This work proposes a quotient-space formulation of diagnostic monitoring in physical systems. The central claim is that anomaly detection is not fundamentally a supervised classification problem in raw observation space, nor does it require identifying the full forward physical model, which may be unknown, nonlinear, high-dimensional, or impractical to learn. Instead, the objective is to recover a representation in which observations that are physically equivalent with respect to the latent state are mapped to the same equivalence class.

The key step is to separate physical-state information from nuisance variability. Measurements are affected by transformations such as amplitude scaling, phase shift, speed variation, sensor offset, installation effects, and additive noise. These transformations modify the observed signal while preserving the underlying physical state. They therefore induce an equivalence relation on the observation space. The resulting quotient space is the natural geometric domain for diagnosis: the inverse problem is no longer posed as explicit recovery of the full generative model, but as recovery of membership in physically meaningful equivalence classes.

Under this formulation, a nominal operating condition is represented not by a single waveform or label, but by a connected quotient-space structure. Operationally, and in the sense of the companion manifold-saturation report, this structure is interpreted as a functional manifold. Anomaly detection becomes the problem of deciding whether an observation belongs to this nominal quotient manifold. Fault classification, when required, becomes a secondary operation: once an observation is detected outside the nominal manifold, it may be assigned to another quotient-space region or fault manifold. This provides a geometric mechanism for the emergence of functional manifolds: they arise by collapsing nuisance orbits that are irrelevant to the physical state.

After establishing the mathematical object, we introduce one computational approximation. Nominal trajectories are generated by Monte Carlo simulation over nuisance variables, a simple one-dimensional convolutional autoencoder is trained only on the synthetic nominal orbit, and the reconstruction error is used as an inverse-problem residual. An adaptive threshold estimates the local boundary of the nominal manifold for each physical asset. Synthetic validation on four parallel bearing signals provides preliminary evidence that quotient-space recovery is operationally feasible under controlled conditions. The experiments do not prove the mathematical formulation itself; they demonstrate that one computational approximation can recover sufficient quotient geometry to support anomaly detection and subsequent signal classification.