The Global Odd Tower: A Unified Structural Framework for Odd Integers, Prime Geometry, Twin Symmetry, Collatz Descent, and Finite-Cell Factorization
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2025
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| _version_ | 1866901187456925696 |
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| author | Matsuua, Yoshihito |
| author_facet | Matsuua, Yoshihito |
| contents | <p>Abstract.</p> <p>We introduce the Global Odd Tower (GOT), a unified layered architecture governing the structure of all odd integers.</p> <p>The tower links four independent phenomena that have traditionally been studied in isolation:</p> <p> </p> <ul> <li>the odd Collatz descent,</li> <li>the ±2 twin-symmetry branching structure,</li> <li>the Hierarchical Symmetric Twin Generator (HSTG), and</li> <li>the Prime Structural Map (PSM), which assigns structural coordinates (H(p),L(p)) to every prime.</li> </ul> <p> </p> <p> </p> <p>These components form a single geometric spine that organizes primes, composites, twin primes, smoothness patterns, and factorization behavior.</p> <p>The GOT framework explains why composite integers occupy lateral “accessory layers,”</p> <p>why prime gaps follow discrete twin-based hierarchies,</p> <p>and why the structural profile \Phi(N) = (H(N),L(N)) enables finite-cell factorization.</p> <p> </p> <p>This preprint provides the global architecture, its mathematical justification,</p> <p>and its consequences for prime distribution, twin primes, the Collatz hierarchy,</p> <p>and finite-depth structural analysis of odd integers.</p> <p>Description.</p> <p>This work presents the Global Odd Tower (GOT), a unified geometric structure for odd integers that integrates Collatz descent, twin symmetry, prime geometry, and finite-cell factorization.</p> <p>The GOT highlights a vertical prime spine surrounded by accessory composite layers, explains the local ±2 twin-branching mechanism, and unifies previously separate mathematical structures into a single finite-depth hierarchy.</p> <p>This preprint consolidates results from the MSHD–HSTG program, offering a global perspective linking primes, composites, twin primes, descent chains, and structural factorization.</p> <p>A full diagrammatic overview (TikZ) is included.</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17828968 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | The Global Odd Tower: A Unified Structural Framework for Odd Integers, Prime Geometry, Twin Symmetry, Collatz Descent, and Finite-Cell Factorization Matsuua, Yoshihito Collatz odd integers MSHD-HSTG prime distribution twin primes factorization structural hierarchy finite descent Prime Structural Map Global Odd Tower <p>Abstract.</p> <p>We introduce the Global Odd Tower (GOT), a unified layered architecture governing the structure of all odd integers.</p> <p>The tower links four independent phenomena that have traditionally been studied in isolation:</p> <p> </p> <ul> <li>the odd Collatz descent,</li> <li>the ±2 twin-symmetry branching structure,</li> <li>the Hierarchical Symmetric Twin Generator (HSTG), and</li> <li>the Prime Structural Map (PSM), which assigns structural coordinates (H(p),L(p)) to every prime.</li> </ul> <p> </p> <p> </p> <p>These components form a single geometric spine that organizes primes, composites, twin primes, smoothness patterns, and factorization behavior.</p> <p>The GOT framework explains why composite integers occupy lateral “accessory layers,”</p> <p>why prime gaps follow discrete twin-based hierarchies,</p> <p>and why the structural profile \Phi(N) = (H(N),L(N)) enables finite-cell factorization.</p> <p> </p> <p>This preprint provides the global architecture, its mathematical justification,</p> <p>and its consequences for prime distribution, twin primes, the Collatz hierarchy,</p> <p>and finite-depth structural analysis of odd integers.</p> <p>Description.</p> <p>This work presents the Global Odd Tower (GOT), a unified geometric structure for odd integers that integrates Collatz descent, twin symmetry, prime geometry, and finite-cell factorization.</p> <p>The GOT highlights a vertical prime spine surrounded by accessory composite layers, explains the local ±2 twin-branching mechanism, and unifies previously separate mathematical structures into a single finite-depth hierarchy.</p> <p>This preprint consolidates results from the MSHD–HSTG program, offering a global perspective linking primes, composites, twin primes, descent chains, and structural factorization.</p> <p>A full diagrammatic overview (TikZ) is included.</p> <p> </p> |
| title | The Global Odd Tower: A Unified Structural Framework for Odd Integers, Prime Geometry, Twin Symmetry, Collatz Descent, and Finite-Cell Factorization |
| topic | Collatz odd integers MSHD-HSTG prime distribution twin primes factorization structural hierarchy finite descent Prime Structural Map Global Odd Tower |
| url | https://doi.org/10.5281/zenodo.17828968 |