Geometric Encoded Medium (GEM) Physics Framework

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🌌 Overview

The GEM Framework is a Rust library that models reality as a single Geometric Encoded Medium. It posits that space-time is a medium with intrinsic impedance ($Z_p$) and a Horn Torus topology. All physical phenomena—Gravity, Electromagnetism, Mass, and Charge—are derived as specific "encodings" on this geometric hardware.

Unlike traditional physics engines that rely on curve-fitting or arbitrary constants, GEM derives fundamental values like Newton's Gravitational Constant ($G$), the Fine Structure Constant ($\alpha$), and the Proton Radius from first-principles geometric axioms.

🧠 Core Philosophy: The Unified Phase Engine

In GEM, the universe operates as a geometric circuit:

• The Hardware: A Horn Torus manifold ($R = r$) where the singularity at the center acts as a topological pump.
• The Phase: Interactions are not just magnitudes; they are complex vectors. Extreme scales (Planck, Black Holes, Muonic states) rotate real linear forces into Imaginary/Rotational components.
• The Whirl: Particle "Spin" is visualized as a Traveling Wave of energy density orbiting the singularity, rather than a point-particle rotating in a void.

Roadmap

Milestone	Tasks
v0.3	Add horn torus simulation; enhanced orbit sims (hydrogen, muonic, neutron star)
v0.4	Interactive CLI; Python bindings; initial multi-body support
v0.5	gemphy-web deployment with live visualizations; basic n-body examples (e.g., Sun-Earth-Moon)
v1.0	Full predictions (e.g., dark matter as impedance, novel deviations in fine structure); arXiv preprint with derivations

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🚀 Quick Start

📦 Installation

[dependencies]
gemphy = "0.2.2"

Note: Re-exports Complex64 from num-complex.

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Example Test Usage

View tests

cargo run --bin orbit_sim

cargo run --bin electron_proton_action

cargo run --bin muon_proton_action

cargo run --bin neutron_star_action

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Stable Orbit Simulation

GEMPHY handles the "Dynamic Stability" of orbits by calculating the interaction between geometric knots.

use gemphy::{knot::GeometricKnot, medium::{GAMMA_P, GeometricEncodedMedium}};
use physical_constants::{ELECTRON_MASS, PROTON_MASS, ELEMENTARY_CHARGE};

fn main() {
    let m1 = ELECTRON_MASS;
    let m2 = PROTON_MASS;
    
    let medium = GeometricEncodedMedium::new();

    // Setup Particles as Geometric Knots
    let electron = GeometricKnot::new(medium.clone(), m1, &[-1.0], 0.0, "Electron");
    let proton = GeometricKnot::new(medium.clone(), m2, &[1.0], 0.0, "Proton");

    let rg1 = (GAMMA_P / (electron.mass * medium.alpha)).powi(2);
    let rg2 = (GAMMA_P / (proton.mass * medium.alpha)).powi(2);
    let d = (rg1+rg2).sqrt();

    let interaction = medium.calculate_interaction(&electron, &proton, d.into());
    
    println!("Result:             {:#?}", interaction);
    println!("Er (eV):            {:#?}", interaction.er1.norm()/ ELEMENTARY_CHARGE);
    println!("Ei (eV):            {:#?}", interaction.ei1.norm()/ ELEMENTARY_CHARGE);
    println!("E  (eV):            {:#?}", interaction.binding_energy.norm()/ ELEMENTARY_CHARGE);
}

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📐 The Geometric Model

I. Fundamental Scaling

The framework uses a fundamental geometric normalization constant to bridge the subatomic and cosmic scales:

• Normalization Constant ($S$): $$({4 \pi})^{1/4} \approx 1.8827925275534296$$

• Mass-Charge Metric ($\phi$): $$\phi = 10^4 \text{ kg}^2 \text{ m}^{-2} \text{ s}^2 \text{ C}^{-2}$$

• Magnetic Scaling ($\Phi$): $$\Phi = \frac{1}{10^7} \text{ H/m}$$

• Primary Impedance ($Z_p$): $$Z_p = \frac{2h}{e^2} \Omega$$

Fine Structure ($\alpha$) Scaling Relationships

• Primary Fine Structure ($\alpha_p $): $$\alpha_p = \frac{4\pi c}{Z_p}$$

• Fine Structure ($\alpha $): $$\alpha = \frac{4\pi c}{Z_p} \Phi$$

• Impedance ($Z_p$, $Z_o$): $$\alpha Z_p \implies Z_0$$

• Permeability ($\mu_p $): $$\mu_p = \frac{Z_p}{c} \implies \mu_0 = \alpha \mu_p$$

• Permittivity ($\epsilon_p$) $$\epsilon_p = \frac{1}{c Z_p} \implies \epsilon_0 = \frac{1}{\alpha} \epsilon_p $$

• Gamma factor ($\Gamma_p$) $$\Gamma_p = \frac{e^2}{\alpha_p} \implies \Gamma = \frac{\Gamma_p}{\alpha}$$

II. Gravitational Unification

GEM derives $G$ as a result of geometric impedance scaling rather than an empirical measurement: Where $Z_0$ is the vacuum impedance and $S$ is the geometric shape factor: $$ G = \frac{Z_0}{c S \phi} [\frac{m^3}{kg s^2}] $$

III. Complex Geometry

A complex rotation relating field geometry to mass-charge equivalence. ($\Xi$) $$\Xi = \sqrt{4\pi \sqrt{2} G \epsilon_0} \left( \cos\frac{\pi}{8} - i \sin\frac{\pi}{8} \right)[C/kg]$$

Force is calculated as a complex vector.

Linked Complex Potentials ($E_r$, $E_i$)

Energy interaction in GEM is calculated as the sum of two body-specific complex potentials. Each potential represents the geometric "tension" localized to that body within the medium:

• $E_r$: Complex potential of Body 1 (e.g., the orbiting mass).
• $E_i$: Complex potential of Body 2 (e.g., the central mass).
• Total Interaction Energy ($E_r + E_i$): .

The relationship between $E_r$ and $E_i$ is intrinsically linked by the medium's impedance. As distance or energy density changes, these values rotate in the complex plane, representing the transition from linear work to orbital/spin action.

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🛠 Visualization: The Horn Torus Manifold

The provided HornTorusManifold component (for Three.js/React) is a Raw Scientific Viewer designed to mirror the Rust engine's output:

• Traveling Waves: Total Energy ($E$) is treated as a complex phase. This phase drives a wave that "chases its tail" around the torus, visually representing particle spin and momentum.
• Singularity Flow: The mesh geometry respects the Horn Torus topology ($R=r$), creating a natural "topological pump" at the center that breathes the vacuum.
• No Fakes: All surface deformations are driven by the Complex64 results of the physical interaction. If the engine calculates zero energy, the manifold remains stagnant.

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⚖️ License

Licensed under the MIT License.

Scientific Attribution: If you use GEMPHY in a research paper or commercial simulation, please cite the framework to preserve the geometric integrity of the medium.

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