Arithmetic Extermination II–IV: Rigorous Descent Proofs for Fermat's Last Theorem in the Energy-Lattice Framework
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2025
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| author | scott, eron |
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| contents | <p> </p> <p><strong>Arithmetic Extermination II–IV: Rigorous Descent Proofs for Fermat’s Last Theorem in the Energy-Lattice Framework</strong></p> <p>This paper presents <strong>complete and rigorous infinite descent proofs of Fermat’s Last Theorem</strong> for the cases<br>( n = 3 ) and ( n = 4 ), together with a <strong>structural reinterpretation of Kummer’s approach for regular primes</strong>.</p> <p>The proofs for ( n = 3 ) and ( n = 4 ) are <strong>fully explicit and self-contained</strong>, relying only on classical tools:<br>Euler–Eisenstein parametrization, Fermat’s original descent for fourth powers, and elementary properties of coprime factorizations.<br>No modern machinery beyond standard algebraic number theory is required.</p> <p>In addition, the paper introduces the <strong>Order of Integer Solutions (OIS)</strong> framework, an axiomatic lens designed to<br>analyze why certain Diophantine equations are structurally incompatible with the integer lattice.<br>The OIS axioms—well-ordering compatibility, lattice closure, and non-recursive reduction—are used <strong>interpretively</strong>, not as independent proof engines.</p> <p>Within this framework:</p> <ul> <li> <p>The ( n = 3 ) and ( n = 4 ) cases are shown to terminate via explicit infinite descent.</p> </li> <li> <p>Kummer’s theory for regular primes is reformulated as a structural obstruction arising from cyclotomic ideal behavior.</p> </li> <li> <p>High-degree equations (( n \ge 3 )) are interpreted as <strong>energy configurations incompatible with integer lattice stability</strong>.</p> </li> </ul> <p>This work <strong>does not claim a new proof of Fermat’s Last Theorem in full generality</strong>.<br>Rather, it consolidates classical descent proofs and situates them within a unified structural perspective,<br>clarifying why infinite descent is not merely a technique but a consequence of deeper order constraints on integer solutions.</p> <p><strong>Keywords:</strong> Fermat’s Last Theorem, infinite descent, Euler proof, Kummer theory, regular primes, Diophantine equations, integer lattice, axiomatic frameworks</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18012088 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | Arithmetic Extermination II–IV: Rigorous Descent Proofs for Fermat's Last Theorem in the Energy-Lattice Framework scott, eron Fermat's Last Theorem infinite descent, Euler proof Kummer theory <p> </p> <p><strong>Arithmetic Extermination II–IV: Rigorous Descent Proofs for Fermat’s Last Theorem in the Energy-Lattice Framework</strong></p> <p>This paper presents <strong>complete and rigorous infinite descent proofs of Fermat’s Last Theorem</strong> for the cases<br>( n = 3 ) and ( n = 4 ), together with a <strong>structural reinterpretation of Kummer’s approach for regular primes</strong>.</p> <p>The proofs for ( n = 3 ) and ( n = 4 ) are <strong>fully explicit and self-contained</strong>, relying only on classical tools:<br>Euler–Eisenstein parametrization, Fermat’s original descent for fourth powers, and elementary properties of coprime factorizations.<br>No modern machinery beyond standard algebraic number theory is required.</p> <p>In addition, the paper introduces the <strong>Order of Integer Solutions (OIS)</strong> framework, an axiomatic lens designed to<br>analyze why certain Diophantine equations are structurally incompatible with the integer lattice.<br>The OIS axioms—well-ordering compatibility, lattice closure, and non-recursive reduction—are used <strong>interpretively</strong>, not as independent proof engines.</p> <p>Within this framework:</p> <ul> <li> <p>The ( n = 3 ) and ( n = 4 ) cases are shown to terminate via explicit infinite descent.</p> </li> <li> <p>Kummer’s theory for regular primes is reformulated as a structural obstruction arising from cyclotomic ideal behavior.</p> </li> <li> <p>High-degree equations (( n \ge 3 )) are interpreted as <strong>energy configurations incompatible with integer lattice stability</strong>.</p> </li> </ul> <p>This work <strong>does not claim a new proof of Fermat’s Last Theorem in full generality</strong>.<br>Rather, it consolidates classical descent proofs and situates them within a unified structural perspective,<br>clarifying why infinite descent is not merely a technique but a consequence of deeper order constraints on integer solutions.</p> <p><strong>Keywords:</strong> Fermat’s Last Theorem, infinite descent, Euler proof, Kummer theory, regular primes, Diophantine equations, integer lattice, axiomatic frameworks</p> <p> </p> |
| title | Arithmetic Extermination II–IV: Rigorous Descent Proofs for Fermat's Last Theorem in the Energy-Lattice Framework |
| topic | Fermat's Last Theorem infinite descent, Euler proof Kummer theory |
| url | https://doi.org/10.5281/zenodo.18012088 |