The Ruler-Free Test: Alcock–Paczyński Ratios from DESI Grade the Shape Alone

Every baryon acoustic oscillation measurement is a ratio to the sound horizon, which makes BAO evidence entangled with the early universe: change the ruler and every distance moves. One combination escapes the entanglement. The Alcock–Paczyński ratio F_AP(z) = D_M(z) H(z) / c, formed from the transverse and line-of-sight BAO measurements at the same redshift, divides the ruler out exactly, and grades the expansion history's shape alone — blind to the Hubble constant, blind to the sound horizon, and therefore immune to every early-universe resolution of the Hubble tension. This paper forms F_AP from the five anisotropic DESI 2024 measurements and scores three shapes against it. The observed ratios run from 0.649 ± 0.022 at redshift 0.51 to 4.661 ± 0.144 at redshift 2.33. The Planck shape scores a chi-squared of 9.72 for five points, the evolving-dark-energy shape 9.65, and the low-matter-fraction shape of the small-ruler combination 13.01. Fitted alone, the ratios return a matter fraction of 0.316, with a one-sigma range of 0.290 to 0.345 — centred on the Planck value and excluding the small-ruler combination's 0.268 at the level the present errors allow. All three shapes share one common failure, the redshift-0.51 ratio, which sits high against every prediction and traces back to that sample's line-of-sight measurement. The ruler-free test's verdict is that the expansion's *shape* at intermediate redshifts is close to the standard one, so the burden of the tension falls where this test cannot see it: on the normalisation and the ruler, jointly. This study is computed from published values; it is not a measurement.