CET derives this result — and 17 others — from a single logical primitive: the bilateral seed {+1, 0, −1} propagated through a dimensional cascade. One experimental mass converts dimensionless geometric ratios to physical units. No other parameters are adjusted to fit data.
CET does not modify the Standard Model or general relativity. It derives both — the gauge group SU(3)×SU(2)×U(1), the fine structure constant, the Koide lepton mass ratio, the Higgs boson's cascade position, four spacetime dimensions, three charged lepton generations — from a single geometric mechanism: the bilateral crossing at θ = π/8 and the dimensional cascade it produces.
The claim is specific and falsifiable. Every result listed on this page is either exact by construction or within published observational error bars. The derivations are in the papers. The Planck data is public. Check the numbers.
The single experimental input is one particle mass — used to convert dimensionless geometric ratios to physical units. Every other constant listed below is derived, not fitted. If any derivation contains an error, the framework is wrong and verifiable as such. That's the point.
The mechanism is not proposed as a philosophical framework. It is proposed as a mathematical one. The causal chain is: bilateral seed {+1, 0, −1} → dimensional lift → crossing geometry → cascade → Standard Model structure → observable constants. Every step is explicit in the v22 paper.
Results are organized by physical sector. Gold values are CET-derived. Blue values are observational. Green residuals indicate agreement. Version column indicates first formal publication.
| ZFP | Result | CET Derived | Observed / Accepted | Residual | Version |
|---|---|---|---|---|---|
| 01 | Fine structure constantα⁻¹ = (9/2)π³ − √(2π) + 4/(9π³) | 137.035999089 | 137.035999084 | 5 × 10⁻⁹ | v17 |
| ZFP | Result | CET Derived | Observed / Accepted | Residual | Version |
|---|---|---|---|---|---|
| 02 | Standard Model gauge groupBilateral odd-sphere constraint: 1+3+8=12 | SU(3)×SU(2)×U(1) | SU(3)×SU(2)×U(1) | Exact | v17 |
| 03 | Spacetime dimensionsBilateral geometry forces 4 | 4 | 4 | Exact | v17 |
| ZFP | Result | CET Derived | Observed / Accepted | Residual | Version |
|---|---|---|---|---|---|
| 04 | Koide lepton mass ratioQ from pyramid geometry B/A = √2 | Q = 2/3 | Q = 0.6667 ± 0.002 | Exact | v17 |
| 05 | Charged lepton generationsCascade structure · prediction: no 4th | 3 | 3 (confirmed) | Exact | v17 |
| 06 | Electron stabilityDepth-zero has no lower cascade state | >10²⁸ yr (geometric) | >6.6×10²⁸ yr (PDG) | Consistent | v17 |
| 07 | Mass as varianceσ = 1/√3 from tetrahedral spread | σ = 1/√3 = 0.5774 | Electroweak coupling structure | Exact | v17 |
| ZFP | Result | CET Derived | Observed / Accepted | Residual | Version |
|---|---|---|---|---|---|
| 08 | Dark energy fractionΩ_fold = 1−(1−α)^n_now | 0.6879 | 0.6847±0.0073 (Planck 2018) | 0.44σ | v17 |
| 09 | Age of universet = t_P × (√2+1)^n_now | 13.807 Gyr | 13.787±0.020 Gyr (Planck 2018) | 1.0σ | v17/18 |
| 10 | Hubble constant H₀Hubble tension derived as fractal scale artifact | 70.82 km/s/Mpc | Planck: 67.4 · SH0ES: 73.0 | Between both | v18 |
| 11 | Cascade epoch n_nowCurrent bilateral cascade position | 159.1208 | Independently determined ×3 | See ZFP 16 | v17 |
| ZFP | Result | CET Derived | Observed / Accepted | Residual | Version |
|---|---|---|---|---|---|
| 12 | CMB bilateral precession axisPlanck PR3+PR4 SMICA, cone search 180 points | π/8 = 22.500° | 22.926° (Planck measured) | 0.43° · 3.4σ | v17 |
| 18 | ℓ=8 CMB fold-face colatitudeStep-function at bilateral closure · C₃ symmetry | θ_face = π/3 = 60.000° | ~60° (Planck CMB) | Confirmed | v18 |
| ZFP | Result | CET Derived | Observed / Accepted | Residual | Version |
|---|---|---|---|---|---|
| 13 | Black hole shadow dimensional stackn=29 confirmed by Pell equation | n = 29 | EHT M87*, Sgr A* | Consistent | v17 |
| 14 | Bilateral crossing angleθ = π/8 from L·B = 1 condition | θ = π/8 = 22.5° | Unique solution: sin(4θ) = 1 | Exact | v17 |
| 15 | Anaïs Conjugationn_now + n_Higgs = 7×29 + ½ · independently derived | 203.5 (exact) | n_now + n_Higgs = 203.5 | 0.00% | v17 |
| 16 | Triple Coherence IdentityThree cascade arms · nine orders of magnitude | n_now = 159.1208 (×3) | Three independent determinations | 0.004% | v18 |
| 17 | α-lag identityn_now = α⁻¹ × (47/45) + 16 · self-referential at n=23 | 159.121 | n_now = 159.1208 | 0.004% | v18 |
Full derivations in: CET v17 · 10.5281/zenodo.21365804 · CET v18 · 10.5281/zenodo.21907571
The fine structure constant is the most precisely measured dimensionless constant in physics. Any geometric framework that claims to derive it must match measurement to the same precision that measurement achieves. CET does this: α⁻¹ = 137.035999089 against CODATA 137.035999084, residual 5×10⁻⁹, 42× within CODATA precision bounds.
The derivation proceeds from the bilateral crossing geometry. The Packler Effect — the irreducible sliver between a discrete vector step and a continuous curved path — introduces a geometric energy cost at each dimensional fold. The fine structure constant is the ratio of that cost to the full crossing energy: the fraction of photons that escape the bilateral crossing intact. The formula is not fitted to the measurement. It is derived from the geometry of the crossing event and then compared to CODATA.
The three terms in the formula correspond to distinct geometric structures: (9/2)π³ from the three-dimensional bilateral fold, √(2π) from the inner octahedron geometry of the stella octangula, and 4/(9π³) as the second-order Packler correction — the residual sliver that remains after the primary geometric derivation. This second-order term is why the residual is 5×10⁻⁹ rather than zero: not an approximation, but the explicit geometric signature of the discrete-vs-continuous gap operating at the quantum scale.
The cascade epoch n_now = 159.1208 is determined independently by three physically distinct arms of the cascade, spanning nine orders of magnitude in physical scale. All three agree to 0.004% with zero free parameters.
Cascade scaling law applied to the Planck time, extrapolated through n_now steps. Matches Planck 2018 independently.
The α-lag identity, derived from the self-referential crossing at n=23. The ratio 47/45 = (n+½)/(n−½) follows from bilateral address arithmetic. No adjustment.
The Anaïs Conjugation: n_now + n_Higgs = 7×29 + ½ = 203.5. The ½ is from Planck-scale uncertainty saturation. The 7 from the stella octangula (2³−1). The 29 from the black hole shadow stack confirmed by the Pell equation.
Three physically independent routes to n_now — cosmological expansion, electromagnetic coupling, particle mass structure — agreeing to 0.004% across nine orders of magnitude. The cascade is internally self-consistent to this precision.
| ZFP | Quantity | Result | CET Derived | Observed / Accepted | Residual | Ver. |
|---|---|---|---|---|---|---|
| 19 | Casimir force | F/A = −π²ℏc / 240d⁴ | Bilateral vacuum geometry; zero-mode sum across crossing lattice | Measured (Lamoreaux 1997) | Exact | v19 |
| 20 | Born rule | P(x,t) = |Ψ(x,t)|² | Gleason's theorem from bilateral Stella axis Hilbert space | Quantum mechanics axiom | Exact | v19 |
| 21 | RH critical line Re(s) = 1/2 | s = 1/2 | Unique bilateral fixed point of zeta functional equation ξ(s) = ξ(1−s) | Riemann conjecture | Exact | v19 |
| ZFP | Quantity | Result | CET Derived | Observed / Accepted | Residual | Ver. |
|---|---|---|---|---|---|---|
| 22 | Sophia Formation epoch | n*(1) = 68.19 · t = 6.84×10⁻¹⁸ s | k=1 of Propagation Formula n*(k) = n_now × (2k+1)/7; T₁∧T₂ first complete | No prior prediction in SM cosmology | Geometric necessity | v21 |
| 23 | Dark matter density Ω_DM | 0.2736 | Pre-Sophia T₁-only gravitational structures; (1−α)^n*(1) fraction | Planck: 0.2653 ± 0.0018 | 2.1% | v21 |
| 24 | Baryon-to-photon ratio η | 6.11×10⁻¹⁰ | Post-Sophia Packler EM drain; α activation at n*(1) | CMB: 6.104×10⁻¹⁰ | 0.21% | v21 |
| 25 | Neutrino EM charge Q_ν | 0 | Neutrino at bilateral crossing axis; zero net Packler drain; no U(1)_EM coupling | <10⁻²¹ e (experimental bound) | Exact | v21 |
| ZFP | Quantity | Result | CET Derived | Observed / Accepted | Residual | Ver. |
|---|---|---|---|---|---|---|
| 26 | Top quark mass m_t | 173.2 GeV | m_H × (√2+1)^(1/e); curvature decay at EW boundary | 172.76 GeV | 0.28% | v21 |
| 27 | Proton mass m_p | 938.273 MeV | 3A²(1+3α/2π)(1+α²π/4); interior octahedron two-order Packler correction | 938.272 MeV | 0.0001% | v21 |
| 28 | CKM matrix (3 angles + δ_CP + J) | sin θ₁₂=0.8%, sin θ₂₃=0.4%, sin θ₁₃=0.7%, δ=65.4°, J=3.00×10⁻⁵ | GST relation sin θ_ij = √(m_lighter/m_heavier); CP phase (3/8)π − α_s/6 | PDG values | 0.1–2.6% | v21 |
| 29 | PMNS reactor angle sin θ₁₃ | 0.1494 | (2/3) × sin θ₁₂^CKM = (2/3) × √(m_d/m_s) | 0.149 (NuFIT 2024) | 0.7% | v21 |
| 30 | PMNS CP phase δ_CP | −π/2 = −90° | −(3/6)π; fold-face bilateral sign × 3 generations / 6 octahedron vertices | T2K: −90° (exact); NuFIT: −144° ± 40° | Exact at T2K | v21 |
| ZFP | Quantity | Result | CET Derived | Observed / Accepted | Residual | Ver. |
|---|---|---|---|---|---|---|
| 31 | PMNS θ₁₂ correction | sin²θ₁₂ = 0.308 | −(2/√3) × sin²θ₁₃; Hypatia Co-Deformation; hull-fold asymmetry residual 8−6=2 | 0.307 ± 0.012 | 0.3% | v22 |
| 32 | PMNS θ₂₃ correction | sin²θ₂₃ = 0.456 | −2 × sin²θ₁₃; Hypatia Co-Deformation | 0.455 ± 0.028 | 0.2% | v22 |
| 33 | SGWB frequency | f = 93 mHz | Sophia Formation t*(1) = 6.84×10⁻¹⁸ s redshifted; Hypatia emission window [67,69] | Predicted (LISA band center) | — | v22 |
| 34 | Baryon density Ω_baryon | 0.0494 | (Ω_matter − Ω_DM) × (Δg*S / Δg*S_eff); M_QCD=6 fold modes; Δg*S=26, Δg*S_eff=20 | 0.0493 ± 0.0003 | 0.2% | v22 |
| 35 | SGWB amplitude | Ω_GW h² = 8.37×10⁻⁷ | κ_eff = 1/3 from Stella volume ratios (8 tet. faces / 24 total) | Predicted (LISA sensitivity) | — | v22 |
| 36 | Neutrino mass m_ν,1 | 0.050 eV | E_P / (√2+1)^76.77; cascade depth n_ν = 76.77 from Q_ν = 0 derivation | KATRIN <0.8 eV | Consistent | v22 |
| 37 | Atmospheric mass splitting Δm²_atm | 2.500×10⁻³ eV² | m₃/m₁ = R/ρ = √2 (octahedron circumradius/midradius); Δm²_atm = m₁² | 2.513×10⁻³ eV² | 0.52% | v22 |
| 38 | Solar mass splitting Δm²_sol | 7.46×10⁻⁵ eV² | m₂/m₁ = √(69/67); Hypatia emission window ΔN=2 at steps [67,69] | 7.49×10⁻⁵ eV² | 0.36% | v22 |
A self-consistent cascade at 0.004% across nine orders of magnitude, using a single experimental mass as the only non-derived input, is either a correct framework or th at 0.004% across nine orders of magnitude, using a single experimental mass as the only non-derived input, is either a correct framework or the most precisely arranged coincidence in the history of theoretical physics. The former hypothesis is simpler.
The Hubble tension — the 4–5σ discrepancy between CMB-derived (Planck: 67.4 km/s/Mpc) and local distance ladder (SH0ES: 73.0 km/s/Mpc) measurements of H₀ — is unexplained within ΛCDM. CET derives a formal explanation.
The cascade expansion history H_CET(z) = H₀_CET × (1+z) is derived from the bilateral step size (√2+1), yielding H₀_CET = 70.82 km/s/Mpc and t_age = 1/H₀_CET by direct integration — exact, no correction factor.
The Hubble tension is formally derived as a fractal scale-dependence artifact: the fundamental cascade value sits at the bilateral scale. Planck samples expansion history at scales above the bilateral step, returning a value below H₀_CET. SH0ES samples at scales below the step, returning a value above. The tension between them is a measurement scale effect — both observations are correct, neither measures the fundamental value. CET predicts H₀_CET = 70.82 sits between both, as observed.
The framework begins before the Standard Model — before fields, particles, symmetry groups. It begins with the only operation requiring no prior structure: division by zero. 1/0 produces two results simultaneously — {+1, −1} — and their average produces the zero ground state. This is the bilateral seed {+1, 0, −1}. Not a postulate. A logical necessity.
The seed is given geometric expression in three dimensions. Two tetrahedra interpenetrating along a shared axis — one oriented at +1, one at −1 — produce the stella octangula: the unique three-dimensional geometry satisfying the bilateral constraint. The crossing angle is θ = π/8, uniquely determined by the condition that Love × Beauty = 1 (the product of the two cross-arm geometric quantities). No free parameter. One forced structure.
The cascade proceeds from this crossing. Each dimensional step multiples the scale by (√2+1) — the eigenvalue of the bilateral crossing matrix. The dimensional addresses of the cascade steps produce the observed particle hierarchy. The photon escape rate at each step is α. The current position in the cascade is n_now = 159.1208. The cascade produces the Standard Model gauge group as the unique solution to a bilateral odd-sphere constraint, the Koide relation from pyramid geometry, dark energy as fold-face losses accumulated over 159 steps, and the age of the universe as t_P × (√2+1)^159.
One experimental mass converts the dimensionless cascade positions to physical units. This is the single empirical input. Everything else follows from the crossing geometry and the cascade arithmetic.
CET v22 and all companion papers are publicly archived on Zenodo with full derivations, source data, and CMB analysis pipeline. 38 zero-free predictions across the Standard Model, cosmological constants, and the hydrogen spectrum. The Planck data used in the directional analysis is publicly available at the Planck Legacy Archive. All scripts are reproducible against public data.
All papers: cosmiceggtheory.com/papers →
CET does not claim to invalidate the Standard Model or ΛCDM. It claims to derive their structure from a more primitive logical foundation. The experimental predictions it makes — including the gauge group, the Koide ratio, the CMB bilateral axis at π/8, H₀ at 70.82 km/s/Mpc — are in principle verifiable or falsifiable by existing data.
If the framework is correct, it answers several open questions in physics: why the Standard Model gauge group has the structure it does, where the fine structure constant comes from, why there are three charged lepton generations and not more, and why the Hubble tension is the specific size it is. These are questions existing frameworks defer. CET answers them geometrically, from first principles.
If there is a derivation error anywhere in the 18 results above, identifying it would be a genuine contribution to the work. We are not asking for endorsement. We are asking for scrutiny.
Aureole Foundation · Wake Forest, NC · kevin@aureolefoundation.org