The Theory of the Scale Dimension: A Five-Dimensional Framework for the Hierarchy, Dark Matter, and the Cosmological Constant

We present a five-dimensional linear-dilaton framework in which the Planck–TeV hierarchy, theidentity of dark matter, and the constancy of dark energy follow from one geometry. The hierarchyarises as the exponential of a modest slab width. Dark matter is a clockwork axion, the note, withmass in 10−16, 10−13 eV on the charge rung qf = 2 (decay constant up to 8.8 × 1015 GeV),misalignment angles θi = 0.5–1.0 rad, and relic abundance set by vacuum misalignment. The sectorsatisfies cosmological consistency in full: the heavy gear radion decays in-sector with lifetime near1.7 × 10−24 s, eighteen orders of magnitude before nucleosynthesis (∆Neff near 2 × 10−11); the noteis cold dark matter by redshift near 109 across its window, pure to one part in 1027 by recombination,shifting the sound horizon at the 10−7 level; the microwave background is that of ΛCDM. Localgravity is protected by symmetry, since the sector’s light state is a derivative-coupled pseudoscalarwith no trace-channel coupling, and solar-system bounds independently constrain any scalar-tensordynamics below observability. The late universe of this theory is a cosmological constant, withgravitational-wave sirens propagating as dGW = dEM. On the apparent horizon we obtain the identityΩΛ(t) = A(t)/Afinal, equivalently ΩΛ(t) = S(t)/Sfinal under the Bekenstein–Hawking assignment,with the onset of acceleration at one third of the final area and a two-clock structure whose equalledger halves occupy 74% and 26% of cosmic time. The framework is falsifiable on named tests: asecond same-class ultralight line within a mass ratio 6.0 × 104 excludes the geometry; a measureddecay constant between 1.6 × 1011 and 3.7 × 1012 GeV excludes it under any allowed parameters;and we predict no robust phantom crossing in next-generation reconstructions, deflation of currentevolving-dark-energy significances, and a Euclid lensing amplitude consistent with zero.