SPEC: Exact Structural Constraints Are Cheap, Specifying Them Is Not

Exactness is the defining property of structure-preserving dynamics models, yet the specification it rests on, which law to impose, under which metric, and in which coordinates, is never itself tested, because a prediction loss cannot tell a correct specification from a wrong one. To address this gap, we present SPEC, a staged framework that makes the specification computable from trajectory data and applies an exact constraint only where the data support one. Its foundation is a per-step identity, $(Ux{)}^{\top }M(Ux)-x^{\top }Mx=w^{\top }{\Sigma }_M(A)w$, which reduces the structural question to the sign of one symmetric matrix and yields, under the Euclidean block metric, a library of five exact constructions together with the boundaries beyond which none exists. Three conditional stages then decide what the identity leaves open, separating a statistical decision taken on finite trajectories from a guarantee that holds at every parameter value. Experiments on analytically labelled systems isolate what each stage contributes. Discover recovers coordinates without a reconstruction weight, at median invariant $R^2$ of $0.73$ to $0.86$ against $0.10$ to $0.22$ for a comparison objective. Refusing unsupported blocks costs $4\%$ more out-of-family error than an oracle router, and enforcing every block costs $39\%$. Over the fitting horizon, exact enforcement matches a penalty of equal capacity in modal-energy error while holding the selected latent law to $10^{-15}$ where the penalty reaches only $10^{-3}$. Accuracy is therefore limited by neither stage, but by what sits behind them: the metric the construction is instantiated in, for which we give the exact form in the damped case, and the decision rule, which measures agreement inside the selection window rather than validity beyond it.