Scientific machine learning models are often checked only against governing partial differential equations. A surrogate can keep PDE residuals small while still breaking constitutive relations, for example by predicting a thermal field that does not match the diffusivity implied by the same equation. This paper presents Moju, an open-source library for physics admissibility auditing with native support for JAX and PyTorch. Moju reports governing-law residuals and constitutive consistency separately, and assigns tiered admissibility scores (HIGH, MODERATE, LOW, and NON-ADMISSIBLE) so users can see which part of the physics fails. The study uses one-dimensional transient slab cooling with Robin convection. Seven fully connected networks are trained for 14,000 L-BFGS steps, with hidden widths from 2 to 128. All models reach at least 99.99% governing-law admissibility on the Fourier conduction residual, but constitutive admissibility spans 17.8% to 93.2%. That gap is not visible from PDE loss alone. The paper reports training trajectories, worst-point constitutive diagnostics, and agreement between training and evaluation grids within 0.5 percentage points. It also discusses how capacity relates to constitutive admissibility in this benchmark. The Moju software is available at https://github.com/IfimoAI/moju and on PyPI as moju. Companion Colab notebooks reproduce the paper training run and a fast audit-only demo from pre-exported states.