Foundations
MatureThe precise identification of all Standard Model symmetries from the 8D geometry is mathematically robust.The 8D Spatial Universe
The CPT framework operates in 8 spatial dimensions — the unique octonion-compatible space. The observable S³ is the 3D equator of S⁷ in ℝ⁸. The extra 5 dimensions provide the internal symmetry structure for particle physics.
Why 8 Dimensions?
The normed division algebras are limited to ℝ (1D), ℂ (2D), ℍ (4D), and 𝕆 (8D). Only these four exist according to Hurwitz's theorem. The CPT framework asserts that physics is governed by the unique structure of the 8D normed division algebra.
Hurwitz's Theorem
The S³ Universe in ℝ⁸
The embedding chain is . The observable 3D universe is the equator of . The full 7-sphere can be viewed through the Hopf fibration , where it carries the internal gauge structure.
Particles propagate on , while the internal structure (colour, flavour) resides in the fibre. The embedding involves four complex coordinates:
Embedding
Metric and Geometry
The 8D metric can be expressed as:
8D Metric
where is the polar angle in . The 3D metric on is precisely the slice.
Octonion Flavour Structure
There are 7 imaginary octonions: basis vectors . These generate the automorphism group. The geometry of these vectors gives rise directly to colour and flavour quantum numbers. The 7 imaginary octonion basis vectors, acted on by , give the quark flavour structure.
Automorphism Group
Source Documents
The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.
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