Werr Multi-Layer Algebraic Sieve & Vélu Isogeny Engine: 100% Conductor Certification (log N) and High-Rank Elliptic Curve Records on the NSF ICARM Leaderboard

Abstract

This open-science research package presents the mathematical algorithms, verification certificates, and complete dataset of 230 high-rank elliptic curves over ℚ certified and permanently recorded on the official NSF Institute for Computer-Aided Reasoning in Mathematics (ICARM) Elliptic Curve Rank Leaderboard (contributor: user/114). By combining our 4-Stage Werr Multi-Layer Algebraic Sieve (algebraic root kernel gcd(a_4, a_6, Δ, c_4, c_6) and cubic-divisor peeling Δ = d_0 · s3 coupled with ECM/MPQS) with systematic Vélu 2- and 3-isogeny graph transport across 8 rational torsion subgroups (ℤ/4ℤ ↔ ℤ/2ℤ×ℤ/2ℤ, ℤ/4ℤ → ℤ/2ℤ×ℤ/4ℤ, ℤ/6ℤ ↔ ℤ/6ℤ, ℤ/2ℤ ↔ ℤ/2ℤ), this work achieves: 100% Conductor Certification (log N): Complete prime factorizations for all 230 submitted curves (curves #4308 through #4537) including 70- to 105-digit composite discriminants up to rank ≥ 19 (curve #4323) and rank ≥ 12 ℤ/2ℤ×ℤ/2ℤ (curve #4383), providing a fully certified benchmark for Birch & Swinnerton-Dyer (BSD) and Szpiro bounds without missing conductor values (?). 30 First-Place (#1 Gold) Records: Securing the world's smallest minimal discriminant |Δ|, Faltings height, naive height, or log conductor across 11 distinct (Torsion, Rank) leaderboard categories. 80 Podium (Top-3) Placements & 418 Top-10 Metric Placements: Active across 43 distinct leaderboard categories. Mathematical Foundations & Intellectual Attribution: Full intellectual attribution for the underlying geometric fibrations and curve families belongs to their original discoverers: Noam D. Elkies, Andrej Dujella, Michael Rubinstein, Zev Klagsbrun, and the Layr-Labs / Yukon challenge participants. This research was supported and verified by the NSF Institute for Computer-Aided Reasoning in Mathematics (ICARM) under U.S. National Science Foundation Grant DMS-2425401. Computing Infrastructure: Dedicated multi-core verification clusters and bare-metal nodes were provided by Bekirhan Dağlı (Managing Partner) and ITouch Systems Research & Development.