The Hubble-Kernel Growth Equation Against DESI DR1: Consistent, Not Yet Discriminated, and the Precision That Decides

The Hubble-kernel growth equation (doi:10.5281/zenodo.22706925) fixes the effective gravitational coupling of linear structure with no free parameter, Geff(z)/G = ν(cH(z)/a0) with (cH0/a0)2 = 8π/(3κ2ΩΛ) = 49, and predicts fσ8 raised over Planck-ΛCDM by 2.7%, 1.9%, 1.1% at z = 0.3, 0.6, 1.0 (canonical footing; 3.8, 2.6, 1.6% alternative). We confront it with the DESI DR1 full-shape growth measurements (DESI 2024 V, Table 9, ShapeFit fσs8/(fσs8)fid in six bins, both sub-panels) and with the DESI modified-gravity parameter μ0 = 0.05 ± 0.22. Result: consistent (χ2 per bin 0.65-0.73 on both footings), not yet discriminated from ΛCDM (|Δχ2| < 0.25; DR1 errors 10-19% vs a 1-4% effect); the present-day coupling excess (0.077 / 0.099) lies 0.1-0.2σ from μ0. A 3σ test needs ~1% on fσ8 at z ≈ 0.3-0.6 (DESI final sample). The prediction is on record before the decisive data; its sign (more growth) is the one the S8 tension does not favour. AI-assisted research programme; not peer reviewed.