Gradient descent avoids strict saddles with a simple line-search method too

The paper proves that stabilized Armijo backtracking gradient descent (including the Riemannian analytic-retraction case) avoids strict saddles for almost all initial step sizes by (i) showing step sizes stabilize along convergent orbits, (ii) applying a center-stable manifold argument that only requires invertibility almost everywhere, and (iii) globalizing via the Luzin N−1 property for each fixed-stepsize map g(i), obtained by a Fubini–Tonelli slicing argument; see Algorithm 1 and Theorem 4.1 with its proof roadmap and use of Theorem 3.11 and Theorem 1.3 . By contrast, the model’s proof hinges on representing the stabilized Armijo method as a time-invariant piecewise-smooth map T: M→M and then invoking a preimage-of-null-sets lemma for T. This is not valid: the stabilized algorithm’s state includes the current step size (αt), so there is no single autonomous map on M capturing the dynamics (a difficulty explicitly noted by the paper) . The model also incorrectly asserts that the “determinant-zero” set is measure zero because it is the zero set of a C^1 function; the paper instead proves the Luzin N−1 property correctly via parameter-slice finiteness and Fubini-type reasoning (Lemmas/Thms in Section 3) . Finally, the model employs a diffeomorphism version of the center-stable manifold theorem and handwaves generic exclusions, whereas the paper uses a version that does not require invertibility at specific points (Hirsch–Pugh–Shub) . Hence, the paper’s argument is sound and complete for its claims; the model’s argument contains key logical gaps.