The correctness term matters here, especially once training can change the answer as well as its confidence. If an answer is correct with probability p and reports q, the expected RLCR reward is p^2 - (q-p)^2. At q=p, the reward is p^2, so a more accurate answer still pays.

With only the negative Brier loss, the optimum would instead be -p(1-p): being reliably wrong and reporting zero confidence would score just as well as being reliably right. The joint objective avoids that trap. Section 3 of the paper discusses this distinction: https://arxiv.org/html/2507.16806v2#S3

That describes the ideal incentive, of course, not a guarantee of calibration after finite training or on a new task.