An Effective Cosmological Constant from a Fifth Coordinate: Size of the Extra Dimension Implied by Observation

A compact fifth coordinate contributes a curvature term that can play the role of a cosmological constant. This paper turns that qualitative statement into a size. Setting the effective density from a compact dimension of radius L, with an order-unity geometric coefficient, equal to the observed dark-energy density gives a radius comparable to the present Hubble radius, not a microscopic length. Conversely, a Planck-length radius overproduces the density by the familiar enormous factor, and a millimetre radius, of the kind laboratory gravity tests discuss, still overproduces it grossly. The extra dimension can therefore mimic dark energy only if its present radius is cosmological in scale or if its coefficient is suppressed by a mechanism that must be named separately. Headline values are: L required m = 1.660e+26, L required Gpc = 5.379e+00, Hubble radius m = 1.373e+26, rho DE obs kg m3 = 5.836e-27, ratio planck length = 1.054e+122. This study is simulated and computed from stated assumptions; it is not a measurement.