sec-approximate-comparator-descent

12.3 Approximate descent and observer transfer

When \(\varepsilon _c\) is not invertible, category theory identifies a failure of descent but supplies no canonical magnitude. A numerical bound requires additional observer structure.

Proposition 12.6 Observer transfer from an approximate comparator

Suppose a realization of \([\mathbb A,\mathcal D]\) carries a metric \(d\), the cumulative evaluation \(L\) is \(K_L\)-Lipschitz, and the numerical comparison observer \(\Omega (a,-)\) is \(K_\Omega \)-Lipschitz in its reference argument. Let \(\bar c=q^\ast \operatorname {Ran}_qc\) be admissible and suppose the realized counit obeys

\[ d(c,\bar c)\le \delta . \]

Then

\[ \left| \Omega (L^{\mathrm{on}},L(c)) - \Omega (L^{\mathrm{on}},L(\bar c)) \right| \le K_\Omega K_L\delta . \]

Proof ▼

Lipschitz continuity of \(L\) gives \(\lvert L(c)-L(\bar c)\rvert \le K_L\delta \). Applying the Lipschitz bound for the second argument of \(\Omega \) proves the claim.

The proposition locates the assumptions precisely. The counit supplies the universal comparison; the metric turns its failure into a size; the evaluation transports that size to value space; and the observer transports it to a reported regret defect. None of the last three steps follows from right Kan universality alone.

For endogenous information, this descent theorem is necessary but not yet sufficient. The object \(c\) must first be generated on its own closed-loop history. Only then may \(\varepsilon _c\) test whether the resulting strategy trajectory belongs to the comparator sketch. This preserves the nonclassical replay boundary of Proposition 9.22.