Hopf Fibration and Z₂ Holonomy: Fiber Bundle Models for Spinor Sign Flips — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Fiber bundles formalize "extra dimensions" attached to each spacetime point; the Hopf fibration S³→S² with U(1) fiber is the canonical example, and a Z₂ bundle over the φ-torus yields holonomy −1, encoding spinor-like sign flips for fundamental constants. | MATH: Hopf fibration: S¹ ↪ S³ → S² (fiber U(1) ≅ S¹); Z₂ bundle: hol(γ_φ) = −1 ∈ {+1,−1}; principal bundle: P → M with structure group G (e.g., U(1) for EM); fiber bundle model: Hookean springs with breaking thresholds, load-sharing dynamics (arXiv:2012.12698). | CONNECTION: The Z₂ holonomy −1 is the geometric root of spin-½ statistics — a double cover, directly analogous to the 2:1 covering SU(2)→SO(3), whose character table and root system (A₁) encode the golden ratio via the icosahedral/dodecahedral symmetry (5-fold axes, φ = 2cos(π/5) = 1.618). The U(1) fiber's circle group has Z₂ subgroup {±1}, linking to the 0.5 ratio (half-integer spin). No direct 0.382/0.618/0.786/2.618 or base-60 appears in these sources. | DEPTH:
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026hopf,
title = {{Hopf Fibration and Z₂ Holonomy: Fiber Bundle Models for Spinor Sign Flips — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23526130},
url = {https://zenodo.org/records/23526130},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hopf Fibration and Z₂ Holonomy: Fiber Bundle Models for Spinor Sign Flips — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23526130
UR - https://zenodo.org/records/23526130
AB - FINDING: Fiber bundles formalize "extra dimensions" attached to each spacetime point; the Hopf fibration S³→S² with U(1) fiber is the canonical example, and a Z₂ bundle over the φ-torus yields holonomy −1, encoding spinor-like sign flips for fundamental constants. | MATH: Hopf fibration: S¹ ↪ S³ → S² (fiber U(1) ≅ S¹); Z₂ bundle: hol(γ_φ) = −1 ∈ {+1,−1}; principal bundle: P → M with structure group G (e.g., U(1) for EM); fiber bundle model: Hookean springs with breaking thresholds, load-sharing dynamics (arXiv:2012.12698). | CONNECTION: The Z₂ holonomy −1 is the geometric root of spin-½ statistics — a double cover, directly analogous to the 2:1 covering SU(2)→SO(3), whose character table and root system (A₁) encode the golden ratio via the icosahedral/dodecahedral symmetry (5-fold axes, φ = 2cos(π/5) = 1.618). The U(1) fiber's circle group has Z₂ subgroup {±1}, linking to the 0.5 ratio (half-integer spin). No direct 0.382/0.618/0.786/2.618 or base-60 appears in these sources. | DEPTH: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -