ora-0164
14.6 Tangent repair certificates
The preceding convergence theorems concern an update that is already well posed. Repair begins one level earlier: a comparison defect \(d:Z\to V\) has been observed, and the learner seeks a variation \(\delta z\in T_zZ\) that removes it without leaving the admitted tangent cone or disturbing protected blocks.
At a state \(z\), a first-order tangent repair for defect \(d(z)\) is an admitted tangent \(\delta z\) satisfying
in the observer’s linearized defect object. It is supported on a block \(B\) when its projections to the protected complementary tangent blocks vanish.
The equation distinguishes feasibility from optimization. If \(-d(z)\notin \operatorname {im}(Dd_z)\) on the admitted tangent cone, then no infinitesimal repair exists inside the current declaration. A larger step or a change of doctrine may still succeed, but the tangent obstruction must not be hidden by an optimizer that merely reduces the residual.
Let \(Z\subseteq \mathbb R^m\), let \(d:Z\to \mathbb R^k\) be continuously differentiable, and suppose its derivative is \(L\)-Lipschitz on the segment from \(z\) to \(z+\delta z\). If \(Dd_z(\delta z)=-d(z)\), then
If the step is conservative on protected blocks, their first-order variations remain zero as well.
The integral remainder formula gives
The first two terms cancel. Lipschitz continuity bounds the integrand by \(Ls\lVert \delta z\rVert ^2\); integration gives the stated inequality. The protected-block claim is the definition of conservative support.
This theorem is local. Repeating tangent repairs requires a trust region, retraction, or other realization map and a proof that the resulting base trajectory remains in the neighborhood where the derivative estimate holds. The homotopy profile from Chapter 14 is also still needed: a linear equation can have many solutions lying in distinct globally inadmissible components.