Certified Minimal Robust Optical Probes for Finite Newton-Ring Reconstruction under Bounded Measurement Error

We study a finite, synthetic inverse problem based on an idealized Newton-ring interference model. The reconstruction task consists of twelve candidate lens geometries and sixteen available radius–wavelength measurement probes, with independent adversarial absolute intensity errors bounded by 0.04. Using exhaustive pairwise separation analysis and finite combinatorial optimization, we establish that two probes are necessary and sufficient for guaranteed reconstruction when illumination at 589.3 nm and 532.0 nm is available. Exactly 35 optimal fixed two-probe configurations are identified. In contrast, the eight probes using only 589.3 nm illumination fail to distinguish six pairs of candidate geometries robustly. The computational results are supported by rational interval enclosures of all 192 ideal intensity values, independently implemented numerical verification, exhaustive subset enumeration, and an analysis of adaptive bounded-error reconstruction. Both the minimum fixed-probe cardinality and the worst-case adaptive reconstruction depth equal two for the specified model. The work provides a reproducible finite benchmark connecting bounded-error inverse problems, optical interferometry, sensor selection, and validated numerical computation. The findings apply exclusively to the prescribed idealized finite model. They do not establish universal optical identifiability, experimental validation, or optimality over continuously varying geometries and measurement configurations. This manuscript is released as an unreviewed mathematical preprint with supplementary Python source code and computational verification materials. AI assistance disclosure: OpenAI ChatGPT was used substantially in mathematical exploration, programming, certificate design, literature discovery, and manuscript drafting. The named author is responsible for reviewing the mathematics, checking the released files, verifying citations and originality claims, and deciding whether to submit.