Can a lower Brier score prove better calibration?

Classifier M1 has a lower Brier score than classifier M2 on the same test set. Which conclusion is the most defensible?

M1 is necessarily both more calibrated and more discriminative than M2

M1 has better overall probabilistic accuracy, but the result alone does not prove that M1 is better calibrated

M1 must have a smaller expected calibration error under every possible binning scheme

M1 must have a better ROC-AUC because the Brier score is rank-based