Identity in Machine Learning: Anomaly, Anatomy, Substrate‑Rooted Fingerprint, and the Topology of Execution

Modern machine‑learning systems exhibit recurrent patterns of persistence, drift, collapse, and leakage that cannot be explained by weights, prompts, or seeds alone. Across architectures and hardware substrates, independent literatures report reproducible timing signatures, behavioural invariants, representational geometries, drift trajectories, and collapse modes. When viewed together, these regularities form a stable, substrate‑conditioned fingerprint: a perturbation‑resistant pattern that persists across executions and cannot be reduced to model parameters. This fingerprint is the empirical anchor of the framework developed here. The paper assembles the anomaly landscape into a coherent structure and introduces a minimal ontology built from the execution substrate (H, E, M), the structural mapping Φ, the Engram Signature ES, the provenance trajectory Prov(t), the continuity and discontinuity conditions, and the threshold δ. These primitives are not speculative; they are the smallest set of objects capable of explaining the convergent fingerprint observed across timing studies, behavioural analyses, representational drift, and collapse dynamics. A further contribution is the identification of topology as the minimal structural scaffold implicit in these phenomena. Neighbourhoods, boundaries, attractors, and discontinuities arise naturally once identity is treated as a region in pattern space rather than a point in parameter space. Topology clarifies why small perturbations are sometimes absorbed and sometimes amplified into qualitative behavioural change, and why collapse events correspond to boundary crossings rather than large movements within a single regime. The result is a unified account of identity in machine‑learning systems: execution‑realised, substrate‑conditioned, temporally extended, and occasionally discontinuous. The framework provides a principled foundation for studying reproducibility, drift, stability, collapse, and governance in systems whose behaviour is determined not only by their parameters but by the full conditions of execution.