lin-0044
2.1 From loss minimization to structural diagnosis
Let \(J=\operatorname {Path}(S)\) be the path category generated by a sketch \(S\), and let \(D:J\to \mathcal C\) be a candidate diagram. Declared path equations induce a quotient \(q:J\to J/{\sim }\). Compositionality is not defined by adding an arbitrary penalty; it is the existence of a factorization.
The base obstruction \(\mathcal O_0(D)\) records the failure of \(D\) to factor through the declared quotient, or to satisfy the specified universal property. Under Axiom 1.1, it is trivial precisely when the declared structural condition holds. A particular observer may fail to detect a nontrivial obstruction.
This formulation separates the categorical failure from its measurement. One may later observe \(\mathcal O_0(D)\) through a norm, likelihood, test statistic, or decision functional, but those scalarizations are not the obstruction itself.