# SDGFT — Six-Dimensional Geometric Field Theory

> SDGFT is a topological approach to fundamental physics that derives 89+ observables
> of the Standard Model and cosmology from two axioms with zero free parameters.
> It is implemented as a modified f(R) = R^{67/48} gravity model within the
> Horndeski scalar-tensor framework, using Bellini-Sawicki parameterization.

## Axioms

- δ = 1/24 — Topological seed from the Dedekind η-function; appears in the bosonic
  string partition function as ζ(−1) = −1/12, with δ = ½|ζ(−1)|. It is the unique
  modular-invariant regulator of the 24-cell polytope lattice.
- Δ = 5/24 — Complement parameter of the 24-cell polytope (the only self-dual regular
  polytope in 4D). Satisfies 6Δ + 6δ = 1 as the fundamental partition of unity.

All physical constants are derived algebraically from these two rational numbers.
There are zero adjustable parameters — no fitting, no tuning.

## Key Derived Quantities

- Effective spacetime dimension: D* = 67/24 ≈ 2.792
- f(R) gravity exponent: n = D*/2 = 67/48 ≈ 1.396
- Fine-structure constant: α_em^{-1} = 2π(D*)³ + δD* ≈ 136.96 (observed: 137.036)
- Strong coupling: α_s(M_Z) = √2/12 ≈ 0.11785 (observed: 0.1179 ± 0.0009)
- Weinberg angle: sin²θ_W = 1/9 + γ_EW ≈ 0.23122 (observed: 0.23122 ± 0.00003)
- Higgs mass: m_H = √(2Δ/φ) · v ≈ 124.95 GeV (observed: 125.25 ± 0.17 GeV)
- Dark energy EoS: w_DE = −67/72 ≈ −0.931 (observed: −1.03 ± 0.03, tension 3.32σ)
- Baryon density: Ω_b = 115/2304 ≈ 0.04993 (observed: 0.0493 ± 0.0003)
- Muon anomalous magnetic moment: Δa_μ ≈ 2.49 × 10⁻⁹ (observed: 2.49 ± 0.48 × 10⁻⁹, 0.07σ)

## Statistical Performance

- Master scorecard: 22 observables with precise measurements
- χ²/ndof = 1.48 (p ≈ 0.07)
- Mean deviation: ⟨n_σ⟩ = 0.63
- Distribution: 12 exact, 25 excellent (<1σ), 3 good (1–2σ), 3 fair (2–3σ), 1 tension (>3σ)
- Comparison data: PDG 2024 (particle physics), Planck 2018 TT,TE,EE+lowE (cosmology)

## Established Physics Context

SDGFT is built on and testable within established frameworks:

- **Horndeski gravity**: The most general scalar-tensor theory with second-order
  equations of motion. SDGFT derives all four Bellini-Sawicki parameters:
  α_T = 0 (exact, confirmed by GW170817), α_M = 19/86 ≈ 0.221, α_B, α_K.
- **Asymptotic Safety**: The UV fixed point D*_UV = 2 matches predictions from
  the asymptotic safety program and Causal Dynamical Triangulations (CDT).
- **QFT bridge**: The gauge group SU(3) × SU(2) × U(1) is derived from the
  symmetry breaking of the dihedral group D₄ of the 24-cell.
- **Gravitational waves**: Predicts c_T = c (tensor speed equals light speed),
  confirmed to |c_T − c| < 10⁻¹⁵ by LIGO/Virgo.

## Falsifiable Predictions

| Observable       | SDGFT         | Falsified if                    | Experiment      | Timeline |
|------------------|---------------|---------------------------------|-----------------|----------|
| w₀ (dark energy) | −0.931±0.010  | w₀ < −0.96 or w₀ > −0.90       | Euclid, DESI    | 2029     |
| r (tensor/scalar) | 0.013±0.003  | r < 0.007 or r > 0.019         | LiteBIRD, CMB-S4| 2033     |
| β_iso            | 0.028±0.008   | β < 0.012 or β > 0.044         | CMB-S4          | 2035     |
| σ₈               | 0.775±0.010   | σ₈ < 0.755 or σ₈ > 0.795       | Euclid, DESI    | 2030     |
| Σm_ν             | 0.05–0.10 eV  | Σm_ν > 0.12 eV                 | KATRIN + cosmo  | 2028     |

The leading tension is w_DE = −0.931 vs observed ≈ −1.03 (3.32σ). This will be
definitively resolved by Euclid/DESI by ~2029. If w₀ < −0.96, SDGFT is falsified.

## Current Tensions (Transparency)

- w_DE: 3.32σ — the theory's strongest falsifiable prediction
- η_B (baryon-to-photon ratio): 2.9σ — with updated Planck 2018 precision
- m_H (Higgs mass): 2.1σ — mild tension with PDG 2024 world average
- |V_cb| (CKM element): 1.86σ — known experimental scatter between inclusive/exclusive

## Documentation

- Website: https://sdgft.org
- Theory: https://sdgft.org/theory.html
- Scorecard: https://sdgft.org/scorecard.html
- Code & Data: https://sdgft.org/code.html

## Papers

- Current (v2): https://doi.org/10.5281/zenodo.18984174
  "Foundations of the Six-Dimensional Geometric Field Theory" (March 2026)
- Original (v1): https://doi.org/10.5281/zenodo.18793846
  First publication (February 2025)

## Open-Source Code

- Core library: https://github.com/cosmologicmind/sdgft (pip install sdgft)
- Cosmology engine: https://github.com/cosmologicmind/sdgft-cosmology-engine
  (Horndeski/hi_class integration)
- ML toolkit: https://github.com/cosmologicmind/sdgft-ml-toolkit
  (GATv2 GNN, R² = 0.9995)
- MCMC sampling: https://github.com/cosmologicmind/sdgft-mcmc
  (cobaya/emcee/dynesty pipelines)
- GW analysis: https://github.com/cosmologicmind/sdgft-gw-analysis
- Observational data: https://github.com/cosmologicmind/sdgft-data
  (Planck, BOSS DR12, Pantheon+, SPARC, GWOSC)
- GitHub organization: https://github.com/cosmologicmind

## Author

David A. Besemer
Email: david.besemer@sdgft.org