A Flat Curve Means Two Opposite Things: Reading Loss-Conditional Training Correctly

Loss-conditional training replaces a hyperparameter sweep with a single network that receives the loss weight as an input. We study this approach in a physics-informed neural eigensolver that learns approximately orthonormal, low-energy spectral coordinates under a learned metric regularized toward isotropy. On Two Moons, concatenation and FiLM both produce responses that are roughly ten times flatter than those of separately trained specialists, yet their absolute representation quality differs substantially: concatenation yields a shared compromise, whereas FiLM matches or exceeds two of the three specialists. On MNIST, the architectural ranking reverses, with concatenation matching the best fixed-weight specialists. Full-test audits find no dead MNIST coordinates, indicating that the flat probe curves are not explained by catastrophic basis collapse. We further observe that conditioned training can avoid representation failure while retaining substantial metric anisotropy, suggesting an optimization effect rather than simple reproduction of the isotropy penalty. A negative control shows that conditioning only the coordinate networks, rather than the regularized metric, produces a confidently flat but uninformative response. These results show that response span must be evaluated together with absolute quality and that the placement of conditioning is essential.