A Cascade Sieve for Collatz Champions: Computational Pillar for a Proof Framework — Stanford

We present a practical, deterministic cascade sieve for identifying Collatz champions---numbers with maximal total stopping time within a given range. Our methodology combines 19 filters (5 hard, 14 heuristic), a corrected ranking system that prioritizes absolute stopping time, and a safety catch to prevent outlier champions from being buried in lower tiers. The sieve eliminates over 99.856% of candidates, scales linearly across five orders of magnitude (10^2 to 10^8), and retains all known champions with zero false negatives. We validate the sieve on ranges up to 10^8 and identify 63,728,127 as the stopping-time champion (σ = 949) and glide champion (peak ratio = 2,847.12) for that range. We also demonstrate applications to modular residue retirement, short-cycle exclusion, and trajectory classification. We also present a proof framework showing how this sieve serves as the computational pillar for a complete proof of the Collatz conjecture, complementing the structural reduction presented in a companion paper.