SEF-Atlas: An Auditable Structural-Fingerprint Framework for Cross-Domain Equation Similarity

Mathematical models arising in different disciplines can share deep structure while using different notation, physical interpretations, and parameterizations. Conversely, equations that appear similar can have sharply different solution theories. This paper presents SEF-Atlas, an auditable software framework for organizing cross-domain equation relationships without confusing structural resemblance with mathematical equivalence. Each registered object is represented by an eleven-dimensional fingerprint covering derivative orders, principal operators, nonlinear mechanisms, dynamical character, conserved quantities, variational structure, symmetries, geometry, lower-order terms, and system architecture. Comparable PDE pairs receive an exact rational Affinity score, while actual transformations, limits, analogies, and explicit rejections are stored separately as referenced Verdicts. Version 1.0.0 contains 52 registered objects, including 50 PDEs, 53 Verdict records, and all 1,225 distinct PDE-pair scores. Affinity is intended as a transparent research-prioritization tool, not as proof of equivalence. The framework aims to support literature triage, cross-domain comparison, mathematical education, knowledge representation, and the generation of research questions requiring expert review.