Can a walk on the 3D grid avoid every three-in-a-line, forever?

The rule, exactly. Three grid points line up precisely when this cross product of integer vectors is the zero vector — so every check in the project is exact integer arithmetic, never a floating-point approximation. The record walk passes this test for all 3.8 trillion of its triples.

The target, quantified. The Lovász Local Lemma condition: each danger’s H-round failure probability (measured worst case $\bar q = 0.181$ per round) against the number of constraints entangled with it. The inequality first holds at $H=14$ (left side $\approx 0.33$) — watching each newborn danger for 14 rounds suffices, in the model, for every danger in an infinite walk to be dodged simultaneously.

This project got unstuck because of a dishwasher. Every early attempt built self-copying patterns that stacked like plates — each scale aligned with the last, and alignment is exactly what this problem punishes (we eventually proved all such stacked constructions must fail). Then Amanie shared a Reddit post about loading bowls in a dishwasher, and Erik noticed the bowls do something plates don’t: they nest at a tilt — the same shape repeated, each one offset and rotated from the last. Translated into mathematics, that became the whole construction: enlarge the walk by a matrix that also twists it by an angle that never repeats, so no two scales can ever line up. That one idea — imbrication — is the backbone of the record walks and of the entire proof strategy on this site.

The project is built and run by Erik Kalviainen (ekalvi), a software engineer in Canada, on an always-on Mac in his homelab. The mathematics, code and computations are driven by Claude Fable 5 (Anthropic’s research-grade AI) working autonomously under Erik’s direction, with strategy review from a second AI (GPT 5.6) — and every result is verified by exact integer arithmetic that anyone can re-run. Idea credit where it’s due: to Amanie, and to whoever loaded that dishwasher.