Relational Attribution of Model Preference: Compositional Interfaces, Representation Contrasts, and Shared-Nuisance Elimination

When adding a dataset changes preference between two models, endpoint minima do not identify the accommodation mechanism. I derive an exact finite-update identity separating the incoming conditional contribution from reoptimisation and show that coalition minima alone do not recover this split. In linear-Gaussian models, anchored preference faces are generalized quadratic forms. For shared nuisance directions, the triple I = (S, C, W) is equivalent to the full Fisher block when W > 0: S is the profiled information, while (C, W) is the interface required for later composition. This state has an exact associative, commutative merge law; S alone is not composition-closed. A representation change is encoded by a mixed finite difference J. Under a fixed-effective-metric interface model, centered stage-by-partner interactions have rank at most the interface dimension d. I also derive the corresponding variable-metric hierarchy: allowing the two endpoint metrics to differ enlarges the feature space to dimension d(d + 3)/2. In the scalar two-stage/three-branch design this bound is saturated, so a nonzero second singular value falsifies the scalar fixed-metric model but cannot by itself establish an interface dimension greater than one. A known positive scalar branch-weight factor can be divided out before centering, restoring the rank-≤d diagnostic. Using public DESI DR2 MAP products, the case study reconstructs J_P = −1.341, J_U = −0.850, and J_Y = −1.405; 95.9% of ||J||² lies in the branch-common direction. The two-stage test requires two new ΛCDM CamSpec-no-lensing minima. Its protocol is frozen but not yet executable because exact updated-YAML byte bindings have not been retrieved for 8 of 24 registry corners and the historical compressed-CMB runtime class is not publicly bound under the frozen contract; no substitute or reconstructed YAML or likelihood implementation is permitted. Once the registry and executable-likelihood dependencies close, exact rank one (released reference values) requires b + 0.535 = −0.13035(a + 0.535); conditional on both Union3 stages remaining negative, the predeclared DES-Y5 prediction is −0.605 < b < −0.494 (central; operational ε = 0.002 band [−0.624, −0.478]). These are prospective predictions, not measurements. No sampling-significance or physical-mechanism claim is made.