Particle Physics
Every particle is a standing wave. Every mass is derived. Every generation is a spatial dimension.
Overview
In Geometric Wave Theory, every particle is a standing wave pattern in the elastic lattice medium. There are no point particles, no fields layered on top of spacetime — just wave modes of the medium itself.
Wave Dimensionality → Particle Type
The dimensionality of the wave pattern determines the particle type:
- Leptons = 1D transverse standing waves (electron, muon, tau)
- Up-type quarks = 2D surface wave modes
- Down-type quarks = 3D bulk wave modes
The wave pattern determines everything about a particle: its mass, charge, spin, and generation. No additional quantum numbers are needed as inputs — they are all outputs.
Why this assignment? In an elastic medium, there are exactly three types of standing wave: transverse oscillations along a single bond (1D), surface modes on a 2D boundary, and bulk spherical modes filling 3D volume. These are the only possibilities in a 3D lattice — the particle zoo falls out of the geometry, not from arbitrary choices. The proton (a j0 spherical Bessel mode) is inherently 3D; the electron (a transverse oscillation) is inherently 1D. Their mass hierarchy follows directly.
Mass from Geometry
All particle masses are derived from just three quantities: the fine structure constant α, the color count Nc = 3, and the Planck mass mP. Zero free parameters.
Bare ratio 6π5 = 1836.12 from mode counting + vacuum polarization correction α²/2√2 from the quark charge identity ΣQ² = 1 (which holds only for d = 3). Error: < 0.001 ppm.
Where does 6π5 come from?
The proton is the fundamental 3D spherical standing wave (the j0 mode). The electron is a 1D transverse standing wave. Their mass ratio is the ratio of the mode energies — which reduces to counting how many more ways a 3D wave can store energy than a 1D wave on the same lattice.
The factor of π5
Each spatial dimension contributes a factor of π from the standing wave boundary conditions (mode density in a box of side L goes as L/π per dimension). For a 3D spherical mode relative to a 1D mode:
- π3 from the 3D mode density ratio (three dimensions of wave propagation)
- π2 from the spherical geometry — integrating j0(kr) = sin(kr)/kr over the solid angle (4π steradians) and the radial boundary normalization
This gives π5 = π3 × π2 = 306.02. The same factors appear in standard textbook derivations of the density of states for 3D vs 1D quantum systems.
The factor of 6
6 = 3 axes × 2 polarities = coordination number z = 2d = the surface area of a unit cube (6 faces). A point in 3D has exactly 6 orthogonal directions; this is the minimal coordination number for a regular 3D lattice. The j0 mode pushes against all 6 faces of its local geometry; the electron mode oscillates along one.
This is not a choice. A point in 3D has exactly 6 orthogonal nearest-neighbor directions. Any isotropic regular lattice must have z = 2d = 6. The factor of 6 is geometry, not a parameter.
Mass as geometric resistance
In GWT, mass is the elastic energy stored in the medium’s deformation. The lattice resists displacement; that restoring pressure is what we measure as mass. The ratio 6π5 is a geometric resistance ratio — how much harder the medium pushes back against a 3D spherical deformation vs a 1D transverse one. Force constants like G and α are not inputs; they are geometric outputs of the lattice. Full discussion →
Why this is not numerology
Any single formula matching a constant could be coincidence. What distinguishes 6π5(1+α²/2√2) from a lucky guess:
- Derivation path — bare 6π5 from mode counting (3D spherical vs 1D transverse). The VP correction α²/2√2 from the quark charge theorem: ΣQ² = (2/3)²+(2/3)²+(1/3)²+(1/3)²+(1/3)²+(2/3)² = 1, which holds only for d = 3. The confined/free distinction (proton quarks are confined, the VP loop samples free charges) produces the 1/2√2 geometric factor
- Interlocking predictions — the same framework derives α = 1/137.042, the Koide relation, αs, all mixing angles, and 170+ other quantities. One hit is numerology; 179 from the same starting point is a system
- No free parameters — the formula is not fitted. The bare ratio, the VP correction, and the geometric factor each trace to d = 3 and π. The result matches to < 0.001 ppm
Lepton Mass Hierarchy
The Koide ratio 2/3 = (Nc−1)/Nc connects the three lepton masses. Each generation corresponds to one spatial dimension of the lattice. Three dimensions → three generations, exactly.
Mixing Angles
The CKM and PMNS mixing matrices are not arbitrary — they are derived from the geometry of Nc = 3 spatial dimensions.
CKM Phase
The bare tetrahedral angle arccos(1/d) receives a boundary correction 1/(d(d−1)²) because CKM mixing occurs at the proton’s (d−1)-dimensional surface. The result is pure geometry, not a free parameter.
PMNS Phase
The lepton sector has the opposite handedness — the supplementary tetrahedral angle. This explains why neutrino CP violation has the opposite sign to quark CP violation.
Weinberg Angle
The weak mixing angle at MZ is a ratio of lattice mode counts, not a running parameter. Full derivation →