Exactness as an Explanatory and Computational Target: A Reduction Architecture for First-Order ODEs, with a Higher-Order Outlook

A typical first-order ODE syllabus is often presented as a sequence of named methods, which can obscure the unity among them. We propose organizing a substantial part of the classical symbolic first-order syllabus around a single computational target: an exact representative. For a regular scalar equation represented by a one-form $\omega$, the reduction problem is to find, on an appropriate coordinate patch, a coordinate change $\Phi$ and a nonvanishing factor $\mu$ such that $\mu\Phi^*\omega=dG$, so that solution curves are level sets $G=C$. This yields a two-floor architecture. The structural floor records the target, transformations, and domain restrictions; the operational floor executes substitutions, normalizations, and integrations. Linear and separable equations arise from the simplest one-variable integrating factors, while homogeneous, Bernoulli, and related classes become reduction paths to the same target. In an AI- and CAS-assisted environment, this shifts emphasis toward recognition, justification, auditing, and interpretation. The extended version also develops reverse construction, a structural exercise laboratory, comparisons of reduction paths, and a higher-order outlook through first integrals, operator factorization, and structural normalization.