Optimizers can fit the same task while acquiring different generalizing relations. We study the training conditions governing these differences in single-hidden-layer ReLU networks, combining composite evidence tasks, parameter-level interventions, and a three-seed strict-parity scan. Our central finding is that nuisance-connected trainability reshapes both shared failures and relative optimizer advantages. In a nuisance-heavy task, all twenty tested optimizer configurations remain near chance on the hardest stage. Retaining every input but fixing nuisance-connected first-layer weights at initialization raises that stage's accuracy from approximately 50\% to 70.56\%, 68.47\%, and 69.00\% for momentum SGD, Adam, and Muon. Masking the same inputs only after full training does not recover this performance. On a separate pairwise-mode task, background freezing reduces Muon's rare-mode advantage over momentum SGD by 12.48 percentage points, while the target and mode frequencies remain fixed. Each intervention is evaluated under a common validation-selection protocol with condition-specific learning rates and checkpoints. A strict-parity sweep over orders 1--20 provides a complementary reference without spurious cues or extra nuisance coordinates: the optimizers separate at orders 9--11, then approach chance despite substantial remaining Bayes predictability. A mixed task establishes a recovery boundary, and CIFAR-10 supplies an external architecture comparison. Together, these findings connect optimizer comparison to the acquisition and use of specified relations, identifying permitted adaptation as a concrete training variable that changes what a fixed architecture learns.

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