Exact finite comparisons of a symmetry orbit and uniform binary constraint relations

The probability law of a random constraint relation matters even when the domain size, relation density and overlap between assignments are identical. We audit an explicit four-value symmetry example using exact arithmetic. Uniformly permuting the rows of a two-block relation gives 24 permutation draws and six distinct relations. Two assignment pairs with the same one-coordinate overlap have joint inclusion probabilities 1/2 and 0 in this law, whereas both have probability 7/30 under uniform sampling of eight of the sixteen possible tuples. Exact second moments for two and three variables remain different after averaging over ordered constraint scopes. A separate checker reproduces all 130 saved finite-instance counts, seed streams and structural checks; a fresh implementation independently re-enumerates the assignments and both local relation laws. The first run's increasing-scope error is retained as a source-alignment failure despite correct arithmetic. These results establish finite distinctions between two fully specified laws and provide a reproducible model audit. They do not specify the source paper's general growing-domain symmetry ensemble, establish algorithmic scaling or resolve P versus NP. The corrected 130-instance stream passes exact solution counts, source-law and seed checks. The first 130-instance stream retains its source-law orientation failure despite passing arithmetic. Exact 24-permutation and 12,870-relation laws and second moments reproduce. This is an author preprint and has not been peer reviewed. The supplement preserves nulls, adverse checks and the stated scope.