Published October 29, 2026
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Emergent Time from Helical Motion on a Compact Matter Dimension: Constants, the Holographic Pinning of the Mass Quantum, and the Vacuum Energy
Description
In a framework with three-dimensional space, one compact matter dimension and no external time coordinate,
the axioms fix structure but not scale, leaving a mass quantum k as an input. We treat the constants. All variations
of the constants are locked to one fibre-strain field σ (rank one), whose operational signature is the exact constancy
of the proton-to-electron mass ratio; the dynamics of σ, its screening branches and the data bounds are worked out.
The hierarchy number is shown to be underivable within the scale-free axioms by four independent executions. A
holographic pinning condition, the saturation of the Cohen–Kaplan–Nelson bound, fixes k = √ mPmH ≈ 4.15 meV,
the geometric mean of the Planck and Hubble masses, with its direction derived from gravitational stability. Applying
Huygens’ principle to the lines of the framework — each point of space re-emits the line it receives, one relay per
cell of the line’s own Compton size, each carrying the endpoint energy k/4π that completes the neutrino spectrum
— gives the vacuum energy ρΛ = k 4/4π, hence w = −1, ΩΛ = 2/3 (a retrodiction) and a local route to the pinning
condition that coincides with the global one wherever ΩΛ = 2/3. The cosmic coincidence is reduced to the selection
of the configuration that contains the observer. The fine-structure constant is identified as a vortex logarithm, 1/α =
2ln(mP/k)−Lc, with one core number left open. Every comparison is computed with the framework’s own equations
from observed quantities. With Ωm = 1/3 the measured CMB acoustic angle gives H0 = 66.3±0.3 km s−1Mpc−1 , in
line with the neutrino value 65.9±0.5, and the neutrino and CMB routes to k then agree to 0.8σ; Ωm = 1/3 agrees
with the supernova samples, while the DESI DR2 BAO shape lies 3.9σ below it, mainly through its Lyman-α points,
and the BAO amplitude requires H0 ≈ 69 at Ωm = 1/3, 3–4σ from both the CMB angle and the neutrino value. The
framework’s own sound horizon, rd = 146.6 Mpc, is the standard one.