Emergent Time from Helical Motion on a Compact Matter Dimension: Rank-One Locking of the Constants and the Underivability of the Hierarchy

In a framework with three-dimensional space, one compact matter dimension and no external time coordinate, the constants of nature are powers of one variable, the radius of the matter dimension in Planck units. All their variations are therefore locked to one field σ, the logarithm of that radius, with fixed coefficients (rank one), in contrast to models in which each constant varies independently. Because all masses change together, the proton-to-electron mass ratio is exactly constant; any measured drift of it falsifies the locking. Self-consistency of the action makes σ dynamical: it is the Kaluza–Klein radion, sourced by the mass density and by the electromagnetic invariant. Chameleon screening is excluded by computation; a stiff branch and a Vainshtein-screened branch survive, and the bounds on the coupling from white dwarfs, atomic clocks, lunar laser ranging and MICROSCOPE are worked out. The response of the fine structure constant has two branches that experiment can separate. The hierarchy number, the ratio N ≈ 2 × 1021 of the electric to the gravitational coupling of the electron, is shown to be underivable within the scale-free axioms by four independent arguments; their one positive output is that fermionic winding modes obey antiperiodic boundary conditions. The scale must therefore be supplied by a separate condition, the holographic pinning of a companion paper, after which the winding numbers of the charged leptons and the range and mass of σ become definite and the derivation of the fine-structure constant reduces to one core number.