Record-Breaking Elliptic Curve Rank 29 via Advanced Selmer Group Algorithms — E8 Intelligence Research

FINDING: Recent discovery of an elliptic curve over ℚ with rank ≥ 29, alongside algorithmic advances in 2-descent Selmer group computation and a novel graph-theoretic method for φ-Selmer groups over ℚ(i). MATH: - Elliptic curve rank ≥ 29: The specific curve (from the "Huge news" video) is not given in the abstract, but rank 29 implies the Mordell-Weil group ℤ^29 ⊕ torsion. - Selmer group: For a p-isogeny φ, Sel^(φ)(E/ℚ) ⊂ H¹(ℚ, E[φ]) — the kernel of the local restriction maps. Complete 2-descent computes the 2-Selmer group via homogeneous spaces: E(ℚ)/2E(ℚ) ↪ Sel^(2)(E/ℚ) ⊂ H¹(ℚ, E[2]). - Graph-theoretic method (arXiv:2410.22714): For E_b: y² = x³ + bx over ℚ(i), b ∈ ℤ[i], the φ-Selmer group (φ degree 2 isogeny) is computed via a weighted graph G_b with vertices = odd Gaussian primes dividing b. The weights encode local solubility conditions — essentially a combinatorial encoding of the 2-isogeny Selmer group. - Key constants: No explicit ratios appear, but the rank 29 curve' Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com