Information Geometry of DGLAP Evolution: From Geodesic Tests to Emergent Small-Circle Geometry

We investigate the geometric structure of leading-order (LO) nonsinglet DGLAP evolution on the Fisher–Rao manifold of normalized valence distributions, asking whether the QCD-selected trajectory is geodesic and how its departure from geodesic motion is organized. Through the isometric embedding $Φ=2\sqrt{p}$, the evolution is represented as a curve on a Hilbert sphere of radius two, with its velocity and acceleration determined directly by the DGLAP operator. For a GRV98-based total-valence trajectory over $2\leq Q^2\leq 2000~{\rm GeV}^2$, we find a nonzero but weakly scale-dependent geodesic curvature. An arc-length-weighted principal-component analysis shows that two affine directions capture more than $99.999\%$ of the trajectory variance. The corresponding affine plane intersects the Fisher sphere in a small circle, whose predicted geodesic curvature closely agrees with that reconstructed from the DGLAP-generated acceleration. An $x$-resolved analysis further reveals that Fisher motion is governed by relative PDF reshaping rather than probability weight alone, with different momentum-fraction regions contributing to motion along the trajectory and its transverse bending. The Mellin representation provides a complementary perspective on the effective dimensional reduction through the collective evolution of the spectral hierarchy. These results establish a quantitative information-geometric characterization of the reference LO DGLAP trajectory and reveal an organized, near-small-circle structure over the finite evolution interval.