The Integrality Gap of the Congruent-Disk Covering Relaxation, with Certified Lower Bounds for the Disk Covering Problem

Let r(n) denote the least radius such that n congruent disks cover the unit disk D. We establish three results. First, we prove that the natural linear-programming relaxation of the covering problem coincides with the fractional covering number of the disk, and that its integrality gap converges exactly to the Kershner constant 2π/√27; consequently no bound derived from this relaxation can asymptotically exceed 90.94% of the optimum, which identifies angular information as the missing ingredient for further progress. Second, we derive from this relaxation and its integer strengthening the first systematic table of certified lower bounds for r(n) in the range 3 ≤ n ≤ 21, improving the elementary area and boundary-arc bounds by 7 to 15 percent; the bounds are exact for n = 3, 4, 7 and come with dual certificates verifiable through a single one-dimensional supremum. Third, we introduce an exhaustive branch-and-bound search over configuration space formulated in polar coordinates, in which the covering condition becomes a one-dimensional interval-covering problem on each concentric circle. The method yields rigorous enclosures of r(n) for 3 ≤ n ≤ 8 accurate to nine significant digits at a cost of between one and 26 917 nodes, independently confirming results previously obtained by hand and thereby validating the search against known ground truth. A self-contained reproduction script requiring only NumPy regenerates all enclosures in approximately twenty seconds.