The Phase-Winding Energy Ladder: Quadratic Cost of Transit in a Fifth Coordinate

Transit in a fifth coordinate that is an angular phase chi requires the phase to advance, and advancing it costs energy. This paper derives the resulting energy ladder from a stated kinetic form, E(n) = E1 n^2 for winding number n, with the unit step fixed at E1 = 0.01 eV, and tabulates which windings a given energy budget can buy. A budget of 1 eV admits windings up to n = 10, a budget of 10 eV admits n = 31, and a budget of 100 eV admits n = 100, while the twentieth winding alone costs 4.0 eV, four hundred times the first. The same calculation fixes how finely the final phase can be labelled: at the stated readout resolution of 0.0667 rad there are 94.2 distinguishable phase positions in one turn, or 6.558 bits of phase information per turn. The cost of transit therefore grows quadratically with distance measured in turns, not linearly, and any scheme that prices long transits as repeated short ones understates the energy by a factor that reaches 400 at twenty turns. All results are computed from the stated assumptions; nothing was measured. SIMULATED, NOT MEASURED.