Emergent Time from Helical Motion on a Compact Matter Dimension: A Candidate Geometric Origin of the Fine-Structure Constant and a Neutrino Test

In a framework with three-dimensional space, one compact matter dimension and no external time coordinate, the form of electrodynamics is derived and its strength is one parameter. A companion paper noted that 1/α ≈ 2ln(mP/k), with k the neutrino mass quantum, carries 97.6% of the measured value. We give this proximity a structure. Mass and charge are rates of one fibre angle in two directions, which the quota treats alike; a unit twist then presents to space the two-dimensional Green’s function of the cross-section of the matter dimension, with coefficient 4π/2π = 2. The scale independence of α, from the electron g−2 to atom recoil, excludes any logarithm accumulated over spatial distances, and with it renormalization-group and induced-coupling readings. The remaining O(1) term is set by the core of the charge. About ninety pre-listed geometric rules fail. One rule built from framework elements — a ring of the reduced Planck radius √ 8π ℓP, where the reduced Compton bound meets a conical deficit of one radian, one Planck cell thick — gives 1/α = 137.024 with no free parameter, 1.7σ below the measured value in units of the uncertainty of k; equivalently 1/α = ln(hc¯ /8πGk2 ) − 0.070. Its arguments were assembled after it led, one of them holds only if the conical deficit does not act on the geometry of the cross-section, and a chance agreement at this level has a probability of about one in five. Inverted, the rule predicts k = 4.1155 meV and ∆m 2 31 = 2.471 × 10−3 eV2 , which reactor-neutrino data will test within a few years.