Dynamic Topology of the Universe

RP³ with Double Counter-Rotation — Interactive Companion
Boris Kriger | Information Physics Institute

1. The Ball Model of RP³

RP³ can be visualized as a solid 3-ball with antipodal boundary points identified: p ~ −p. Interior points are distinct. Every straight line through the center is a closed geodesic — a projective line topologically equivalent to a circle.

Drag to rotate. The white dot is a traveler: it departs from pole P, passes through center O, reaches the antipodal point P', and — because P' ~ P — returns to P. Every diameter is a closed loop.

2. The 4D Projective Sphere with Hopf Fibration

S³ lives in ℝ⁴. Stereographic projection maps it to ℝ³, then perspective projection maps to the screen. The colored curves are Hopf fibers — circles S¹ that fill S³ without intersection. The XW, YW, ZW sliders control rotation through the fourth dimension.

3. Double Rotation in 4D

In 3D, rotation fixes an axis. In 4D, rotation occurs in a plane, and there are 6 rotation planes. A general rotation decomposes into at most 2 simultaneous independent rotations in orthogonal planes. Three modes:

Simple: one plane rotates, one is fixed (3D analog). Double: both rotate at different speeds (θ₁ ≠ θ₂). Isoclinic: both rotate at equal speeds (θ₁ = θ₂) — every point traces an identical orbit. This is the Clifford rotation generating the Hopf fibration.

4. QGVP Initial State and Expansion

The initial state: RP³ at nuclear density (~2.3 × 10¹⁷ kg/m³), radius ~15 AU, filled with Quark-Gluon-Vacuum Plasma. Time is frozen (v_rot = c, v_time = 0). The Kriger Limit prevents singularity: above nuclear density, the chiral condensate is destroyed and α → 0.

v²_rot + v²_time = c²   |   Initial: v_rot = c, v_time = 0

At epoch 0: QGVP state, 4 RP³ poles visible, nuclear density, time rate ~7%. Press Play to watch expansion: density drops, temperature falls, α decreases, time accelerates.

5. Summary of the Construction

One topology → six solutions

RP³ = S³/ℤ₂ with two counter-rotations at v = c in orthogonal planes, L₁ = −L₂, L_total = 0.

SO(4) ≅ (SU(2) × SU(2)) / ℤ₂ → two independent conserved momenta

Cosmological constant: Λ₀ from QCD sigma terms, factor 1.65, zero parameters.
Dark energy: Centrifugal expansion + vacuum gravitation via α.
Dark matter: Vacuum captured in bound structures.
Singularity: Five barriers — Kriger Limit, membrane elasticity, double rotation, centrifugal force, time absence.
Baryon asymmetry: η = α⁴ = 6.2 × 10⁻¹⁰. CPT from antipodal identification.
Inflation: Centrifugal force at v = c. Exit via angular momentum conservation.

Banach fixed-point theorem: ∃! x* = Φ(x*) → all constants are properties of x*

Boris Kriger | Information Physics Institute | Institute of Integrative and Interdisciplinary Research, Toronto
ResearchGate | ORCID: 0009-0001-0034-2903