Perfect Cuboid Existence via Rational Points on Elliptic Curves — E8 Intelligence Research

FINDING: Perfect cuboid existence is equivalent to a rational point on a specific elliptic curve; the rank-zero case is now provably constrained via irreducibility of the degree-8 cuboid polynomial | MATH: Cuboid eqs: a²+b²=d², a²+c²=e², b²+c²=f², a²+b²+c²=g². Sharipov specialization → Pa,u(t) monic degree-8; proved irreducible in ℤ[t] for u²≠a² (arXiv:2510.11768v3). Diophantine triple {a,b,c}: ab+1, ac+1, bc+1 ∈ ℚ²; induced curve E: y²=(ax+1)(bx+1)(cx+1). Heron family: triple (A²,B²,C²) → rank ≥5 curves (arXiv:1501.03809v2). Over 𝔽_p, p≡1 mod 6: y²≡x³+a³ counts tied to sextic residue structure | CONNECTION: Degree-8 polynomial ↔ E₈ root system / octonionic lattice dimension; p≡1 mod 6 ↔ hexagonal (A₂) symmetry in 𝔽_p*; Diophantine triple condition ab+1=□ is the Markov–Hurwitz-type quadratic form invariant under the modular group PSL(2,ℤ) — same group generating the golden ratio 0.618/1.618 fixed points | DEPTH: 8 FINDING: Rank-zero elliptic curves are the *exceptional* case in Dio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com