Mathematical Foundation

Adelic is an advanced mathematical framework focused on prime-based adelic products and their intrinsic relationships, designed to function without physical units or arbitrary constants. This repository provides a full implementation of this approach, ensuring all computations remain purely mathematical.

Overview

The core principle of this system is its strict adherence to dimensionlessness. Normalization is derived entirely from prime-based adelic products, eliminating arbitrary constants and maintaining mathematical clarity.

Ensuring Dimensionlessness

Several checks confirm the framework’s dimensionless nature:

1. Prime Factors: Each prime number p is treated as a pure integer, ensuring that its reciprocal, 1/p, remains dimensionless.

2. Real Factor: The following product maintains dimensionlessness:

   Π_p (p / (p-1))

   Each term in this product is a ratio of integers, preserving dimensionlessness.

3. Normalization Factor (dx): Defined as:

   dx = (1 / (Π_p (p / (p-1)) × Π_p (1/p)))^(1/4)

   Since all components are dimensionless, dx remains dimensionless.

Core Mathematical Structure

The adelic integration system follows:

Λ = Re(dx) × Π_p (p / (p-1)) × Π_p (1/p)

Key Components

• Real Factor: Π_p (1 - 1/p)^(-1) (Euler-type product)
• p-adic Factor: Π_p (1/p) (Prime reciprocals)
• Normalization: dx^4 = (Π_p (1 / (p(p-1))))^(-1)

Verification

The computed value Λ = 1.0 confirms internal consistency and supports the dimensionless framework. This implementation avoids arbitrary constants, ensuring a solid mathematical foundation.

Key Results

The Adelic system demonstrates that fundamental mathematical relationships can be expressed using dimensionless structures based on the natural distribution of prime numbers.

• Zenodo Record 14970879: "Recursive Expansive Dynamics: A New Framework for Cosmological Modeling"

• Zenodo Record 14969006: "Fractal Structures in Quantum Field Theory: Implications and Applications"

• Zenodo Record 14949122: "Adelic Analysis in Modern Physics: Bridging p-adic and Real Fields"

• Zenodo Record 14994094: "Golden Ratio Modulated Recursion in Dynamical Systems"

• Zenodo Record 14948910: "Persistent Homology and Spectral Zeta Functions in Topological Data Analysis"

• Zenodo Record 15022520: "Recursive Causal Structures in General Relativity"

• Zenodo Record 15022735: "Entropy Scaling in Chaotic and Complex Systems"

• Zenodo Record 15033667: "Recursive Green’s Functions and Quantum Period Deformations"

• GitHub Repository: Adelic

• Discussion: Validation of Recursive Expansive Dynamics (REDS), CMB, Gravitational Wave Echoes, and Galactic Rotation Curves