D1: the stage, not the mechanism
The base D1 is ordinary one-dimensional observable space. It carries no special structure of its own — it is where motion is measured, not where the physics originates.
Why D1 x S1
Not spacetime with an extra dimension bolted on — a minimal structure chosen so that one local rule is enough to reproduce relativistic kinematics.
S1: a clock built into the geometry
The fiber S1 is a closed internal direction of fixed circumference λ0, attached to every point of the base. It must be periodic, so phase motion can cycle indefinitely, and compact, so only standing waves with a discrete spectrum of harmonics are physically allowed. A circle is the minimal manifold with both properties — nothing simpler would do the job, and nothing more complex is required for kinematics alone.
One constraint, not a postulated metric
The entire kinematical content rests on a single local rule: the wave's total propagation speed, split between its base and fiber components, equals c. This is a pointwise relation on the wavefield, not an a priori statement about how different observers' measurements must relate — the relativistic structure is recovered as a consequence of this rule, not assumed alongside it.
Pre-Minkowskian, precisely
The constraint is metric-like on the space of wave-vectors — it has Lorentzian signature built in. What is not assumed is a 4D Minkowski spacetime as a primitive object, nor the Lorentz transformations as primitive symmetries. The Minkowski structure is recovered in Theory as the geometry of consistency among different observers' descriptions of the same bundle, not introduced at the foundation.
Not Kaluza-Klein
Both frameworks add a compact internal dimension, and the resulting wave operator is formally similar. But Kaluza-Klein starts from a 4D Minkowski (or curved) spacetime taken as primitive, with the internal dimension generating gauge symmetries. Beta assumes no spacetime metric at the outset; the internal dimension encodes rest mass and inertia, not gauge interactions. The distinction is ontological, not mathematical.
What comes next
D1 x S1 is the minimal structure for relativistic kinematics in (1+1) dimensions. Spin and electric charge require extending the fiber to S3, since a circle cannot carry half-integer representations — that extension is ongoing work beyond the scope of the manuscript this site is currently built around.