The Dark-Fraction Ledger Is Geometry: One Universal Concentration Reproduces the Host-Mass Dependence, and the Lyman-alpha Forest Closes the Late-and-Puffy Component

A MOND kernel on the de Sitter-locked acceleration scale passes galaxies, clusters and the CMB together only if a real cold component is present with a fraction of its LambdaCDM halo mass that rises with host mass: below 0.105 inside three disc scale lengths of a 1.2e10 solar-mass spiral, 0.14 inside 30 kpc of the Milky Way, 0.576 inside R500 of X-COP clusters (f ~ M^0.16). We show these numbers are geometry: the measurement radii sit at 0.50, 1.19 and 2.71 LambdaCDM scale radii, so a component whose haloes share one universal concentration c* shows exactly such a rise; the fit gives c* = 0.40 and f = 0.086, 0.165, 0.704, all within a factor 1.6 of the targets. The cost is that c* = 0.4 lies far below the concentration of any collapsed halo (3-4): the component would have no structure below roughly a megaparsec, i.e. a linear spectrum cut at k <~ 1 h/Mpc. Whether nonlinear top-down fragmentation could regenerate the Lyman-alpha forest power from such a spectrum was an open door; a phase-matched particle-mesh comparison closes it: at k = 5 h/Mpc and z = 2.2 the regenerated power is 0.04, 0.45, 0.81 of LambdaCDM for k_cut = 1, 2, 4 h/Mpc, against a forest tolerance of 0.9, so the forest requires k_cut > 4 while the puffy profile requires k_cut <~ 1. With the acoustic route closed by the clock stability theorem (doi:10.5281/zenodo.22717950) and the decay/kick route by a lifetime pincer, the three physical routes to a host-mass-dependent depletion are each computed to failure, and the ledger's geometric shape is the exact target any future mechanism must reproduce. AI-assisted research programme; not peer reviewed.