βBeta Hypothesis

← Back to Theory

SR vs Beta: Logical Sequencing

Special relativity builds outward from two physical postulates toward geometry. Beta inverts this: it starts from a single geometric postulate and derives both the kinematic structure and, in later work, the internal quantum numbers as consequences.

Comparative table

# SR (orthodox) Beta
1Postulate 1: principle of relativityPostulate: substrate D¹×S¹ (later extended to D³×S³)
2Postulate 2: invariance of cPostulate: internal phase / winding on S¹ = rest mass
3Derived: Lorentz transformations (from 1+2)Derived: dispersion cone (from phase-velocity relation)
4Derived: Minkowski metric (invariance of the quadratic form)Derived: Gudermannian bridge (circular ↔ hyperbolic geometry)
5Derived: dispersion relation / mass-shell condition (final link)Derived: velocity composition, SO(1,1)
6—Derived: Lorentz transformations
7—Derived: Minkowski metric
8—Postulate: extension to S³≅SU(2) (Beta 2)
9—Derived: covariance lemma (SU(2) action: postulate → consequence) (Beta 2)
10—Derived: Peter-Weyl decomposition → SU(2)L/SU(2)R = spin/charge (Beta 2)
11—Derived: photon as gauge boson of the Hopf U(1) (Beta 3)

Rows 7–11 (dimmed) belong to Beta 2 and Beta 3, not yet covered elsewhere on this site — included here to show where the program is headed.

The conceptual hinge

SR postulates two physical axioms — relativity and the invariance of c — and derives all subsequent geometry from them. Beta postulates a single geometric axiom — substrate plus winding number — and derives from it both the kinematic structure (dispersion, Lorentz, Minkowski) and, in later work, the internal quantum numbers (spin, electric charge): one foundational act generating more structure, at the cost of a less immediately intuitive manifold (S³ vs. flat space).

The key reversal: in SR, the dispersion cone is the last link in the chain — a consequence of an already-postulated metric and transformation group. In Beta, the dispersion cone is the first non-trivial geometric object, and the Lorentz/Minkowski structure is what follows from it.

Classification follows the Derived/Postulated/Open convention used across the Beta Hypothesis manuscripts. See Theory for the full Beta 1 narrative this table summarizes.