lin-0097

7.4 The infinitesimal decision obstruction

Assume a computational realization supplies comparison morphisms

\[ \chi ^L_v: \dot L_v^{\mathrm{ext}} \longrightarrow \dot L_v^{\mathrm{dir}}, \]

and

\[ \chi ^R_v: \dot U_v^{\mathrm{ext}} \longrightarrow \dot U_v^{\mathrm{dir}}, \]

with directions adapted as necessary to the variance of the realization. LINCS retains their failures rather than immediately reducing them to a single loss.

Definition 7.3 Infinitesimal decision obstruction

Let \(\mathcal O_{\mathrm{iso}}(\chi )\) denote the universal obstruction object for a sketch declaration that the comparison \(\chi \) is invertible. The infinitesimal decision obstruction is the typed family

\[ \mathcal O_{\mathrm{ID}}(\theta ,v) = \bigl( \mathcal O_{\mathrm{iso}}(\chi ^L_v), \mathcal O_{\mathrm{iso}}(\chi ^R_v) \bigr). \]

Its first component localizes failure in candidate generation; its second localizes failure in consistency or selection.

In an additive numerical realization, observations of these components may be residuals

\begin{align*} \Omega ^L_v & = \dot L_v^{\mathrm{dir}}-\dot L_v^{\mathrm{ext}},\\ \Omega ^R_v & = \dot U_v^{\mathrm{dir}}-\dot U_v^{\mathrm{ext}}. \end{align*}

The types of the two terms must first be aligned by the declared comparison. These residuals are computational witnesses, not the definition of the obstruction.

The separation matters. A rollout can vary smoothly while the final decision jumps because an active constraint changes. Conversely, the final action can remain fixed while the candidate landscape becomes unstable. The latter is invisible if one differentiates only the selected action.