lin-0222

18.5 Transverse repair

Suppose compatibility is represented locally by a regular constraint \(g:M\to \mathbb R^m\), with descent locus \(Z=g^{-1}(0)\). Given a predictive direction \(u_x\), choose \(a_x\) so that

\[ Dg_xa_x=-g(x)-Dg_xu_x, \]

then retract \(x^+=\operatorname {Ret}_x(u_x+a_x)\).

Proposition 18.5 First-order repair

If \(g\) is twice continuously differentiable, \(Dg_x\) has a locally bounded right inverse, and the retraction is first-order accurate, then

\[ g(x^+) = O\! \left(\lVert u_x+a_x\rVert ^2\right). \]

The linear compatibility defect is removed without descending \(\lVert g(x)\rVert ^2\).

Proof ▼

Taylor expansion cancels the base and linear terms by the repair equation.

If the linear solve has residual at most \(\varepsilon _x\), the same argument gives

\[ \lVert g(x^+)\rVert \leq \varepsilon _x+c\lVert u_x+a_x\rVert ^2. \]

Solver error, statistical uncertainty, and linearization error therefore remain separately auditable.

Boundary

Exact categorical notation does not make an estimated restriction exact. A repair is admitted only when its reduction exceeds uncertainty in the overlap witness and the predicted second-order remainder.