Restricted Minimax Certificates for Integerized-Geometric Estimation

This paper gives a restricted minimax certificate for a geometric estimation problem formulated in the language of Integerized Geometry (IG). The objects are planar radial traces represented by radial functions. The loss is Hausdorff distance. The data are noisy samples of the radial function on a uniform angular grid. The estimator is an explicit IG decoder: it records a finite list of grid-aligned knots and an integer payload of quantized local-polynomial coefficients, then decodes those records into a set estimate. The main theorem identifies the minimax scale, up to constants, for the declared contract. For a Hölder smoothness parameter s > 0, sample size n, and Gaussian noise level σ, the worst-case Hausdorff risk is bounded below by Ψn,σ = σ 2 log n n s/(2s+1) ∨ n −s . The first term is the statistical price of uniform recovery under noise, and the second is the deterministic sampling barrier. A tuned IG local-polynomial decoder attains the same rate on the certified restricted regime. The result is not a universal claim about all shapes or all sensing models. It is a fully scoped certificate: once the object class, observation model, loss, event information, and admissible decoder are declared, the theorem says exactly what can and cannot be guaranteed in worst-case Hausdorff risk.