The cascade series tests one hypothesis with zero free parameters. From a single axiom (orthogonality), the series derives the cosmological constant, quantum mechanics, general relativity with d=4 and Lorentzian signature, the Standard Model gauge group and its symmetry breaking, three fermion generations, precision mass and coupling predictions, and the background cosmological parameters including a Planck-compatible Hubble constant and a universe age of 13.88 Gyr.
One hypothesis. Zero free parameters. Every prediction below is a test of the hypothesis.
Mathematical uniqueness proofs leave no alternative. These are not approximations.
| Prediction | Value | Status | Source |
|---|---|---|---|
| Spacetime dimension | d = 4 | Confirmed | Lovelock ∩ Clifford (III) |
| Metric signature | (−,+,+,+) | Confirmed | Propagator + Clifford (III) |
| Gauge group | SU(3) × SU(2) × U(1) | Confirmed | Adams + Bott (IVa) |
| Symmetry breaking | SU(2) broken; SU(3), U(1) exact | Confirmed | Hairy ball theorem (IVa) |
| Fermion generations | Exactly 3 | Confirmed | Bott periodicity + d1=19 (IVa) |
| Dark energy EoS | w = −1 exactly | Confirmed | Fixed geometric constant (III) |
| Strong CP phase | θQCD = 0 | Confirmed | π3(S11) = Z2 (IVa) |
| No supersymmetry | — | Confirmed (LHC) | No pairing mechanism (IVa) |
| No dark matter particles | — | Confirmed (null results) | Geometry provides content (V) |
| No extra Higgs bosons | — | Confirmed (LHC) | One hairy ball zero (IVa) |
| No gravitons | — | Not yet testable | Metric is state property, not quantised field (II=III, III) |
Numerical predictions from cascade geometry. Formulas are exact; deviations reflect leading-order truncation.
| Observable | Formula | Predicted | Observed | Dev. |
|---|---|---|---|---|
| ρΛ / M4Pl,red | 18 · Ω19 · Ω217 / π3 | 0.6996 × 10−120 | 0.7150 × 10−120 | −2.2% |
| ΩΛ | (π−1)/π | 0.6817 | 0.685 ± 0.007 | −0.5% |
| Ωm | 1/π | 0.3183 | 0.315 ± 0.007 | +1.1% |
| Ωr | 1/(4π7) | 8.28 × 10−5 | 8.27 × 10−5 | +0.1% |
| TCMB | from Ωr, H0 | 2.642 K | 2.7255 K | −3.1% |
| H0 | from ρΛ, ΩΛ | 66.78 km/s/Mpc | 67.4 ± 0.5 | −0.9% |
| t0 | ΛCDM integral | 13.88 Gyr | 13.80 ± 0.02 | +0.6% |
| mH / mW | π/2 | 1.5708 | 1.559 | +0.8% |
| mμ / me | exp(ΔΦ) · 2√π | 206.50 | 206.77 | +0.13% |
| me | geometric-topological | 0.514 MeV | 0.511 MeV | +0.6% |
| mμ | geometric-topological | 106.2 MeV | 105.66 MeV | +0.5% |
| αs(MZ) leading | α(12) · exp(ΔΦ) | 0.1159 | 0.1179 ± 0.0009 | −1.7% |
| sin2θW leading | Radon-Hurwitz ratio | 0.2286 | 0.23121 | −1.1% |
Seven observables close within experimental error via δΦ = α(d*)/χk shifts sourced at Part 0's distinguished dimensions. Three shift-observable pairs reuse the same correction across independent quantities.
| Observable | Shift source | Predicted | Observed | Residual |
|---|---|---|---|---|
| αs(MZ) | α(14)/χ | 0.11792 | 0.1179 ± 0.0009 | +0.02σ |
| mτ / mμ | α(14)/χ | 16.8173 | 16.8170 ± 0.0011 | +0.24σ |
| mτ absolute | α(19)/χ | 1776.82 MeV | 1776.86 ± 0.12 | −0.31σ |
| sin2θW | α(5)/χ3 | 0.23123 | 0.23121 ± 0.00004 | +0.40σ |
| Ωm | −α(5)/χ3 | 0.31474 | 0.315 ± 0.007 | −0.04σ |
| θC (Cabibbo) | −α(7)/χ2 | 13.04° | 13.04 ± 0.05° | +0.03σ |
Specific predictions testable by current or near-future experiments (DESI, Euclid, CMB-S4, SH0ES).
| Observable | Predicted | Current data | Status |
|---|---|---|---|
| H0 | 66.78 km/s/Mpc (Gram-corrected ≈ 67.5) | Planck: 67.4 · SH0ES: 73.0 | Planck-side of Hubble tension; incompatible with SH0ES |
| rd (sound horizon) | ≈147.75 Mpc | Planck: 147.60 Mpc | Essentially equal to Planck; cascade and ΛCDM share a ruler |
| DESI DR2 BAO fit | χ2/n = 2.35 (cascade) vs 1.90 (Planck) | Two shared outliers at z=0.510, z=0.706 | Cascade fits slightly worse than Planck; both face same anomalies |
| DESI w ≠ −1 signal | w = −1 exactly (structural theorem) | DESI DR2: w ≈ −0.76 | Challenges cascade and ΛCDM equally; no ruler-based explanation |
Results where the argument has acknowledged gaps or needs strengthening.
| Observable | Issue |
|---|---|
| Ωb = 1/(2π2) | “One unit of content on S3” argument needs strengthening |
| ns, As | Primordial spectrum not yet derived |
| Correction selection rule | Observable-to-source assignment not fully derived from first principles |