Expansion Does Not Dilute a Vacuum: The Equation of State the Dilution Story Would Require

A widely repeated informal claim holds that the vacuum energy began large and was diluted by cosmic expansion to the small value observed. This paper tests the claim arithmetically. Granting an initial Planck-cutoff density of 1.40 x 10^74 GeV^4 and the full expansion since the Planck era, a factor of 5.19 x 10^31, a component dilutes as a^{-3(1+w)}. For a true vacuum, w = -1, the dilution factor is exactly one. Reaching the observed density of 2.52 x 10^-47 GeV^4 requires w = 0.27, close to the equation of state of radiation. The dilution story therefore requires the vacuum to have redshifted like light while remaining vacuum, which is a contradiction, and the claim is shown to fail on its own generous terms. The paper is a computation from stated assumptions, not a measurement.