Foundations
EstablishedFrozen axiom — constitutionally non-negotiableThe S³ Universe
The universe is a 3-sphere. This single axiom eliminates spatial infinity, provides finite total mass, and hosts the Hopf fibration that underlies all particle structure.
Last updated: August 2026
The Core Claim
The spatial universe is the 3-sphere , embedded in with a fixed radius — the cosmological radius. This is Axiom A1 of the ontology freeze. It is not a model to be tested against alternatives: it is the geometric substrate on which everything else is constructed.
The S³ Metric
In standard hyperspherical coordinates , the round metric on with radius is:
Round metric on S³ · χ ∈ [0, π], θ ∈ [0, π], φ ∈ [0, 2π)
Why S³ — and not flat ℝ³?
The choice of over flat is not cosmetic. It has three direct physical consequences that the CPT framework requires:
No spatial infinity
S³ is compact. There is no boundary at infinity, no need for boundary conditions at large r, and no infrared divergences from an unbounded volume. Total energy is finite.
Finite total mass
The finite volume of S³ with radius R_U gives a finite total mass for the universe. This is required for Mach's principle: the inertial frame is defined by the totality of mass, which must be finite for the definition to be well-posed.
Hopf fibration is globally defined
The Hopf fibration π: S³ → S² (with fibre S¹) is a global topological structure of S³ — it cannot be defined on ℝ³. It is the geometric object that hosts particle spin and the exchange statistics of the FR theorem.
The Hopf Fibration on S³
The Hopf fibration is the map that fibres over with fibre . Concretely, writing as pairs with , the map is:
Each point on has a fibre — the “seat” of a particle in the CPT language. Exchanging two seats traces a path in that picks up a phase — the topological origin of half-integer spin.
See Hopf Fibration for the full exchange construction and FR theorem application.
The Cosmological Radius R_U
The parameter is the radius of the universe. It enters the framework as the single free geometric parameter. In the CPT derivation of Newton’s , it appears through the Mach closure condition:
where is the total mass of the universe. This is the Schwarzschild condition on the universe itself — fixing in terms of and . Together with the Weber double-copy relation, this closes the system.
See Mach Closure and Deriving G.
Integrity Status
Source Documents
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