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.

Roadmap

📦 Installation

cargo add gemphy

Install via Cargo:

[dependencies]
gemphy = "0.2.1"

Note: Re-exports Complex64 from num-complex.

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

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 . $$Q = M \Xi$$

4. Interaction Examples

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).

Kappa ($\kappa$) in GEM is similar to Einstein's curvature in general relativity, coupling mass to geometry via the complex unification factor $\Xi$. It derives as $\kappa = \Xi \frac{M + m}{Q_{sh} + q_{sh}}$, where $Q_{sh} = \Xi M$ (shadow charge), simplifying to a phased unit complex in gravity-dominated regimes.

Contributing

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

⚖️ License

This project is licensed under GPL-3.0.