In theoretical ML, the teacher-student paradigm is often employed as an effective metaphor for real-life tuition. The above scheme proves particularly relevant when the student network is overparameterized as compared to the teacher network. Under these operating conditions, it is tempting to speculate that the student ability to handle the given task could be eventually stored in a sub-portion of the whole network. This latter should be to some extent reminiscent of the frozen teacher structure, according to suitable metrics, while being approximately invariant across different architectures of the student candidate network. Unfortunately, state-of-the-art conventional learning techniques could not help in identifying the existence of such an invariant subnetwork, due to the inherent degree of non-convexity that characterizes the examined problem. In this work, we take a leap forward by proposing a radically different optimization scheme which builds on a spectral representation of the linear transfer of information between layers. The gradient is hence calculated with respect to both eigenvalues and eigenvectors with negligible increase in terms of computational and complexity load, as compared to standard training algorithms. Working in this framework, we could isolate a stable student substructure, that mirrors the true complexity of the teacher in terms of computing neurons, path distribution and topological attributes. When pruning unimportant nodes of the trained student, as follows a ranking that reflects the optimized eigenvalues, no degradation in the recorded performance is seen above a threshold that corresponds to the effective teacher size. The observed behavior can be pictured as a genuine second-order phase transition that bears universality traits.