Probe-Restricted Validation and Structural Non-Identifiability in Theoretical Machine Learning: Topological–Categorical Obstructions Beyond ERM

Validation in machine learning is typically operationalized as agreement on a chosen set of observables (losses, benchmarks, probe tasks). We argue that this induces a structural non-identifiability: for a restricted probe interface, there can exist many mechanistically distinct models that are indistinguishable under those probes, even at near-zero empirical risk. The ”circle-versus-interval” example makes the issue concrete: arbitrarily accurate local linear fits can coexist with global failure by collapsing the fundamental loop; persistent homology supplies stable probes that detect this failure. Finally, we outline (conditionally) how quantum-assisted extraction of topological profiles could act as a distinct inductive bias in high-dimensional regimes where classical topological pipelines become combinatorially bottlenecked.