GEMSTone (Cradian Form)

(GEMSTone means Geometry-Energy-Mass-Space-Time is One interrelated Set of Ratios)

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GEMSTone := The invariant ratio of Energy to Mass expressed as velocity–geometry coupling in Cradian space:

E/m = dx/dt x C/2D

Meaning:
GEMSTone encodes the unified structure of Energy–Mass–Space–Time as a single geometric interaction. It characterizes all waveform‑structures propagating at v=c, where:

• dx/dt is the interaction rate; c=dx/dt is an invariant ratio.

• C/2D is the Cradian geometric ratio: pi/2 Radians.

• c = pi/2 in Cradian units - an invariant ratio; pi = C/D is a geometric invariant ratio.

• E/m = pi^2/2^2 = pi^2/4 is an invariant ratio.

This is the static invariant of the universe — the “resting geometry” of light-like propagation.

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GEMLight (Cradian Phase Invariant)

Complex curves of the general form:

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GEMLight := The dispersive phase invariant of waveform propagation in 3D, expressed as:

phi = 2(E/m) / (dx/dt)

Meaning:
GEMLight describes the propagating form of the same invariant ratio. It governs:

• inverse‑square dispersion

• phase rotation

• waveform interaction

• coherence propagation

GEMLight is the dynamic invariant — the “moving geometry” of light-like interaction.

Derivation:

(E/m)/(dx/dt) = pi/2 → pi = (2*E/m)/(dx/dt)

From the Euler Identity, substituting for pi, in Cradians exp((i*2*E/m)/(dx/dt)) = -1.

Unified Interpretation

Together, GEMSTone and GEMLight encode the entire observable universe of:

• Energy

• Mass

• Space

• Time

• Geometry

• Waveform propagation

• Waveform interaction

 

They are two projections of the same underlying invariant:

• GEMSTone = static ratio (structure)

• GEMLight = dynamic ratio (phase)

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Both collapse into the Cradian identity:

v = dx/dt = c = pi/2 = C/2D

This is the closure:
velocity, geometry, and light-like propagation are the same invariant expressed differently.

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Subluminal Case (Unified with Cradians)

For v < c:

KE/m = v^2/2

Substituting the Cradian form:

KE/m = dx/2dt x C/2D

and equivalently, where v=dx/dt and pi=C/D:

KE/m = v/2 x pi/2

This unifies:

• kinetic energy

• mass

• space

• time

• geometry

…into the same Cradian structure that governs v=c.

We now have all velocities — subluminal and luminal — expressed in one coherent system.