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| Format: | Recurso digital |
| Jezik: | angleščina |
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Zenodo
2026
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| Online dostop: | https://doi.org/10.5281/zenodo.19577121 |
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- <p>This document explains the mathematical and physical structure of the uploaded dataset.</p> <p>1. Foundational Mathematics</p> <p>Lucas Numbers: A sequence of integers similar to the Fibonacci sequence. The sequence begins with 2 and 1. The seventh number in this sequence is 29.</p> <p>Galois Field (GF): A system in mathematics containing a finite number of elements where you can add, subtract, multiply, and divide. The size of the field is often a prime number raised to a power. In this dataset, the base number is the seventh Lucas number (29). The field size is 29 squared, which equals 841. This creates the field GF(841).</p> <p>Multiplicative Group: If you take all the elements in a Galois Field and remove the zero element, the remaining elements form a group under multiplication. For GF(841), this group has exactly 840 elements.</p> <p>Discrete Logarithm (dlog): A way to index elements in the multiplicative group. Every element can be generated by taking a single starting element (the generator) and raising it to successive powers. The exponent used to reach a specific element is its discrete logarithm.</p> <p>2. Decomposition and Metrics</p> <p>Chinese Remainder Theorem (CRT): A mathematical theorem that allows a large number to be broken down into smaller, independent components based on its prime factors.</p> <p>CRT Components (z3, z5, z7, z8): The multiplicative group order is 840. The number 840 factors cleanly into 3 times 5 times 7 times 8. Using the Chinese Remainder Theorem, every element's discrete logarithm can be mapped to four independent coordinates based on these factors. These coordinates are labeled z3, z5, z7, and z8.</p> <p>Golden Ratio (phi): A specific irrational mathematical constant approximately equal to 1.618.</p> <p>Effective Dimension (D_eff): A calculated metric for each element in the dataset. It is generated by a "Master Equation" that combines a base integer (17), the CRT components, and successive powers of the Golden Ratio. The equation is 17 + (z3 * phi) + (z5 * phi^2) + (z7 * phi^3) + (z8 * phi^4).</p> <p>Keith Sequence: A specific sequence of integers used to categorize states in this model. The relevant numbers in this sequence are 17, 33, 57, and 107. These numbers serve as classification levels for the elements.</p> <p>3. Transformations and Physics</p> <p>Frobenius Automorphism: A mathematical operation specific to Galois Fields. In GF(841), this operation takes an element and raises it to the power of 29.</p> <p>Conjugate Pairs and Self-Conjugate: When the Frobenius Automorphism is applied, it maps an element to a new element. These two elements form a conjugate pair. If the operation maps an element to itself, that element is called self-conjugate.</p> <p>Birch and Swinnerton-Dyer (BSD) Conjecture: A famous unsolved problem in mathematics concerning the arithmetic of elliptic curves. This dataset categorizes elements into "BSD Ranks" and "BSD Forms" (such as Classical or Tamagawa-1) to map the field elements to properties of this conjecture.</p> <p>Quantum Chromodynamics (QCD) Vacuum: A concept from physics describing the lowest energy state of the strong nuclear force. The author of this dataset maps each element in GF(841) to a dual meaning. Each element represents both an arithmetic BSD formula and a physical QCD vacuum mode. The Frobenius Automorphism is mapped to the physical concept of charge conjugation.</p> <p>Strata: A system of layers used to group the elements based on their mathematical and physical properties.</p> <p>4. File Explanations</p> <p>Now that all terms have been defined, the contents of the uploaded files can be explained clearly.</p> <p>Structure</p> <p>This file acts as the theoretical foundation for the dataset. It defines the Galois Field GF(841) based on the seventh Lucas number. It establishes the multiplicative group size of 840. It presents the Chinese Remainder Theorem breakdown, the Master Equation for the Effective Dimension, and the Keith sequence numbers. Finally, it explicitly states the physical meaning of the dataset, connecting the mathematical states to both BSD formulas and the QCD vacuum.</p> <p>GF(841) Dictionary</p> <p>This is the primary catalog of the entire system. It lists all 841 elements. For each element, it provides the discrete logarithm, the algebraic coordinates, the four CRT components, and the calculated Effective Dimension. It also identifies the Keith sequence level, the Frobenius Automorphism target, whether the element is self-conjugate, and its assigned Stratum.</p> <p>27 Coarse BSD</p> <p>This file provides a simplified view of the system. It groups the elements into 27 broad categories. These categories are mapped to specific BSD Forms (Classical, Tamagawa-1, and Mixed). It also shows the base Effective Dimension and Keith sequence level for each of these 27 coarse states.</p> <p>Frobenius Pairs</p> <p>This file focuses entirely on the Frobenius Automorphism. It lists the elements side by side with their calculated conjugate pairs. It details the algebraic coordinates and CRT components for both the starting element and the target element, allowing a direct comparison of their Effective Dimensions.</p> <p>Keith Distribution</p> <p>This file summarizes the entire catalog based on the Keith sequence. It counts how many elements belong to the 17, 33, 57, and 107 levels. It provides the percentage of the whole for each level and assigns a specific BSD Rank and a specific physics concept (such as Particle physics or Nuclear binding) to each tier.</p>