Source Clarifications and Proof-Route Note

Theorem 1(iii): calibration reading

• Source reading: the source’s calibration display at score values with positive probability, $\Pr(Y=y\mid q_y=c)=c$, uses the standard conditional-expectation meaning on every measurable score event $B$: $\mathbb E[1_{\{q_y\in B\}}1_{\{Y=y\}}]=\mathbb E[1_{\{q_y\in B\}}q_y]$.
• Scope: this is the ordinary calibration definition intended by the source proof, including continuous score distributions. It is a source clarification rather than an additional theorem assumption.
• Proof route: a direct calibration inequality replaces the paper’s mass-transformation argument; the bound itself is unchanged on that domain.

Theorem 2(iii) weighted-objective wording

• The main-text “unless gamma=1 and D agrees” wording → for 0 <= gamma < 1, optimality of an independent rule D for the weighted objective O_N^gamma implies almost-sure agreement with argmax. It does not require gamma=1 for an agreeing rule or assert that agreement suffices for optimality.
• Appendix B.1’s one-sample switch proves this necessary-condition reading for the paper’s dataset-dependent reference family: positive-probability disagreement permits a strict improvement. The proof also includes gamma=0; no additional statistical assumption is used.

Pareto-optimal terminology

• Source → retained helper: maximizing a weighted objective for some $0<\gamma\le1$ → nondominance. The definition correspondence is uncredited; Theorem 2(iii)’s domination and weighted-objective comparisons are checked separately.
• Why distinguish them: among metric pairs $(1,0),(0,1),(0.4,0.4)$, the last is nondominated but never maximizes a positive weighted sum. This separates the generic definitions; it is not a counterexample to Theorem 2(iii).