The Periodic Table of Elementary Particles from S³ Geometry

Q: What is the Periodic Table of Elementary Particles?

The Periodic Table of Elementary Particles (PTEP) is an interactive chart organizing all known elementary particles — quarks, leptons, bosons, hadrons, and their antiparticles — with their masses, charges, spins, and S³ geometry formulas. It shows how 63 physical quantities can be derived from just two inputs: the electron mass and the muon mass.

Q: How are particle masses calculated in S³ geometry?

In the S³ geometry framework, all elementary particle masses are derived using resonance patterns of the three-sphere (S³). The theory uses Bessel function zeros, Koide-type formulas, Clifford algebra dualities, and π-harmonic towers to compute masses from just mₑ (electron mass) and mμ (muon mass). The typical deviation from experimental values is less than 1%.

Q: What particles are included in PTEP?

PTEP includes all Standard Model particles: 6 quarks (up, down, charm, strange, top, bottom), 6 leptons (electron, muon, tau, and their neutrinos), gauge bosons (photon, gluon, W, Z), the Higgs boson, hadrons (proton, neutron, J/ψ, Ω⁻, B meson, Σ⁺), pions, dark sector predictions (χ scalar, X17 boson), and all corresponding antiparticles.

Q: What is the mass of the Higgs boson?

The experimentally measured Higgs boson mass is 125.1 GeV. In the S³ geometry framework, it is predicted as MW · j₃/j₁ = 80,343 × 6.988/4.493 = 124,946 MeV (124.9 GeV), with a deviation of only 0.12% from experiment.

Q: What is the Koide formula?

The Koide formula is a remarkable empirical relation among the masses of the three charged leptons (electron, muon, tau): Q = (mₑ + mμ + mτ) / (√mₑ + √mμ + √mτ)² = 2/3. In the S³ geometry framework, this formula is generalized to predict neutrino masses and other particle mass ratios with high precision.

Q: What are Bessel function zeros in particle physics?

Bessel function zeros (j₁ ≈ 4.493, j₂ ≈ 5.763, j₃ ≈ 6.988) appear as fundamental constants in the S³ geometry framework. When multiplied by the electroweak scale Λₑw ≈ 17,880 MeV, they produce particle masses that match the top quark (~169 GeV), W boson (~80 GeV), and Higgs boson (~125 GeV).

Dmitry Makaryev

The Periodic Table of Elementary Particles

All elementary particle masses obey S³ geometry — the result of resonance and harmonic synthesis of the sphere, torus, and membrane.

IIIIII

Quarks

Up-type

Down-type

Leptons

Charged

Neutrino

Bosons

Gauge Boson

Scalar Boson

Antimatter — Quarks

Antimatter — Leptons

Antimatter — Bosons

Hadrons

Antimatter

Dark Sector

Pion (outside S³)

< 0.5% < 1% < 3% Predicted Antimatter Pion (outside S³)

Bessel Resonance Grid

M(l,n) = Λ_ew × j_l,n

l \ n	n=1	n=2	n=3	n=4
l=0	56 GeV ★	112 GeV ★	169 GeV Top quark (≈mₜ)	225 GeV ★
l=1	80 GeV W boson	138 GeV ★	195 GeV ★	252 GeV ★
l=2	103 GeV ★	163 GeV ★	220 GeV ★	277 GeV ★
l=3	125 GeV Higgs boson	186 GeV ★	245 GeV ★	303 GeV ★
l=4	146 GeV ★	209 GeV ★	269 GeV ★	327 GeV ★

Known particleNew prediction

The π-Harmonic Dark Tower

m_n = m_χ · πⁿ

n=+6

51.8 GeV

near_S01_56GeV★

n=+5

16.5 GeV

★

n=+4

5.2 GeV

B±✓

n=+3

1.7 GeV

Ω⁻✓

n=+2

531.48 MeV

near_eta_prime★

n=+1

169.17 MeV

near_pion_scale★

n=0

53.85 MeV

χ dark scalar★

n=-1

17.14 MeV

X17 (Atomki)

n=-2

5.46 MeV

nuclear_anomalies★

n=-3

1.74 MeV

nuclear_anomaly_scale★

Heavy Compound Resonances

TeV-scale predictions from products of Bessel roots

1.62 TeV

MW × j₁²

ATLAS Wγ excess ~1.6 TeV

2.08 TeV

MW × j₁·j₂

Accessible at HL-LHC

3.24 TeV

MW × j₂·j₃

Accessible at HL-LHC

Constants, Mixing & Topology

All from five geometric building blocks: √2, j₁, j₂, j₃, π, and 3

Coupling Constants

1/α(M_Z)

0.01% ⭐

Exp:127.951

S³:127.97

j₁·j₂·j₃ / √2 = 4.493 × 5.763 × 6.988 / 1.414 = 127.97

sin²θ_W

0.29% ⭐

Exp:0.2312

S³:0.2319

3/8 − 1/j₃ = 0.375 − 0.143 = 0.232

α_s(M_Z)

0.04% ⭐

Exp:0.1179

S³:0.1178

√2 / 12 = 1.414/12 = 0.11785

CKM Matrix

|V_us|

1.74%

Exp:0.2265

S³:0.2225

1/j₁ = 1/4.493 = 0.2225

|V_cb|

0.22% ⭐

Exp:0.04053

S³:0.04044

√(2/3) / j₁² = 0.8165/20.19 = 0.04044

|V_ub|

2.41%

Exp:0.003820

S³:0.003730

√(2/3)·(√2−1) / j₁³ = 0.00373

δ_CKM(rad)

0.33% ⭐

Exp:1.144

S³:1.148

arctan(π/√2) = arctan(2.221) = 1.148 rad = 65.8°

PMNS Angles

sin²θ₁₂

0.83%

Exp:0.3040

S³:0.3015

1/3 − 1/(j₁·j₃) = 0.333 − 0.032 = 0.302

sin²θ₂₃

0.12% ⭐

Exp:0.4510

S³:0.4505

1/2 − 1/j₁² = 0.500 − 0.050 = 0.450

sin²θ₁₃

0.82%

Exp:0.02226

S³:0.02208

N²/j₁⁴ = 9/407.7 = 0.02208

Topology

Q (Koide)

0.01% ⭐

Exp:0.6666

S³:0.6667

2/3 (exact from Clifford torus)

N_gen

exact

Exp:3

S³:3

χ(S²)/Q = 2/(2/3) = 3

δ (Koide phase)

S³:0.2222

χ(S²)/N² = 2/9 ≈ 0.2222

μ² (Koide scale)(MeV)

S³:313.84

((√mₑ + √mμ + √mτ)/3)² = 313.84 MeV

Nucleon Sector

M_W/m_p

0.39% ⭐

Exp:85.67

S³:85.33

2⁸/3 = 256/3 = 85.33

m_c^const(MeV)

0.10% ⭐

Exp:1550

S³:1548.8

μ² · π²/2 = 313.84 × 4.935 = 1548.8 MeV