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.
Read another way: the universe is the asymptotic resolution of an infinite-dimensional starting object (B∞) — what that object looks like partway through resolving itself, dimension by dimension. Time is the rate at which the resolution proceeds. Three vocabularies describe the same rate: mathematical (the slicing recurrence extracts one dimension's worth of finite content from B∞ at each step), physical (each Planck tick adds one cascade layer; slicing is dimensional collapse is time), and geometric (the observer's S3 shell asymptotically falls toward a 5D black-hole pseudo-horizon, never completing). See the cover sheet for the thought experiment.
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) |
| Free Yukawa couplings | Zero — every fermion mass equals the per-layer gauge-coupling amplitude m(d) = R(d)/2; layer position from Bott + Adams (IVa) | Confirmed (charged-lepton sector <1.2%; mτ/mμ +0.24σ; mτ abs −0.31σ; θC +0.03σ; b/s 0.014%) | Chirality halving + Berezin partition (IVb): m(d) = R(d)/χ from Poincaré-Hopf on even-sphere Dirac layers; no Yukawa parameter ever introduced |
| Dark energy EoS | w = −1 exactly | Confirmed | Fixed geometric constant (III) |
| Strong CP phase | θQCD = 0 | Confirmed | π3(S11) = 0 (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 axion | — | Confirmed (null results) | θQCD = 0 topologically (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 · exp(δΦ) | 0.7145 × 10−120 | 0.7150 × 10−120 ± 0.013 | −0.07% (≈ −0.04σ) |
| ΩΛ | (π−1)/π | 0.6817 | 0.685 ± 0.007 | −0.5% (≈ −0.47σ) |
| 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% (descent-dependent) |
| H0 | from ρΛ, ΩΛ | 66.78 km/s/Mpc (Gram-corrected ≈ 67.5) | 67.4 ± 0.5 | −0.9% leading; ≈Planck after Gram |
| 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% |
| θC leading | arctan(tan(arccos(N(13)/N(12))) · exp(−p(13)/2)) | 13.26° | 13.04 ± 0.05° | +1.7% |
| θC (Cabibbo) closed | −α(7)/χ2 (channel-count rule, k=2) | 13.04° | 13.04 ± 0.05° | +0.03σ |
| θ23 (CKM) closed | −α(7)/χ4 (channel-count rule, k=4) | 2.380° | 2.38 ± 0.06° | +0.005σ |
| b/s closed | −α(7)/χ4 (channel-count rule, k=4) | 44.7436 | 44.75 | 0.014% |
| αs(MZ) closed | +α(14)/χ (correction family, k=1) | 0.11792 | 0.1179 ± 0.0009 | +0.02σ |
| mτ / mμ closed | +α(14)/χ (correction family, k=1) | 16.8173 | 16.8170 ± 0.0011 | +0.24σ |
| mτ absolute closed | +α(19)/χ (correction family, k=1) | 1776.82 MeV | 1776.86 ± 0.12 | −0.31σ |
| ℓA closed | +α(19)/χ (correction family, k=1) | 301.44 | 301.6 ± 0.09 | −0.16σ |
| sin2θW closed | +α(5)/χ3 (correction family, k=3) | 0.23123 | 0.23121 ± 0.00004 | +0.40σ |
| Ωm closed | −α(5)/χ3 (correction family, k=3) | 0.31474 | 0.315 ± 0.007 | −0.04σ |
| 1/αem | 1/α(13) + π/α(14) + 6π (chirality theorem, three Dirac layers) | 137.028 | 137.036 | 0.006% |
| mν (heaviest) | m29 · α(21)/χ8 (cascade neutrino chain) | 0.0493 eV | √Δm2atm = 0.0495 eV (PDG 2024) | −0.4% (≈ −0.7σ vs PDG; −2.9σ vs NuFit 6.0) |
| mK / mπ | dV/√N(0) = 5/√2 (cascade pseudoscalar octet, Part IVb rem:cascade-beta0) | 3.5355 | 3.5371 | −0.05% |
| mK (charged) | ΛPDG · 5√2/3 = Λ · dV·√N(0)/Nc | 495.0 MeV | 493.68 MeV | +0.27% |
| mη′ / mη | d0/(Nc+1) = 7/4 (η-η′ double-Adams; Part IVb thm:axial-anomaly-mass) | 1.7500 | 1.7482 | +0.10% |
| mη′ | ΛPDG · √(Nc·d0) = Λ · √21 (double-Adams: ρ(12)−1 × ρ(8)−1) | 962.3 MeV | 957.78 MeV | +0.48% |
| mη | ΛPDG · (Nc+1)·√(Nc/d0) = Λ · 4√(3/7) | 549.9 MeV | 547.86 MeV | +0.37% |
| mπ / ΛPDG | N(0)/Nc = 2/3 (cascade chiral physics; rem:cascade-beta0) | 0.6667 | 0.6646 | +0.31% |
| fπ / mπ | N(0)/Nc = 2/3 (chiral, scheme-invariant) | 0.6667 | 0.6597 | +1.06% |
Specific predictions testable by current or near-future experiments (DESI, Euclid, CMB-S4, SH0ES, lattice QCD, PDG meson masses).
| 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 |
| β0 (QCD 1-loop) | (Nc2+Nc−1) − N(0)·nf/Nc (cascade-primitive identity; rem:cascade-beta0) | β0 = 7 at nf=6 (QCD definition) | Exact match across nf windows; cascade form is structural identification of QCD's 1-loop coefficient |
| β1 (QCD 2-loop) | exact match with two cascade-primitive forms | QCD MS-bar value at any nf | Exact match; cascade-internal disambiguation between the two forms is the open structural piece |
| ΛQCD cascade-native | MZ · exp(−2π) ≈ 170 MeV; with cascade↔MS-bar scheme factor √(Nc/N(0)) → 208.6 MeV | PDG MS-bar (nf=5): 210 ± 14 MeV | Within PDG band (−0.7%); cascade-internal derivation of √(Nc/N(0)) scheme factor is the Tier 2 promotion target |
| χtop1/4 (QCD topological susceptibility) | Witten-Veneziano with cascade-primitive mη, mη′, mK, fπ | 181 MeV (cascade inputs) · 179 MeV (PDG inputs) | Lattice average ≈ 178 MeV; cascade reproduces lattice within +1.7% (cascade) / +0.7% (PDG) |
| Vector mesons (ρ, ω, K*, φ) | Best cascade-primitive fits 1–3% from PDG; structural form ambiguous across the nonet | PDG values | Diagnostic: cascade grammar reaches PS Goldstone octet, stops at J=1; missing machinery is cascade-native hyperfine splitting (open structural direction) |
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 |
| Lighter neutrino masses, solar Δm2, PMNS | Single-source diagonal form gives m2 ≈ 3×10−4 eV and m3 ≈ 3×10−6 eV, too small for the observed solar splitting; cascade analogue of inter-generation mixing not yet derived |