lin-0028

0.14 From structural failure to learning signal

We can now state the transition that motivates the book. Let a sketch \(\mathbb S\) declare a structural promise and let \(D\) be a candidate model. Write

\[ \operatorname {Fact}_{\mathbb S}(D) \]

for the space, category, or type of witnesses that \(D\) satisfies the declaration. Under the representability and triviality axiom introduced in Chapter 1, LINCS associates a base obstruction object

\[ \mathcal O_0(D)=\mathcal O_{\mathbb S}(D), \]

which is trivial exactly when the declared factorization exists. After an admissible tangent lift, the corresponding infinitesimal obstruction is

\[ \mathcal O_1(D)=\operatorname {INC}(D) = \mathcal O_{\mathbb S}(TD). \]

The obstruction object is intentionally abstract. Depending on the application, it may represent missing fillers, incompatible families, cohomology classes, route differences, failed closure conditions, or other typed diagnostics. To compute with it, one chooses an observer

\[ \omega :\mathcal O_k(D)\longrightarrow Z \]

and perhaps a scalarization \(\sigma :Z\to \mathbb R\). The composition \(\sigma \circ \omega \) can become a loss or test statistic. It is evidence about the obstruction, not its definition.

Design principle

The order of construction is:

\[ \begin{aligned} \text{typed declaration} & \longrightarrow \text{factorization problem} \longrightarrow \text{obstruction}\\ & \longrightarrow \text{observation} \longrightarrow \text{scalar diagnostic}. \end{aligned} \]

Reversing this order risks inventing a structural story for a convenient number after the fact.

A repair is similarly typed. It may change an adapter, a policy component, a relational chart, a local predictive section, or a Toulmin warrant. The application must say which changes are allowed, which variations are null, and what separate evidence admits the result.