Classical statistical mechanics distributes energy among microstates according to the Boltzmann distribution. The derivation is clean: maximize entropy subject to a constraint on expected energy, and the exponential distribution falls out. But wavefunctions don't have definite energies — they are superpositions. The Boltzmann recipe cannot be directly applied.
What distribution should a wavefunction ensemble follow at thermal equilibrium? The maximum entropy approach suggests: find the least-biased distribution consistent with what we know. Constraining the expected energy and the shape of the eigenstate distribution seems like enough. It is not.
This paper shows that these constraints fail to yield a valid equilibrium state. A third constraint is needed: the measurement entropy must equal the Rényi divergence of the ensemble with respect to the Gibbs state. Without it, the maximum entropy solution is either inconsistent with thermal equilibrium or underdetermined.
The Rényi divergence — a generalization of the Kullback-Leibler divergence — enters not as a mathematical convenience but as a physical necessity. It measures how much the wavefunction ensemble deviates from the thermal state in a way that the energy constraint alone cannot capture. The result (the Scrooge ensemble) is the unique maximum-entropy wavefunction distribution at thermal equilibrium, and it requires all three constraints simultaneously. Two is not enough. The missing piece was a divergence measure that nobody suspected was physically load-bearing.