Calculation: The Three Master Equations
Deriving c, ℏ, and G from three lattice constants — step by step with actual numbers.
1. The Starting Point (The Axiom)
The entire framework begins with one geometric identity. In Planck units:
The number 2/π = 0.6366... is not chosen. It is the average of |sin(x)| over a full cycle:
Why 2/π?
Any oscillating system on a periodic lattice has this average displacement. The lattice’s stiffness and inertia are both set by this single wave-mechanical number. There was never a choice.
In SI units, these become three measurable constants:
| Constant | Symbol | Value | Meaning |
|---|---|---|---|
| Lattice stiffness | k | 4.77 × 1078 N/m | Spring constant of each bond between nodes |
| Lattice spacing | a | 1.616 × 10−35 m | Distance between adjacent nodes (= Planck length) |
| Inertial response | η | 1.385 × 10−8 kg | How the medium resists displacement (like ε0) |
In Planck units, k = η = 2/π and a = 1. The lattice has zero free parameters. Every constant of nature is a geometric consequence of wave mechanics on a discrete elastic medium.
2. Speed of Light: c = a√(k/η)
The wave speed on any elastic medium equals the spacing times the square root of stiffness over inertia — the same formula as a vibrating string (v = √(T/μ)).
Step 1: Compute k/η
k / η = 3.444 × 1086 s−2
Units check: (N/m) / kg = (kg·m/s²/m) / kg = s−2. Correct — this is a squared frequency.
Step 2: Take the square root
√(k / η) = 1.856 × 1043 s−1
This is the Planck angular frequency ωP — the fastest oscillation the lattice supports.
Step 3: Multiply by lattice spacing
c = (1.616 × 10−35 m) × (1.856 × 1043 s−1)
c = 2.998 × 108 m/s
Result
EXACT. The speed of light is the wave speed on the lattice. Same formula as any mechanical wave: v = √(tension/density). Light is a transverse wave in the elastic medium.
In Planck Units (sanity check)
When k = η, the ratio is exactly 1. The speed of light equals 1 in Planck units because the medium is perfectly impedance-matched — stiffness equals inertia.
Physical Meaning
Light is not “just fast” — it is the maximum signal speed set by the lattice’s mechanical properties. Nothing can exceed it because no disturbance can outrun the medium’s own response time. Relativity is a consequence of wave mechanics, not an axiom.
3. Planck’s Constant: h = π²ka³/c
The minimum action quantum of the lattice. One cell volume (a³) times the spring constant gives an energy; the geometric factor π² comes from the standing-wave boundary conditions; dividing by wave speed gives an action.
Step 1: Compute a³
a³ = 4.221 × 10−105 m³
One cubic Planck volume — the smallest volume in the lattice.
Step 2: Compute ka³
ka³ = 2.013 × 10−26 N·m² = 2.013 × 10−26 J·m
Units check: (N/m)×m³ = N·m² = J·m. This is an energy times a length — about to become an action when divided by speed.
Step 3: Multiply by π²
π² × ka³ = 9.8696 × 2.013 × 10−26
π² × ka³ = 1.987 × 10−25 J·m
Step 4: Divide by c
h = 6.626 × 10−34 J·s
Result
And the reduced Planck constant:
EXACT. Energy is quantized because the lattice is discrete. The minimum action equals the spring constant times one cell volume divided by the wave speed.
In Planck Units (sanity check)
ℏ = h / 2π = 2π / 2π = 1 ✓
In Planck units h = 2π and ℏ = 1, exactly as required. The factor of 2π traces to the full-cycle geometry of standing waves on the lattice.
Physical Meaning
Planck’s constant is not a mysterious fundamental quantity — it is the energy stored in one lattice cell at the minimum excitation, converted to an action by dividing by the wave speed. Energy is quantized for the same reason guitar strings have harmonics: the medium is discrete and waves must fit integer half-wavelengths into a bounded region.
4. Gravitational Constant: G = 2c4/(πka)
Gravity is the longitudinal (compression/rarefaction) response of the lattice. Two standing waves each compress the medium slightly; the overlap region has reduced lattice pressure, producing a net inward force. G measures how efficiently this pressure deficit converts to acceleration.
Step 1: Compute c4
c4 = 8.077 × 1033 m4/s4
Step 2: Compute ka
ka = 7.708 × 1043 N
Units check: (N/m)×m = N. This is the total restoring force per node at unit displacement — an enormous number that reflects the lattice’s extraordinary stiffness.
Step 3: Multiply by π
πka = 2.421 × 1044 N
Step 4: Combine
G = 2 × (8.077 × 1033) / (2.421 × 1044)
G = 1.615 × 1034 / 2.421 × 1044
G = 6.674 × 10−11 m³/(kg·s²)
Result
EXACT. Gravity is a pressure deficit in the elastic medium. The factor 2/π is the wave-mechanical average of |sin(x)| — the same number that sets k and η.
In Planck Units (sanity check)
The factor π in the denominator cancels the π from k = 2/π, leaving G = 1 exactly. This cancellation is not a coincidence — it is the wave-averaging that produces 2/π in the first place.
Physical Meaning
Gravity is not a mysterious “force at a distance.” Every standing wave (mass) creates a slight compression of the lattice around it. When two such compressions overlap, the region between them is slightly less compressed than the regions on the outside. The net lattice pressure pushes the two waves together.
G encodes how efficiently the medium converts this pressure deficit into acceleration: it is the ratio of wave speed (c4) to lattice stiffness (ka), modulated by the wave-averaging factor (2/π).
5. Constant Equivalence: {k, a, η} ↔ {c, ℏ, G}
The lattice constants and the Planck constants are not independent sets — they are the same three degrees of freedom expressed in different unit systems. Each set can be derived from the other.
Lattice → Planck (the master equations above)
Forward Direction
Planck → Lattice (inverting the equations)
Solving for k
k = hc / (π²a³) = 2πℏc / (2π × πa³) = ...
k = 2ℏc / (πa³)
Verification: k = 2 × (1.055 × 10−34) × (2.998 × 108) / (π × (1.616 × 10−35)3) = 4.77 × 1078 N/m ✓
Solving for η
η = ka²/c²
Substitute k = 2ℏc/(πa³):
η = (2ℏc/(πa³)) × a² / c²
η = 2ℏ / (πac)
Verification: η = 2 × (1.055 × 10−34) / (π × 1.616 × 10−35 × 2.998 × 108) = 1.385 × 10−8 kg ✓
Solving for a
G = 2c4 / (π × 2ℏc/(πa³) × a) = 2c4πa³ / (2πℏca) = c³a²/ℏ
Solve for a: a² = ℏG/c³
a = √(ℏG/c³) = lP
The lattice spacing is the Planck length — derived, not assumed.
The Key Insight
These are not independent derivations. They are the same three degrees of freedom written in two different languages:
Like Celsius and Fahrenheit — same temperature, different scale. Three constants in, three constants out. The universe has exactly three mechanical degrees of freedom, no more, no less.
6. Gravity from First Principles
The lattice pressure-deficit mechanism gives a concrete force formula. Two standing waves with energies E1 and E2 separated by distance r experience a net inward force:
Since E = mc² and G = 2c4/(πka), this reduces to Newton’s law: F = Gm1m2/r². Let us verify with actual numbers.
Example 1: Two Hydrogen Atoms, 1 m Apart
Step 1: Energy of one hydrogen atom
E = mHc² = (1.673 × 10−27) × (2.998 × 108)²
E = 1.503 × 10−10 J
Step 2: Lattice force formula
F = 2 × (1.503 × 10−10)² / (π × 7.708 × 1043 × 1²)
F = 2 × 2.259 × 10−20 / 2.421 × 1044
F = 4.518 × 10−20 / 2.421 × 1044
F = 1.866 × 10−64 N
Step 3: Compare with Newton
F = 6.674 × 10−11 × 2.799 × 10−54
F = 1.868 × 10−64 N ✓
Example 2: Two 1 kg Masses, 1 m Apart
Step 1: Energy of 1 kg
Step 2: Lattice force formula
F = 2 × (8.988 × 1016)² / (π × 7.708 × 1043 × 1²)
F = 2 × 8.078 × 1033 / 2.421 × 1044
F = 1.616 × 1034 / 2.421 × 1044
F = 6.674 × 10−11 N
Step 3: Compare with Newton
F = 6.674 × 10−11 N ✓
Verification Complete
The lattice force formula F = 2E1E2/(πkar²) reproduces Newton’s gravitational law exactly for any two masses at any separation. This is not a fit — it is an algebraic identity: substituting E = mc² and G = 2c4/(πka) gives F = Gm1m2/r² automatically.
7. Why Gravity Is Weak (The Hierarchy Problem Solved)
The “hierarchy problem” asks why gravity is roughly 1036 times weaker than electromagnetism. In the lattice picture, the answer is immediate: standing waves barely dent the incredibly stiff medium.
Define the disruption ratio as the wave energy divided by the lattice bond force:
When δ « 1, the wave is a tiny perturbation on an enormously rigid background. Gravity — a second-order effect (overlap of two tiny compressions) — is proportional to δ². This is why it is so weak.
Hydrogen Atom
ka = 7.708 × 1043 N
δ = E / (ka) = 1.503 × 10−10 / 7.708 × 1043
δ = 1.95 × 10−54
A hydrogen atom displaces the lattice by roughly one part in 1054. Almost nothing.
1 Kilogram Mass
ka = 7.708 × 1043 N
δ = E / (ka) = 8.988 × 1016 / 7.708 × 1043
δ = 1.17 × 10−27
Even a kilogram — an enormous collection of standing waves — barely registers on the lattice.
The Earth
E = Mc² = 5.37 × 1041 J
δ = E / (ka) = 5.37 × 1041 / 7.708 × 1043
δ ≈ 10−2
The Earth compresses the lattice by about 1%. Still a small perturbation, but enough that general relativistic corrections become measurable (GPS satellites need them).
The Sun
E = Mc² = 1.79 × 1047 J
δ = E / (ka) = 1.79 × 1047 / 7.708 × 1043
δ ≈ 103
The Sun’s total mass-energy exceeds the bond force of a single node — but spread across ~1057 atoms, each individual node is still barely perturbed. The collective effect, however, bends starlight and holds planets in orbit.
The Hierarchy Problem: Solved
Gravity is weak because it is a second-order pressure-deficit effect in an extraordinarily stiff medium. The lattice bond stiffness k ≈ 1078 N/m dwarfs any particle energy by dozens of orders of magnitude. Two standing waves must both compress the lattice and then have their compressions overlap to produce a gravitational force.
In equation form:
The proton is not “too light” — the lattice is too stiff. There is no hierarchy “problem.” There is only a very stiff medium and a very small wave.
Summary
Three lattice constants → three constants of nature. Every intermediate number shown above, every digit, follows from {k, a, η} with zero free parameters.
| Equation | Formula | GWT Value | Observed | Error |
|---|---|---|---|---|
| Speed of light | c = a√(k/η) | 2.998 × 108 m/s | 2.998 × 108 m/s | exact |
| Planck’s constant | h = π²ka³/c | 6.626 × 10−34 J·s | 6.626 × 10−34 J·s | exact |
| Gravitational constant | G = 2c4/(πka) | 6.674 × 10−11 N·m²/kg² | 6.674 × 10−11 N·m²/kg² | exact |
What This Means
The speed of light, Planck’s constant, and Newton’s gravitational constant are not fundamental inputs to physics. They are mechanical outputs of a 3D elastic lattice with stiffness k, spacing a, and inertial response η.
In Planck units, all three reduce to trivial arithmetic: c = 1, ℏ = 1, G = 1. The lattice has zero free parameters. The constants of nature were never a choice — they are geometry.
Every other result in GWT — the fine structure constant, particle masses, dark energy, the strong force — is built on top of these three equations. This page is the foundation. The full derivation chain starts here.