4.4 Why infinitesimal?

The word infinitesimal describes the resolution of the diagnosis, not the permitted magnitude of the eventual repair. Tangent structure is important to LINCS because it turns an undifferentiated failure into a family of typed directional questions. Given an admissible path of models \(D_t\) with initial direction \(v=\dot D_0\), the tangent lift asks how each declared composition responds along \(v\). Different directions can expose different arrows, interactions, or contexts even when a scalar loss assigns them the same value.

The construction has three further advantages. It is compositional because tangent transport respects composition. It is localizable because a direction can be supported on a declared component or overlap. And it makes null variation explicit: symmetry, reparameterization, or decision-irrelevant directions can be quotiented before repair effort is assigned. These properties make infinitesimal analysis a diagnostic microscope for structural failure.

When the relevant directions integrate, local repairs can assemble into a finite trajectory:

In such cases an apparently abrupt macroscopic change may be the accumulated effect of many locally intelligible changes. But tangent data alone neither guarantees that \(v_t\) integrates nor that \(D^{+}\) lies in the same connected repair regime.

First-order analysis also has familiar blind spots. An obstruction varying as \(t^2\) has zero first derivative at \(t=0\) without being locally constant. Higher Weil probes, jets, brackets, and curvature can expose some such effects. They still cannot manufacture a path to a disconnected model, a new primitive, or a new theory whose vocabulary is absent from \(\mathbb S\).