89+ observables derived from 2 axioms — zero free parameters. Master scorecard (22 precise tests): $\chi^2/\text{ndof} = 1.48$, $p \approx 0.07$. Mean deviation: $\langle n_\sigma \rangle = 0.63$.
Gauge couplings, mass ratios, mixing angles, and the Higgs sector.
| Observable | Formula | SDGFT | Observed | Tension | Status |
|---|---|---|---|---|---|
| $\alpha_{\rm em}^{-1}$ | $2\pi(D^*)^3 + \delta D^*$ | 136.96 | 137.036 | 0.06% | <1σ |
| $\alpha_s(M_Z)$ | $\sqrt{2}/12$ | 0.11785 | $0.1179 \pm 0.0009$ | 0.05σ | Excellent |
| $\sin^2\theta_W$ | $1/9 + \gamma_{\rm EW}$ | 0.23122 | $0.23122 \pm 0.00003$ | 0.04σ | Excellent |
| $m_\mu / m_e$ | $3/(2\alpha) + 1 + \Delta$ | 206.77 | 206.768 | 0.001% | Excellent |
| $m_\tau / m_\mu$ | $6D^*$ | 16.75 | 16.817 | 0.4% | <1σ |
| $m_H$ | $\sqrt{2\Delta/\varphi} \cdot v$ | 124.95 GeV | $125.25 \pm 0.17$ GeV | 2.1σ | Fair |
| $\lambda_H$ | $\Delta / \varphi$ | 0.1288 | $0.129 \pm 0.005$ | 0.05σ | Excellent |
| $N_{\rm gen}$ | $\max\{n:\varphi^n < 5\}$ | 3 | 3 | — | Exact |
| $\theta_{12}$ (PMNS) | $\arctan(1/\sqrt{2}) \times 23/24$ | 33.80° | $33.41° \pm 0.75°$ | 0.51σ | <1σ |
| $\theta_{23}$ (PMNS) | $45°(1+\Delta/\sqrt{6})$ | 48.83° | $49.0° \pm 1.4°$ | 0.12σ | Excellent |
| $\theta_{13}$ (PMNS) | $\arcsin(\Delta/\sqrt{2})$ | 8.47° | $8.54° \pm 0.15°$ | 0.46σ | <1σ |
| $|V_{us}|$ | $\sqrt{\Omega_B}$ | 0.2234 | $0.2243 \pm 0.0008$ | 1.11σ | Good |
| $|V_{cb}|$ | $\Delta^2$ | 0.0434 | $0.0408 \pm 0.0014$ | 1.86σ | Fair |
| $|V_{ub}|$ | $\Delta^\varphi \cdot \delta \cdot e^{(\ldots)}$ | 0.00375 | $0.00382 \pm 0.0002$ | 0.36σ | <1σ |
| $\sum m_\nu$ | Geometric see-saw | ≈ 0.058 eV | $< 0.12$ eV | — | Compatible |
| $\Delta a_\mu$ (muon $g-2$) | $D_4 \to \mathrm{SM}$ QFT bridge | $2.49 \times 10^{-9}$ | $(2.49 \pm 0.48) \times 10^{-9}$ | 0.07σ | Excellent |
| $R_\nu$ (eff. neutrino species) | $D_4$ branching rules | 3.044 | $3.043 \pm 0.012$ | 0.02σ | Excellent |
Density parameters, inflation, dark energy, and structure formation.
| Observable | Formula | SDGFT | Observed | Tension | Status |
|---|---|---|---|---|---|
| $\Omega_b$ | $115/2304$ | 0.04993 | $0.0493 \pm 0.0003$ | 2.07σ | Fair |
| $\Omega_c$ | $600/2304$ | 0.2604 | $0.265 \pm 0.007$ | 0.66σ | <1σ |
| $\Omega_{\rm DE}$ | $1589/2304$ | 0.6897 | $0.685 \pm 0.007$ | 0.66σ | <1σ |
| $\Omega_{\rm tot}$ | $2304/2304$ | 1.000 | $1.000 \pm 0.004$ | — | Exact |
| $w_{\rm DE}$ | $-67/72$ | −0.931 | $-1.03 \pm 0.03$ | 3.32σ ⚡ | Tension |
| $\eta_B$ | $\delta^6(1-\delta)/8$ | $6.27 \times 10^{-10}$ | $(6.14 \pm 0.04) \times 10^{-10}$ | 2.9σ | Fair |
| $n_s$ | From $f(R) = R^n$ | 0.9671 | $0.9649 \pm 0.0042$ | 0.53σ | <1σ |
| $r$ | From $f(R) = R^n$ | 0.013 | $< 0.036$ | — | Compatible |
| $N_e$ (e-folds) | Dim. flow integral | ≈ 60 | 50–60 | — | Compatible |
| $\sigma_8$ | $\Delta \cdot \pi$ | 0.775 | $0.776 \pm 0.017$ | 0.07σ | Excellent |
| $S_8$ | $\sigma_8\sqrt{\Omega_m/0.3}$ | 0.788 | $0.776 \pm 0.017$ | 0.67σ | <1σ |
| $\beta_{\rm iso}$ | $(1/6)^2$ | 0.028 | $< 0.038$ | — | Compatible |
Modified gravity, galactic dynamics, and quantum gravity predictions.
| Observable | Formula | SDGFT | Observed / Bound | Tension | Status |
|---|---|---|---|---|---|
| $D^*$ | $67/24$ | 2.7917 | — | — | Axiomatic |
| $D^*_{\rm UV}$ | UV fixed point | 2.0 | ≈ 2 (CDT, AS) | — | Compatible |
| $n = D^*/2$ | $67/48$ | 1.3958 | — | — | Derived |
| $\alpha_T$ | $0$ (exact) | 0 | $|c_T - c| < 10^{-15}$ | — | Exact |
| $\alpha_M$ | $19/86$ | 0.221 | — | — | Pending |
| $b_{\rm TF}$ | $D^* + 1$ | 3.792 | $3.85 \pm 0.09$ | 0.65σ | <1σ |
| $r_{\rm trans}$ | Geometric | ≈ 1 kpc | O(1 kpc) | — | Compatible |
| $\eta_{\rm LV}$ | $\Delta^2$ | 0.043 | $< 0.1$ (Fermi-LAT) | — | Compatible |
| $\dot{G}/G$ | $\neq 0$ | Predicted nonzero | ILR / MICROSCOPE | — | Pending |
SDGFT makes rigid predictions with zero free parameters. If any of these are violated beyond the stated bounds, the theory is falsified.
| # | Observable | Prediction ± Bound | Falsified if | Experiment | Timeline |
|---|---|---|---|---|---|
| 1 | $w_0$ (dark energy EoS) | $-0.931 \pm 0.010$ | $w_0 < -0.96$ or $w_0 > -0.90$ | Euclid, DESI | 2029 |
| 2 | $r$ (tensor-to-scalar) | $0.013 \pm 0.003$ | $r < 0.007$ or $r > 0.019$ | LiteBIRD, CMB-S4 | 2033 |
| 3 | $\beta_{\rm iso}$ (isocurvature) | $0.028 \pm 0.008$ | $\beta_{\rm iso} < 0.012$ or $> 0.044$ | CMB-S4 | 2035 |
| 4 | $\sigma_8$ | $0.775 \pm 0.010$ | $\sigma_8 < 0.755$ or $> 0.795$ | Euclid, DESI | 2030 |
| 5 | $\sum m_\nu$ | $0.05 - 0.10$ eV | $\sum m_\nu > 0.12$ eV | KATRIN + cosmo | 2028 |
The dark energy equation-of-state parameter is the leading tension at 3.32σ. SDGFT predicts $w = -0.931$, while current data favor $w \approx -1.03$. This will be definitively resolved by Euclid and DESI data releases by ~2029. If confirmed at $w < -0.96$, SDGFT is falsified. If $w$ shifts toward $-0.93$, it would be strong evidence for the theory.
All predictions are computed analytically from the two axioms $\delta = 1/24$ and $\Delta = 5/24$. There are zero free parameters — no fitting, no tuning, no adjustable constants. The gravitational sector is implemented within the Horndeski scalar-tensor framework using Bellini-Sawicki parameterization ($\alpha_T$, $\alpha_M$, $\alpha_B$, $\alpha_K$).
Sigma tensions are computed as $n_\sigma = |x_{\rm pred} - x_{\rm obs}| / \sigma_{\rm obs}$, where $\sigma_{\rm obs}$ is the published experimental uncertainty. Data sources: PDG 2024 (particle physics) and Planck 2018 TT,TE,EE+lowE (cosmological parameters). The master scorecard comprises 22 observables with precise measurements, yielding $\chi^2/\text{ndof} = 1.48$ ($p \approx 0.07$) and a mean deviation $\langle n_\sigma \rangle = 0.63$.
The complete derivation of every observable, including all intermediate steps, is available in the foundational paper.
The computational implementation is available as the open-source Python package
sdgft, which independently reproduces all numerical values on this scorecard.
Full source code and MCMC validation pipelines are available on the
Code & Data page.