Proof of `Coupled scalar recipes give consistent atom gradients` (2nd statement)
What this proof establishes
no assumptions
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
The two role equations turn the opposite coupled equation into equality of the cross row totals. The two self totals already equal the prescribed diagonal sums. Construct the symmetric matrix with those row sums; its symmetry converts them into exactly the two list-gradient requirements.