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Proof of the Non-existence of Perfect Cuboids via Mordell-Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.AG — Algebraic Geometrymath.LO — Logic

Contributed by Jonathan 𝑓(n) Reed ↗

Version 1 / Oct 07, 2026 / CC BY 4.0

Abstract

This manuscript establishes the non-existence of the Perfect Cuboid---a rectangular parallelepiped with integer edges, face diagonals, and space diagonal. By performing a rational sectioning of the governing quadratic forms, we demonstrate that the problem reduces to finding a non-trivial rational point on a family of hyperelliptic curves of Genus 3. We prove that the Jacobian of these curves possesses a Mordell-Weil rank of zero and that the perfection locus is an irrational algebraic singularity of degree d=4d = 4, precluding any solution in the integer domain Z3\mathbb{Z}^3. The non-existence of rational solutions is further verified via formal methods in Lean 4, demonstrating that the intersection of the Mordell-Weil torsion set and the degree-4 perfection locus is empty.

Provenance statement

The author acknowledges the assistance of a large language model, Gemini, for its role as a formalization and editing tool in thepreparation of this manuscript. The AI was used under the direct control of the author. All intellectual and creative decisions, as well as final editorial responsibility, rest with the author.

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References

  1. Cassels, J. W. S. Arithmetic on Curves of Genus 1. IV. Proof of the Hauptvermutung. Journal für die reine und angewandte Mathematik, vol. 211, pp. 95–112. 1962
  2. Leonardo de Moura; Sebastian Ullrich. The Lean 4 Programming Language and Theorem Prover. International Conference on Automated Deduction, pp. 625–635. 2021
  3. Faltings, Gerd. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Inventiones Mathematicae, vol. 73, pp. 349–366. 1983
  4. Lenstra, Arjen K.; Lenstra, Hendrik W.; Lovász, László. Factoring polynomials with rational coefficients. Mathematische Annalen, vol. 261, pp. 515–534. 1982
  5. Mordell, L. J. On the rational solutions of the indeterminate equations of the third and fourth degrees. Proceedings of the Cambridge Philosophical Society, vol. 21, pp. 179–192. 1922
  6. Nagell, Trygve. Solution de quelques problèmes dans la théorie arithmétique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo. 1937
  7. Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer. 2009
  8. Weil, A. L'arithmétique sur les courbes algébriques. Acta Mathematica, vol. 52, no. 1, pp. 281–315. 1929DOI

Version history

  1. v1Submitted by Jonathan 𝑓(n) ReedInitial depositCurrentOct 07, 2026