How Large Can the Circle Be? A Radius Bound for the Fifth Coordinate from Its Excitation Gap

A phase that lives on a circle has a size, and the size has a spectrum. This paper derives the relation between the circle radius R of the fifth coordinate and the energy gap to its first excitation, E_1 = hbar c / R, and converts physical requirements into bounds on R. Requiring the gap to exceed the 0.010 eV transit energy scale gives R <= 1.973e-05 m; requiring a 1 eV gap gives R <= 1.973e-07 m, and a 1 MeV gap gives R <= 1.973e-13 m. In the other direction, a circle of one micrometre radius would carry a gap of only 0.1973 eV, while a Planck-length circle would carry 1.221e+28 eV. The transit scale therefore permits a fifth coordinate that is small by everyday standards yet enormously larger than the Planck length, a window of some twenty-eight orders of magnitude. Computed; not measured. SIMULATED, NOT MEASURED.