A certified opaque barrier for the unit disc of length 4.79984937…

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.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-07-30
DOI
https://doi.org/10.5281/zenodo.21701081
Citations
2
Primary Topic
Spectral Theory in Mathematical Physics
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
preprint

A certified opaque barrier for the unit disc of length 4.79984937…

Vincent Gonzalez
2 citations
Zenodo (CERN European Organization for Nuclear Research)
Spectral Theory in Mathematical Physics
preprint

A certified opaque barrier for the unit disc of length 4.79984937…

Vincent Gonzalez
preprint en
2 citations

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.

Zenodo (CERN European Organization for Nuclear Research)
Sustainable cities and communities
Spectral Theory in Mathematical Physics
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.

A certified opaque barrier for the unit disc of length 4.79984937… — Vincent Gonzalez · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS