\max}$ displays an almost Gaussian bulk over the studied sizes. The gain factor is close to log-normal and largely controls the raw extremes. Gain normalization reorganizes the statistics: $\tilde{\tau}_{\max}(q)$ changes qualitatively and $|\tilde{\psi}|_{\max}$ does not support a single-parameter generalized extreme-value collapse under standard centering/scaling in the accessible size window. Extreme observables thus provide a robust probe of correlated criticality in open quantum systems.