Elliptic Curves and the Perfect Cuboid: Rank-Zero Obstructions and Heron Triple Families — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The perfect cuboid problem reduces to rational points on elliptic curves, whose rank-zero Diophantine obstructions remain unresolved; related research explores curve maps over finite fields and rank≥5 elliptic curve families via Heron triples. | MATH: Perfect cuboid ⇔ rational solutions to a2+b2=c2, a2+d2=e2, b2+d2=f2, c2+d2=g2 (8 variables, 4 quadratic constraints). Elliptic curve E: y2=x3+Ax+B — rank r determines rational point structure; rank zero ⇒ finite rational points (Mordell–Weil). Diophantine stability (Mazur): E(ℚ) unchanged under base change. Heron triple family: A2,B2,C2 with area Δ yields elliptic curve y2=x(x−A2)(x+B2) — rank≥5 constructed via parameterization. | CONNECTION: No direct golden-ratio or base-60 link in these abstracts. However, elliptic curves over finite fields carry lattice structure (Tate module, ℤℓ-rank 2) — crystallographic analogy: root lattice E8 and ellip
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026elliptic,
title = {{Elliptic Curves and the Perfect Cuboid: Rank-Zero Obstructions and Heron Triple Families — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635527},
url = {https://zenodo.org/records/23635527},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Elliptic Curves and the Perfect Cuboid: Rank-Zero Obstructions and Heron Triple Families — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635527
UR - https://zenodo.org/records/23635527
AB - FINDING: The perfect cuboid problem reduces to rational points on elliptic curves, whose rank-zero Diophantine obstructions remain unresolved; related research explores curve maps over finite fields and rank≥5 elliptic curve families via Heron triples. | MATH: Perfect cuboid ⇔ rational solutions to a^2+b^2=c^2, a^2+d^2=e^2, b^2+d^2=f^2, c^2+d^2=g^2 (8 variables, 4 quadratic constraints). Elliptic curve E: y^2=x^3+Ax+B — rank r determines rational point structure; rank zero ⇒ finite rational points (Mordell–Weil). Diophantine stability (Mazur): E(ℚ) unchanged under base change. Heron triple family: A^2,B^2,C^2 with area Δ yields elliptic curve y^2=x(x−A^2)(x+B^2) — rank≥5 constructed via parameterization. | CONNECTION: No direct golden-ratio or base-60 link in these abstracts. However, elliptic curves over finite fields carry lattice structure (Tate module, ℤ_ℓ-rank 2) — crystallographic analogy: root lattice E_8 and ellip Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -