Not Free: Three Computed Floors Refuting the Claim of Zero-Energy Phase Transit

A claim circulates in popular discussions of extra coordinates: because the fifth coordinate is 'only a phase', advancing it costs no energy and transit along it is free. This paper states the claim plainly and refutes it with three independent floors, each computed. Recording the arrival address of 6.558 bits at 300 K costs at least 1.883e-20 J by the Landauer limit. Supplying the unit turn energy of 0.010 eV (1.602e-21 J) with a recovery efficiency of 99 percent still dissipates 1.602e-23 J per turn, or 1.602e-17 J over a million transits. Completing a turn in one second requires an energy scale of at least 3.313e-34 J by the quantum speed limit. Each floor is small; none is zero. 'Free transit' is therefore not an approximation that improves as technology improves; it is excluded in principle, and the honest claim is that phase transit can be cheap. Computed; not measured. SIMULATED, NOT MEASURED.