A complete derivation hierarchy: from two topological axioms to 57 observables, with zero circular dependencies.
The theory is strictly one-directional: each level depends only on previous levels. No circular dependencies.
$\delta = 1/24$ (lattice tension) and $\Delta = 5/24$ (Fibonacci–lattice conflict) — from the 24-cell polytope.
The golden ratio $\varphi = (1 + \sqrt{\Delta/\delta})/2 = (1+\sqrt{5})/2$ and half-opening angle $\theta_{\max} = 30°$.
The spectral dimension $D^*$ follows from the cone geometry:
A self-referential fixed-point iteration yields $D^*_{\rm fp} \approx 2.797$.
The $f(R)$ exponent emerges as $n = D^*/2 = 67/48$, defining the gravitational action:
This simultaneously gives Horndeski parameters $\alpha_M = 19/86$, $\alpha_T = 0$ (consistent with GW170817), and the growth index.
$f(R)$ inflation with $n = 67/48$ predicts spectral index, tensor ratio, and e-folds:
All gauge couplings, mass ratios, mixing angles, and cosmological density parameters emerge at this level.
Tully–Fisher relation, galactic rotation curves, dark matter reinterpretation, and structure growth ($\sigma_8$, $S_8$).
Spacetime's effective dimensionality is not fixed — it runs with scale, governed by a beta-function analogous to Newton's cooling law.
This governs how the effective spectral dimension $D^*(k)$ flows between two fixed points:
The flow interpolates smoothly between quantum ($D=2$) and classical ($D \approx 2.79$) regimes.
Newton's constant runs with scale — weakening at high energies, ensuring asymptotic safety. At galactic scales, this reproduces flat rotation curves without dark matter particles.
At the Planck scale, spacetime is effectively 2-dimensional. As we zoom out, dimensions "grow" to $D^* \approx 2.79$ — never reaching 3. This fractional dimension is the origin of all modified gravity effects.
Gauge couplings, mass ratios, mixing angles, and the Higgs sector — all from $\Delta$, $\delta$, and $D^*$.
All three gauge couplings are derived purely from geometric quantities. The electroweak correction $\gamma_{\rm EW}$ is computed from Standard Model RG equations evaluated at the geometric scale.
| Angle | Formula | Predicted | Observed | Tension |
|---|---|---|---|---|
| $\theta_{12}$ (solar) | $\arctan(1/\sqrt{2}) \times 23/24$ | 33.80° | $33.41° \pm 0.75°$ | 0.51σ |
| $\theta_{23}$ (atmos.) | $45°(1 + \Delta/\sqrt{6})$ | 48.83° | $49.0° \pm 1.4°$ | 0.12σ |
| $\theta_{13}$ (reactor) | $\arcsin(\Delta/\sqrt{2})$ | 8.47° | $8.54° \pm 0.15°$ | 0.46σ |
| Element | Formula | Predicted | Observed | Tension |
|---|---|---|---|---|
| $|V_{us}|$ | $\sqrt{\Omega_B}$ | 0.2234 | $0.2243 \pm 0.0008$ | 1.11σ |
| $|V_{cb}|$ | $\Delta^2$ | 0.0434 | $0.0408 \pm 0.0014$ | 1.86σ |
| $|V_{ub}|$ | $\Delta^\varphi \cdot \delta \cdot \exp(\ldots)$ | 0.00375 | $0.00382 \pm 0.0002$ | 0.36σ |
The golden ratio raised to the $n$-th power must fit within the Fibonacci–lattice conflict bound $\Delta \cdot 24 = 5$. This yields exactly 3 generations — no more, no less.
Density parameters, baryon asymmetry, and dark energy — all predicted from the 24-cell geometry.
All cosmic density fractions are exact rational numbers with denominator $24^2 = 2304$:
*"Dark matter" in SDGFT is reinterpreted as a geometric effect of the running $G(r)$ — no new particles are needed.
This is the single most falsifiable prediction of SDGFT. Current observations give $w = -1.03 \pm 0.03$, yielding a 3.32σ tension. If Euclid and DESI confirm $w < -0.96$, the theory is falsified.
The observed value is $(6.14 \pm 0.19) \times 10^{-10}$ — a remarkable 0.67σ match.
Observed: $0.776 \pm 0.017$, i.e. 0.07σ. This helps resolve the well-known $S_8$ tension in cosmology.
A power-law $f(R)$ gravitational action reproduces galactic dynamics and connects to UV quantum gravity.
The exponent $n = D^*/2 = 67/48 \approx 1.396$ is uniquely fixed by the effective dimension. This is not a free parameter — it is derived.
Observed: $3.85 \pm 0.09$ (0.65σ). The baryonic Tully–Fisher relation emerges naturally from the modified gravitational law.
Below this scale, Newton's gravity applies. Above it, the running $G(r)$ produces "galaxy rotation curve" effects.
At the smallest scales (Planck length), the theory exhibits:
SDGFT does not predict dark matter particles. Instead, the gravitational effects attributed to dark matter arise naturally from the scale-dependent Newton constant $G(r)$. The "dark matter fraction" $\Omega_c$ represents a geometric effect, not a new particle species.
The 24-cell partitions naturally into six conical sectors, each with half-opening angle $\theta_{\max} = 30°$.
The 24-cell's symmetry group naturally partitions upper-dimensional space into six congruent cones, each subtending a solid angle corresponding to $\theta_{\max} = 30°$.
This geometry is the origin of:
Intellectual honesty demands acknowledging the boundaries of the current framework.
SDGFT derives ratios (e.g., $m_\mu/m_e$, $m_\tau/m_\mu$) but not absolute masses. The Planck mass $M_P$ enters as an external input — it is not derived from $\Delta$ and $\delta$.
The CKM and PMNS complex phases are not yet derived. The theory predicts magnitudes of mixing matrix elements but is silent on CP-violating phases.
The smallness of the QCD $\theta$-parameter ($\theta_{\rm QCD} < 10^{-10}$) is not addressed. A geometric mechanism may exist but has not been found.
While the UV fixed point $D^*_{\rm UV} = 2$ is established, a complete non-perturbative UV definition (analogous to a lattice formulation) is not yet available.
All derivations, proofs, and numerical evaluations are available in the foundational paper and the comprehensive monograph.