lin-0111

8.7 A four-level audit

For a declared relation \(p\sim q\), Deep LINCS can retain four observations:

\begin{align*} E_0 & = \omega _0\bigl(D(p),D(q)\bigr),\\ E_1 & = \omega _1\bigl(TD(p),TD(q)\bigr),\\ E_U & = \omega _U\bigl(U_p,U_q\bigr),\\ E_r & = \omega _r\bigl(r_p,r_q\bigr). \end{align*}

The first measures finite forward disagreement. The second measures directional disagreement. The third compares parameter updates, and the fourth compares requests passed across module boundaries.

Level	Question	Typical computation	Common blind spot
Base	Do the declared forward routes agree?	output residual or factorization witness	nearby instability
Tangent	Do their admissible local variations agree?	JVP, VJP, or higher probe	wrong probe semantics
Update	Do the routes propose compatible parameter changes?	optimizer-state comparison	gauge and parameter sharing
Request	Do they send compatible credit backward?	interface cotangent comparison	agreement only at outputs

Table 8.1 Deep LINCS keeps the four levels visible rather than merging them into one training loss.