Emergent Time from Helical Motion on a Compact Matter Dimension: The Holographic Pinning of the Mass Quantum, the Vacuum Energy and the Hubble Constant

In a framework with three-dimensional space, one compact matter dimension and no external time coordinate, the vacuum energy is fixed by a laboratory mass. Huygens’ principle applied to the lines of the framework — each point of space relays the line it receives, one relay per cell of the line’s Compton size, each carrying the endpoint energy k/4π that completes the neutrino spectrum — gives ρΛ = k 4/4π and w = −1. With the mass quantum k from the reactor-neutrino splittings, ρΛ 1/4 = 2.199±0.008 meV, with no cosmological input. The zero-point energy, larger by 10122, does not gravitate: space always moves helically, that motion is space itself and not a source within it, and only the heads of relayed rays carry energy; the mode sum is also excluded by the derived direction of the holographic bound. The same k fixes the Hubble constant. A pinning condition, the Cohen–Kaplan–Nelson bound as an equality (direction derived here, equality in a companion paper), gives k = √ mPmH, the geometric mean of the Planck and Hubble masses, and with the relay count ΩΛ = 2/3. Independently of the relay identification, least action along the lines gives Ωm = 1/3, which turns the relay count into a cross-check. Reactor neutrinos then measure the Hubble constant in the laboratory: the JUNO splittings give H0 = 65.9±0.5 km s−1Mpc−1 , the CMB acoustic angle at Ωm = 1/3 gives 66.3±0.3, and the two routes to k agree to 0.8σ; six years of JUNO data would fix H0 to 0.2%. The framework has no time coordinate, yet every quantity of the description with one follows by conversion, with no parameter fitted to astronomical data: what that description calls the age of the universe comes out as 13.9 ±0.1 Gyr from the reactor data. Through the pinning condition 1/α is tied to ln(RH/ℓP), which explains Eddington’s coincidence. Computed with the framework’s own equations from observed quantities, Ωm = 1/3 agrees with the supernova samples; the tensions with the BAO data are quantified