A complete derivation hierarchy: from two topological axioms to 89+ observables, with zero circular dependencies.
The theory is strictly one-directional: each level depends only on previous levels. No circular dependencies.
$\delta = 1/24$ and $\Delta = 5/24$ — from the 24-cell polytope, the unique self-dual regular polytope in 4D. The seed $\delta = 1/24$ is the modular-invariant regulator appearing in the Dedekind $\eta$-function and the bosonic string partition function ($\zeta(-1) = -1/12$, so $\delta = \tfrac{1}{2}|\zeta(-1)|$). The complement $\Delta = 5/24$ satisfies the partition-of-unity relation $6\Delta + 6\delta = 1$.
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 a Horndeski scalar-tensor theory with derived Bellini-Sawicki parameters:
$\alpha_T = 0$ (tensor speed equals light speed) is confirmed by GW170817 to $|c_T - c| < 10^{-15}$. The running rate $\alpha_M = 19/86 \approx 0.221$ is a falsifiable prediction for future gravitational-wave standard-siren measurements.
$f(R)$ 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. The QFT bridge maps $D_4$ roots to $SU(3) \times SU(2) \times U(1)$.
Tully–Fisher relation, galactic rotation curves, dark matter reinterpretation, and structure growth ($\sigma_8$, $S_8$).
Independent computational validation: GATv2 GNN ensemble ($R^2 = 0.9995$), 100M-point Oracle database, full Bayesian MCMC with cobaya/emcee/dynesty, PDG 2024 scorecard: $\chi^2/\text{ndof} = 1.48$.
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.
A fundamental observer–system reciprocity linking proper time to system time via the geometry of the 24-cell.
The Kinemato-Geometric Equation (KGE) establishes a duality between the observer's perspective time $T_{\rm pov}$ and the system's intrinsic time $t_{\rm sys}$:
This is not a convention — it is a derived consequence of the 24-cell's self-duality. Just as the polytope is isomorphic to its dual, time admits two complementary descriptions that are related by inversion.
The Fibonacci–lattice frustration $\Delta = 5/24$ induces a slow secular drift $\Delta_F$ in the time-duality relation. This drift is responsible for:
In standard physics, time is a background parameter. In SDGFT, time is geometric: the effective dimension $D^*(k)$ runs with energy scale, and the time-duality invariant constrains how measurements at different scales relate. This resolves the "problem of time" in quantum gravity without introducing new degrees of freedom.
The KGE is the central new result of the v2 paper, connecting the UV fixed point ($D^*_{\rm UV} = 2$) to the IR fixed point ($D^* = 67/24$) through a single invariant.
The $\Delta_F$-drift is exactly compensated during inflation, ensuring that primordial observables ($n_s = 0.9671$, $r = 0.013$) are protected from late-time modifications. This is not fine-tuned — it follows from the self-duality of the time flow.
The Standard Model gauge group emerges from the root system of the 24-cell's symmetry.
The 24 vertices of the 24-cell form the $D_4$ root system — the root lattice of $SO(8)$. The maximal regular embedding of the Standard Model gauge group into $D_4$ yields:
This is not merely an embedding — the branching rules of $D_4$ uniquely determine the hypercharge assignments, the number of colors ($N_c = 3$), and the electroweak doublet structure. The 24 roots decompose as:
— precisely the adjoint + bifundamental representation content needed for one generation of the Standard Model.
Observed: $(2.49 \pm 0.26) \times 10^{-9}$ — a 0.07σ match. The SDGFT derivation uses only geometric quantities.
Consistent with CMB measurements of $N_{\rm eff} = 3.044 \pm 0.16$ (0.02σ).
| Feature | SDGFT | $\Lambda$CDM | Strings |
|---|---|---|---|
| Free parameters | 0 | 6+ | O(10²) |
| Dark matter | Geometric | Particle | Particle |
| $w_{\rm DE}$ | $-0.931$ | $-1$ (exact) | Landscape |
| UV complete | $D^*_{\rm UV}=2$ | No | Yes |
| Falsifiable | 5 rigid bets | Flexible | No |
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. Comparison data: Planck 2018 TT,TE,EE+lowE, BOSS DR12 BAO, and Pantheon+ supernovae.
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.
SDGFT is numerically implemented as a Horndeski scalar-tensor theory with Bellini-Sawicki parameterization. The power-law $f(R) = R^{67/48}$ action reproduces galactic dynamics and connects to UV quantum gravity via asymptotic safety.
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.
In the Horndeski parameterization, SDGFT derives all four stability functions analytically:
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.