Geometric Encoded Medium (GEM) Framework

A Geometric Encoded Medium (GEM) Impedance Framework for Physics in Rust

"The universe is a perfect geometric circuit."

🌌 Overview

The GEM Framework is a Rust library that models reality not as a collection of arbitrary forces and constants, but as a single Geometric Encoded Medium. It posits that space itself has impedance ($Z_p$) and geometry (Horn Torus topology), and that all physical phenomena—Gravity, Electromagnetism, Mass, and Charge—are simply different "encodings" running on this geometric hardware.

This library does not approximate physics; it derives it. By defining a few geometric axioms, it mathematically derives Newton's Gravitational Constant ($G$), the Fine Structure Constant ($\alpha$), and the Proton Radius from first principles. Validations match CODATA with high precision (e.g., α rel error -5.47e-10, G rel error 3.86e-6).

For a visual of the horn torus topology:

🧠 Core Philosophy

In GEM, the universe is treated as a software system:

The Hardware: A "Horn Torus" topology representing the vacuum medium.
The Operating System: The Impedance Field ($Z_0, Z_p$) that dictates how information moves.
The Software:
- Gravity: The interaction when the medium is encoded with Mass (Shadow Charge $Q = \Xi M$).
- Electromagnetism: The interaction when the medium is encoded with Electric Charge ($q$).

Complex phases handle extreme scales (e.g., Planck or black holes), rotating real forces into imaginary (rotational/spin) components.

Quick Start

Clone and build:

git clone https://github.com/troydeville/gemphy.git
cd gemphy
cargo test  # Run validations
cargo doc --open  # View API docs

Install via Cargo:

[dependencies]
gemphy = "0.1.0"

Basic usage:

use gemphy::GeometricEncodedMedium;

fn main() {
    let medium = GeometricEncodedMedium::new();
    println!("Derived α: {:.4e}", medium.alpha);
    println!("Derived G: {:.4e} m³ kg⁻¹ s⁻²", medium.g);
}

Roadmap

Milestone	Tasks
v0.3	Add horn torus simulation; publish crate
v0.4	Interactive CLI; Python bindings
v1.0	Full predictions (e.g., dark matter as impedance)

📦 Installation

Add to Cargo.toml:

[dependencies]
gemphy = "0.1.0"

Note: Re-exports Complex64 from num-complex.

🚀 Usage Guide

1. Initialize the Medium

use gemphy::GeometricEncodedMedium;

fn main() {
    let medium = GeometricEncodedMedium::new();
    println!("Vacuum Impedance (Z0): {:.4} Ohms", medium.z_o);
    println!("Fine Structure (α):    {:.4e}", medium.alpha);
    println!("Gravitational G:       {:.4e} m³ kg⁻¹ s⁻²", medium.g);
}

2. Create Geometric Knots

use gemphy::{GeometricEncodedMedium, GeometricKnot};
use num_complex::Complex64;

fn main() {
    let medium = GeometricEncodedMedium::new();
    let electron = GeometricKnot::new(medium.clone(), 9.109e-31, Complex64::new(-1.602e-19, 0.0), 0.0, "Electron");
    let earth = GeometricKnot::new(medium, 5.972e24, Complex64::new(0.0, 0.0), 6.371e6, "Earth");

    println!("--- Particle Topology ---");
    // electron.print_analysis();  // Add this method if needed
    // earth.print_analysis();
}

Output Note: Electron sizes to Bohr Radius; Earth to Schwarzschild Radius.

3. Constants & Model Specification

I. Fundamental Physical Constants

These are the fixed, integer-defined values representing the bedrock of the SI system used in the model.

Planck Constant ($h$): The quantum of action. $$h = \frac{662607015}{10^{42}} \text{ J/Hz}$$
Speed of Light ($c$): The universal speed limit. $$c = 299792458 \text{ m/s}$$
Elementary Charge ($e$): The fundamental unit of charge. $$e = \frac{1602176634}{10^{28}} \text{ C}$$

II. Scaling & Geometric Factors

These parameters act as scaling bridges between the quantum/electromagnetic scale and the gravitational scale.

Magnetic Scaling Constant ($\Phi$): A scaling factor with dimensions of inductance per meter. $$\Phi = \frac{1}{10^7} \text{ H/m}$$
Mass-Charge Metric ($\phi$): A density-like scalar connecting mass and charge squared. $$\phi = 10^4 \text{ kg}^{2 \text{ m}}{-2} \text{ s}^{2 \text{ C}}{-2}$$
Alpha-Lambda Function ($\Lambda_\alpha(n)$): A rational function governing specific integer-ratio field interactions. $$\Lambda_\alpha(n) = \frac{\alpha_\delta n}{\alpha_\gamma + 2 \alpha_\delta n^2}$$

III. Primary Field Parameters (Planck Scale)

These variables (-subscript) represent the theoretical "maximum" impedance and field density before fine-structure scaling is applied.

Primary Impedance ($Z_p$): $$Z_p = \frac{2h}{e^2}$$
Primary Fine Structure ($\alpha_p $): The geometric inverse of impedance. $$\alpha_p = \frac{4\pi c}{Z_p}$$
Primary Field Constants: Permeability ($\mu_p$), Permittivity ($\epsilon_p$), and the Gamma factor ($\Gamma_p$). $$\mu_p = \frac{Z_p}{c}$$ $$\epsilon_p = \frac{1}{c Z_p}$$ $$\Gamma_p = \frac{e^2}{\alpha_p}$$

IV. Vacuum Field Parameters (Observable)

These are the standard observable vacuum constants, derived by scaling the primary parameters by the fine structure constant ($).

Fine Structure Constant ($\alpha $): $$\alpha = \alpha_p \Phi$$
Vacuum Impedance ($Z_0$): $$Z_0 = \alpha Z_p$$
Vacuum Permeability & Permittivity ($\mu_0$, $\epsilon_0$): $$\mu_0 = \frac{Z_0}{c}$$ $$\epsilon_0 = \frac{1}{c Z_0}$$
Vacuum Gamma ($\Gamma$): $$\Gamma = \alpha \Gamma_p$$

V. Gravitational Unification

The model derives the Gravitational Constant ($G$) not as a fundamental arbitrary value, but as a result of geometric impedance scaling.

Gravitational Constant ($G$):

$$ G = \frac{Z_0}{c S \phi} [\frac{m^3}{kg s^2}] $$

Complex Unification Factor ($\Xi$): A complex rotation relating field geometry to mass-charge equivalence.

$$ \Xi = \sqrt{4\pi \sqrt{2} G \epsilon_0} \left( \cos\frac{\pi}{8} - i \sin\frac{\pi}{8} \right) [C/kg] $$

VI. Mass & Length Scales

Definitions of the Planck scale and the specific derivation of the Proton Mass.

Planck Units ($m_P$, $l_P$): $$m_P = \sqrt{\frac{ch}{2\pi G}}$$ $$l_P = \sqrt{\frac{hG}{2\pi c^3}}$$
Proton Mass ($m_p$): A derivation scaling the Planck mass by exponential and geometric corrections. $$m_p = e^{{-14\pi} \left( 1 - \frac{1}{2}{11}-4} \right) m_P \left( 1 - \frac{4 \alpha_\gamma}{\alpha_\delta} \right)$$
Mass-Charge Equivalence ($Q$): A unified charge definition based on mass and the complex factor . $$Q = M \Xi$$

To visualize force scaling (Newtonian vs. GEM-derived, log-log plot from Sun-Earth system):

GEM

4. Interaction Examples

cargo test -- --nocapture

Result

running 9 tests
--- THE DECODER TEST ---
Gravity Force: 2.5686e-47-2.5686e-47i
Coulomb Force: 8.2424e-8+0.0000e0i
--- GEM HYDROGEN DECODER ---
--- GEM MASS HARMONIC SCANNER ---
--- GEM HYDROGEN INTERACTION ANALYSIS ---
1. Electron Field Energy:   13.60569 eV (Source Potential)
2. Proton Coupling Energy:  0.00741 eV (Interaction Term)
3. Net Observable Energy:   13.59828 eV (Measured State)
   Target:                  13.5983 eV
State n=1: Radius = 5.2947e-11 m, Energy = 13.5983 eV
State n=2: Radius = 2.1179e-10 m, Energy = 3.3996 eV
--- RECOIL DERIVATION (Mathematica In[204-208]) ---
Term 1 (Ideal Me):    13.60569 eV (Target: 13.6057)
Term 2 (Recoil Me/Mp):0.00741 eV (Target: 0.0074)
Difference:           13.59828 eV (Target: 13.5983)
>> HARMONIC 48 (Proton Candidate): Ratio = 1836.1318
State n=3: Radius = 4.7652e-10 m, Energy = 1.5109 eV
--- GEM COMPLEX ENERGY ANALYSIS ---
Real Part:      9.61544 eV
Imaginary Part: 9.61544 eV
Magnitude:      13.59829 eV
Target:         13.5983 eV
Derived Proton/Electron Ratio: 1836.1318
test tests::verify_derivations_and_constants ... ok
test tests::test_dynamic_system_orbit ... ok
test tests::test_decoder_philosophy ... ok
test tests::verify_hydrogen_component_interaction ... ok
test tests::verify_hydrogen_recoil_derivation ... ok
test tests::test_mass_resonance_integers ... ok
test tests::test_hydrogen_resonance_decoder ... ok
test tests::verify_complex_energy_components ... ok
test tests::verify_mass_ratio_derivation ... ok

cargo run --bin electron_proton_action

electron: GeometricKnot {
    name: "Electron",
    mass: 9.1093837139e-31,
    charge: Complex {
        re: -1.602176634e-19,
        im: -1.602176634e-19,
    },
    geometric_radius_a: 5.291772104738458e-11,
    physical_radius: 2.817940320834912e-15,
}
proton: GeometricKnot {
    name: "Proton",
    mass: 1.67262192595e-27,
    charge: Complex {
        re: 1.602176634e-19,
        im: 1.602176634e-19,
    },
    geometric_radius_a: 2.8819891620872986e-14,
    physical_radius: 1.5346982642700954e-18,
}
Result: GemInteractionResult {
    q1: Complex {
        re: 8.624716672621998e-41,
        im: -3.5724746174253846e-41,
    },
    q2: Complex {
        re: 1.5836296575938094e-37,
        im: -6.559608819516165e-38,
    },
    q_total: Complex {
        re: 1.5844921292610717e-37,
        im: -6.56318129413359e-38,
    },
    af1: Complex {
        re: 3.3694829747370465e-13,
        im: -3.3694829747370465e-13,
    },
    af2: Complex {
        re: 6.186885172111745e-10,
        im: -6.186885172111745e-10,
    },
    g1: Complex {
        re: 1.5352496334655168e-20,
        im: -1.5352496334655168e-20,
    },
    g2: Complex {
        re: 2.818952718857127e-17,
        im: -2.818952718857127e-17,
    },
    force: Complex {
        re: 2.567892198741124e-47,
        im: -2.567892198741124e-47,
    },
    curvature: Complex {
        re: 1.0000000000000002,
        im: 0.0,
    },
    g_o: Complex {
        re: 6.674325731836407e-11,
        im: 0.0,
    },
    g_recovered: Complex {
        re: 6.674325731836409e-11,
        im: 0.0,
    },
    mqr1_ev: Complex {
        re: 469136044.7141288,
        im: -469136044.7141288,
    },
    mqr2_ev: Complex {
        re: 255499.47534587674,
        im: -255499.47534587674,
    },
    ratio1: Complex {
        re: 0.9994553833012333,
        im: -0.0005440241291649849,
    },
    ratio2: Complex {
        re: 0.0002723084703826605,
        im: -0.00027216024727941913,
    },
    binding_energy_ev: Complex {
        re: 9.62067644577312,
        im: 9.62067644577312,
    },
    schwarzschild_radius: 2.485594236043889e-54,
    ratio_mr_d: 4.697091670283908e-44,
    is_complex: false,
}
(result.q1 / m2).abs(): 5.58125294864566e-14
(result.q2 / m1).abs(): 1.8816952313861778e-7
 ((result.q2 / m1).abs()/(result.q1 / m2).abs()).sqrt(): 1836.1526734215263
Abs[binding_energy_ev] = 13.60569110881573

A. Microscopic Interaction (Electron-Proton)

At the quantum scale, the medium handles interactions via the Coulomb protocol (Electric Charge) and the Gravity protocol (Mass/Shadow Charge) simultaneously.

use gemphy::{GeometricEncodedMedium, GeometricKnot, ForceProtocol};
use num_complex::Complex64;

fn main() {
    let medium = GeometricEncodedMedium::new();
    let q1 = Complex64::new(-medium.e, -medium.e);
    let q2 = Complex64::new(medium.e, medium.e);
    let electron = GeometricKnot::new(medium.clone(), 9.109e-31, q1, 0.0, "Electron");
    let proton = GeometricKnot::new(medium.clone(), 1.672e-27, q2, 0.0, "Proton");
    let d = 5.29e-11; // Bohr Radius

    // Decode as Electromagnetism (Strong Force)
    let f_coulomb = medium.decode_force(&electron, &proton, d, ForceProtocol::Electromagnetism);
    
    // Decode as Gravity (Weak Force)
    let f_gravity = medium.decode_force(&electron, &proton, d, ForceProtocol::Gravity);

    println!("--- Microscopic Interaction ---");
    println!("Coulomb Force: {:.4e} N", f_coulomb); // ~ 8.2e-8 N
    println!("Gravity Force: {:.4e} N", f_gravity); // ~ 3.6e-47 N
}

B. Macroscopic Interaction (Earth-Sun)

At the cosmic scale, the Gravity protocol dominates.

use gemphy::{GeometricEncodedMedium, GeometricKnot, ForceProtocol};

fn main() {
    let medium = GeometricEncodedMedium::new();
    let earth = GeometricKnot::new(medium.clone(), 5.972e24, 5.972e24 * medium.xi, 6.371e6, "Earth");
    let sun = GeometricKnot::new(medium.clone(), 1.989e30, 1.989e30 * medium.xi, 6.96e8, "Sun");
    let d = 149.6e9; // 1 AU

    let f_gravity = medium.decode_force(&earth, &sun, d, ForceProtocol::Gravity);

    println!("--- Macroscopic Interaction ---");
    println!("Gravitational Force: {:.4e} N", f_gravity); // ~ 3.5e22 N
}

5. The Unified Force (Complex Phase Engine)

What happens inside a Black Hole or at the Planck Scale? Standard physics breaks down. GEM handles this by rotating the force vector into the Imaginary Plane.

Real Force: Linear Acceleration (Push/Pull).
Imaginary Force: Rotational Action (Spin/Memory).

use gemphy::{GeometricEncodedMedium, GeometricKnot};

fn main() {
    let medium = GeometricEncodedMedium::new();
    let mp = medium.m_p; // Planck Mass
    
    // Create two particles at the Planck Scale
    let p1 = GeometricKnot::new(medium.clone(), mp, mp * medium.xi, 0.0, "Planck A");
    let p2 = GeometricKnot::new(medium.clone(), mp, mp * medium.xi, 0.0, "Planck B");
    
    // Distance inside the Schwarzschild radius
    let d = 0.5 * medium.l_p; 

    let f_total = medium.calculate_total_force(&p1, &p2, d);

    println!("--- Unified Force (Planck Scale) ---");
    println!("Force Vector: {:.4e} + {:.4e}i N", f_total.re, f_total.im);
    
    if f_total.im.abs() > 0.0 {
        println!("Status: Phase Transition (Information Storage / Spin)");
    }
}

🧪 Testing the Laws of Physics

This library includes a rigorous test suite that validates the framework against CODATA observations.

Run the verification suite:

cargo test

What is tested?

The Golden Loop: Verifies $\frac{h}{2\pi c} \equiv \frac{\Gamma}{\alpha}$.
- Grand Unification: Verifies the geometric scaling from Electrons to the Universe.
- Force Parity: Checks that GEM-derived Gravity matches Newtonian Gravity to $10^{-13}$ precision.
- Phase Change: Ensures forces transition to complex numbers correctly inside event horizons.

Contributing

See CONTRIBUTING.md for details on issues, PRs, and development setup.

⚖️ License

This project is licensed under GPL-3.0.

gemphy 0.1.0