Core Derivations
The trunk of the tree. Every result in GWT traces back to these derivations. Section pages link here rather than re-deriving.
The Geometric Axiom
In Planck units, the entire framework reduces to a single geometric identity:
2/π = the average of |sin(x)| over a full cycle. Not chosen — forced by wave mechanics on a periodic medium.
a = 1 = the lattice spacing defines the unit of length. The Planck length is the pixel size of space.
k = η = stiffness equals inertia. The medium is perfectly impedance-matched — waves propagate without distortion. This is not a coincidence; it is the only self-consistent configuration.
The constants of physics are not inputs to the theory. They are geometric outputs of a 3D wave medium. There was never a choice.
The Three Master Equations
From the axiom k = η = 2/π, a = 1, the three constants of nature follow as algebraic consequences of wave propagation on a discrete elastic medium:
The wave speed on an elastic lattice equals spacing times the square root of stiffness over inertia — the same formula as any mechanical wave (v = √(T/μ) for a string). Light is a transverse wave in the lattice.
The minimum energy quantum of the lattice. One lattice spacing cubed (a³) times the spring constant gives an energy; dividing by wave speed gives the action quantum. This is why energy is quantized — the lattice is discrete.
Gravity is the longitudinal response of the lattice (compression/rarefaction). The factor 2/π comes from the average of |sin(x)| over a full cycle — a wave mechanics signature.
Constant Equivalence
The sets {k, a, η} and {c, ℏ, G} carry the same information — they are interconvertible:
In Planck units: k = η = 2/π, a = 1. The lattice is perfectly impedance-matched — stiffness equals inertia in natural units.
α = 1/137.042 — The Fine Structure Constant
The coupling strength of electromagnetism, derived from the geometry of the lattice's configuration space. Not a fit — a geometric inevitability.
Why α Is Geometric
In Planck units k = η = 2/π, so the lattice is impedance-matched. Electromagnetic coupling is not a force strength — it's the fraction of wave amplitude that couples between transverse modes. This fraction is determined by the shape of the configuration space.
The 5 Degrees of Freedom
Each lattice node has exactly 5 independent degrees of freedom:
| DOF | Count | Origin |
|---|---|---|
| Spatial displacement | 3 | Nc = 3 independent oscillation directions |
| Internal state | 2 | Yin/yang: each node has + or − phase (charge, matter/antimatter) |
| Total | 5 | The configuration space is 5-dimensional |
The Unique Domain: DIV(5)
The lattice wave equation imposes four constraints on the configuration space:
By Cartan's classification, the unique bounded symmetric domain that is 5-dimensional with a quadratic constraint is DIV(5) — the type-IV domain in 5 complex dimensions.
Wyler's Theorem
The coupling constant of any field on a bounded symmetric domain equals the ratio of characteristic volumes. For DIV(5):
αdressed = (9 / 16π³) × (π / 5!)^(1/4) = 1/137.036 (Wyler cross-check, dressed)
This is a volume ratio — like π for circles, it's the same number every time you compute it. The physical factors:
| Factor | Value | Physical Origin |
|---|---|---|
| 9/16π³ | 0.01814 | Ratio of DIV(5) volume to its Shilov boundary |
| (π/120)1/4 | 0.4022 | 4th root of S4 area / codimension correction |
| α | 0.007297 | = 1/137.036 (Wyler, dressed, 0.0001%) |
Why the Wave Picture Matters
Others have attempted to derive α from vacuum structure using a particle picture — treating the 5 DOF as a collection of independent point modes. These efforts are valuable and point in the right direction, but the particle picture limits precision.
The key insight: treat the configuration space as a bounded symmetric domain (a wave picture) rather than a set of point modes. The same 5 degrees of freedom, the same physical starting point — but letting the geometry speak as a wave gives:
1/αdressed = 137.036 (0.0001% — Wyler cross-check)
The numbers speak for themselves.
αs(MZ) = 0.1179 — Strong Coupling
The QCD coupling at the Z mass scale, derived from lattice wave mechanics.
From Nc = 3
With Nc = 3 colors and Nf = 2Nc = 6 flavors:
The QCD beta function coefficient determines how the coupling runs with energy scale.
αs = 1 at Confinement
At the confinement scale, the trinary medium (three states per node) produces a Gibbs phenomenon — the overshoot of a square wave decomposed into its Fourier components:
Running this up to MZ with standard QCD beta function: αs(MZ) = 0.1179 (0.08% accuracy vs PDG 0.1180).
Unification
Three couplings (α, αs, αW) converge at a single point — a direct consequence of Nc = 3.
sin²θW = 15/64 — Weak Mixing Angle
The electroweak mixing angle at the Z mass scale:
Observed: 0.2312. Accuracy: 1.4%.
Mass Formulas
Every particle mass is derived from {α, Nc, mP} — the Planck mass and the geometric constants above.
Electron Mass
Proton-to-Electron Mass Ratio
The bare ratio 6π5 = 1836.12 comes from mode counting (3D spherical vs 1D transverse). The vacuum polarization correction α²/2√2 comes from the quark charge identity ΣQ² = 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. Not a coincidence — a derivation.
Lepton Masses (Koide Relation)
The three charged lepton masses satisfy:
The Koide ratio 2/3 appears because the three leptons correspond to the three spatial dimensions (Nc = 3), and the ratio (Nc−1)/Nc governs the mass splitting.
| Lepton | GWT | Observed | Error |
|---|---|---|---|
| me | 0.511 MeV | 0.511 MeV | exact |
| mμ | 105.658 MeV | 105.658 MeV | 0.005% |
| mτ | 1776.97 MeV | 1776.86 MeV | 0.006% |
Proton Mass — The Mass Gap
The factor of 4 comes from: virial theorem in d+1 = 4 dimensions × RMS geometry (0.532) × Gibbs normalization (αs = 1). This solves the Clay Millennium mass gap problem.
Quark Masses
| Quark | GWT | Observed | Error |
|---|---|---|---|
| Up m(13, 31) | 2.21 MeV | 2.16 MeV | 2.5% |
| Down m(5, 30) | 4.78 MeV | 4.67 MeV | 2.4% |
| Strange m(4, 28) | 98.6 MeV | 93.4 MeV | 5.5% |
| Charm m(11, 27) | 1271 MeV | 1271 MeV | 0.02% |
| Bottom m(7, 26) | 4.31 GeV | 4.18 GeV | 3.1% |
| Top m(12, 24) | 176.5 GeV | 172.8 GeV | 2.2% |
Boson Masses
| Boson | GWT | Observed | Error |
|---|---|---|---|
| W | 79.4 GeV | 80.4 GeV | 1.2% |
| Z | 90.7 GeV | 91.19 GeV | 0.5% |
| Higgs | 124.8 GeV | 125.09 GeV | 0.2% |
| Higgs vev | 245.5 GeV | 246.22 GeV | 0.3% |
Neutrino Masses
| Observable | GWT | Observed | Error |
|---|---|---|---|
| Δm²31 | 2.523×10−3 eV² | 2.534×10−3 | 0.4% |
| Δm²21 | 7.49×10−5 eV² | 7.53×10−5 | 0.6% |
| Nν (generations) | 3 | 2.984±0.008 | 0.5% |
ΩΛ = 2/3 — Dark Energy Fraction
The fraction of the universe's energy budget that is dark energy, derived from dimensionality alone.
The Dimensional Argument
In a d-dimensional lattice, each node displacement has:
- 1 longitudinal mode (compression along displacement direction) → gravity
- (d−1) transverse modes (perpendicular to displacement) → dark energy (restoring force)
Observed: 0.685. The 2.7% gap is ΛCDM model bias — ΛCDM uses GN everywhere, but GWT predicts Geff = 6.8 GN in galactic halos. Correcting this closes the gap.
This is the same ratio as the Koide relation — (Nc−1)/Nc = 2/3 — appearing in both lepton mass harmony and cosmic energy budget. Not a coincidence: both come from the dimensionality of the lattice.
Geff = 6.8 GN — Enhanced Gravity
In galactic halos, the effective gravitational constant is enhanced by the ratio of total matter to baryonic matter:
This replaces dark matter particles entirely. The medium itself, when compressed by large-scale structure, increases its effective gravitational coupling. Observed gravitational lensing data yields 5–7 GN — consistent.