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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.20153885 |
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Table of Contents:
- <p>We establish the geometric foundations of the CosmicCircle framework: the proposal that <br>physics takes place on S³ with the Hopf fibration S¹→S³→S² and Berger metric ds²_B = <br>λ²σ₁ ² + σ₂ ² + σ₃ ² (squashing parameter λ = 1/2), with a single free physical parameter H₀ <br>= 70.94 km/s/Mpc. The paper proceeds in eight sections. <br>The GPS Theorem (§2) establishes that the composite map Φ = arctan∘ σ∘ π is the unique <br>natural map between S³_Berger and Minkowski spacetime: smooth, invertible on spinorial <br>sectors, and conformally flat. The Berger curvatures (§3) K_fh = λ² and K_hh = 4−3λ² are <br>computed exactly via Koszul connection and verified via O’Neill submersion; Corollary G2 <br>fixes λ = 1/2 uniquely from the identity k_eff − k_round = K_fh. Theorem H (§4) derives β = <br>0 from the trivial holonomy of the Hopf bundle at λ = 1/2, making the Friedmann expansion <br>law unconditional. Theorem I (§5) derives the Koide lepton mass ratio K = 2/3 from the <br>Borromeo topology of three Z₃ sections of the Hopf bundle with c₁ = 1; a GPS derivation <br>via O’Neill and Bär establishes b/a = √λ = 1/√2 from first principles, yielding K = 1/(1+λ) = <br>2/3 exactly. The Born rule (§6) is derived from the geometry of the local fibre S¹_local: five <br>operational axioms (normalisation, additivity, rotational covariance, global phase, <br>quadratic homogeneity) uniquely force P = |c_κ|², with no quantum postulates imported <br>(Theorem BR). The logistic equation and BAO scale (§7) follow from the curvature partition <br>field q(z) = K_h/(K_h+K_v); the geometric prediction r_d = 147.07 Mpc matches Planck 2018 <br>to 0.011%. The Diagnostic Principle (§8) assembles cross-domain evidence: the GPS map <br>simultaneously simplifies Maxwell, Dirac, QED, cosmology, and gravity — structural <br>evidence that S³ is the natural arena. <br>All results are machine-verified. The single free physical parameter H₀ propagates <br>correctly across all sectors, with no additional inputs required. Papers II and III extend <br>these foundations to the cosmological and particle sectors respectively.</p>