A 3,125-Point Four-Step Lattice Walk in Z^3 with No Collinear Triple: An Exact Computational Certificate

This record supplies an exact computer-assisted certificate for a finite Gerver--Ramsey lattice-walk lower bound. For the fixed step set S = {(1,0,0), (1,1,1), (1,1000,10^6), (1,10^6,10^12)}, the deposited 3,124-step word generates 3,125 distinct lattice points in Z^3 with no collinear triple. The witness was checked with exact integer arithmetic by two independent verifiers using different primitive-direction deduplication methods. The record includes the paper, complete LaTeX source, exact witness, intermediate checkpoints, search programs, verification programs, and SHA-256 checksums. This is a finite lower-bound result only. It does not construct an infinite walk and does not resolve Erdős Problem 193. Priority as a record lower bound has not been independently confirmed. AI-assistance disclosure: OpenAI ChatGPT (GPT-5.6) assisted with literature organization, mathematical presentation, LaTeX preparation, and development/checking of search and verification code. The sole human author takes responsibility for the deposited content. The AI system is not an author.