Some pairwise residue distances do not agree with the triangle inequality in 1D3Z, 2K39, and 6V5D.

For three saved ubiquitin NMR model sets (PDB entries 1D3Z, 2K39, and 6V5D), we calculated residue distances after separate proper Kabsch fits for each model pair, using equal weights and fit masks 1–70 and 1–76. Across 420 residue distance matrices, all had a negative centered Gram eigenvalue at the stated tolerance, and 346 had at least one triangle violation. We counted 4,721,705 violations in 160,549,200 model-triple/residue checks. All 456 archived source residue scores agreed with our calculations within 3.552713678800501e-15 Å; this limited comparison is not a reproduction of the source workflow. The results apply to these saved coordinates, masks, and weights. Pairwise distances do not measure protein motion or establish coordinate variance, biological uncertainty, equilibrium probabilities, or causes. We found no contradiction of the source result for the tail residues 71–76.

ubiquitin · NMR ensembles · Kabsch alignment · Euclidean distance matrices · triangle inequality · positive semidefinite Gram matrix · structural geometry