Sharp line traces of derivative ranges for bi-Lipschitz Hilbert-space maps

For a bi-Lipschitz map of a Hilbert space, a derivative can be bounded below without being onto. We determine the possible complexity of membership in such derivative ranges along affine lines. For every Π⁰₃ set A ⊆ ℝ, we construct a global bi-Lipschitz map g of a separable real Hilbert space, Gâteaux differentiable in norm everywhere, and a nonempty open cone Ω of directions. For each u ∈ Ω there is an affine line ℓᵤ consisting entirely of points where Dg is not onto, such that u ∈ ran Dg(ℓᵤ(z)) if and only if z ∈ A. The map g is fixed as u varies, and g ∘ ℓᵤ is affine. Equivalently, the inverse has a norm directional derivative in direction u along the image line exactly at the parameters in A. A general Π⁰₃ upper bound shows that the realization is sharp; in particular, the range-incidence sets can be Π⁰₃-complete. Source repository: https://github.com/eulogica/sharp-inverse-range-realization