A certified opaque barrier for the unit disc of length 4.79984937…
Abstract
An opaque set for the unit disc is a set meeting every straight line that meets the disc; the infimum of the lengths of such sets is the beam detection constant, whose value is unknown. The best published upper bound, realised by a three-piece barrier of Faber–Mycielski type, is 4.79989. I exhibit an opaque barrier of length 4.799849374678549857544655408276…, and prove its opacity. The barrier is that same three-piece construction evaluated at its true stationary parameters rather than at the rounded parameters underlying the published figure. The opacity proof given here is exact: it rests on a closed-form apex identity and a convexity argument, and requires no numerical sweep over directions. Its only numerical inputs are two chord parameters, certified by interval arithmetic to lie in (0, 1). The length is likewise a certified interval enclosure of width below 10⁻⁵⁹. I claim no new construction and no optimality for the beam detection constant itself.
Cite it
Gonzalez, V. (2026). A certified opaque barrier for the unit disc of length 4.79984937…. Zenodo. https://doi.org/10.5281/zenodo.21568684