Quintessence-Like Dynamics of the Fifth Coordinate: The Allowed Drift of the Dark-Energy Equation of State

If the radius of a fifth coordinate changes slowly with expansion, the effective dark energy it supplies acquires an equation of state that drifts from minus one. This paper parametrises the drift by the fractional change of the radius per e-fold of expansion, derives the corresponding present equation of state and its redshift evolution in the standard two-parameter form, and confronts the result with the published constraint that the present value lies within a few hundredths of minus one when combined data are used. Only a radius changing by less than roughly nine percent per e-fold survives. That bound converts future improvements in dark-energy measurements directly into limits on extra-dimensional dynamics. Headline values are: alpha bound approx = 9.0e-02, w at alpha bound = -9.400e-01, w if radius doubled since z1 = -7.200e-01. This study is simulated and computed from stated assumptions; it is not a measurement.