Proof of `Coupled scalar recipes give consistent atom gradients` (1st 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
First construct the compatible symmetric atom matrix from the numerical role and p data. Then realize its difference from the fixed tester matrix using one allowed formal ordered L/R product per distinct atom pair.