Roy Lewallen wrote:

What modeling program are you using that gives those results? A half
wave vertical over perfect ground has about 5.15 dBi gain at the
horizon. . .

Correction -- a *quarter* wavelength ground-mounted vertical has the
gain I quoted. A half wavelength ground-mounted vertical has a slightly
narrower pattern, resulting in a little more gain over perfect ground.
It's about 6.9 dBi at the horizon.

Roy Lewallen, W7EL