Each figure matches one step of the argument that an infinite no-three-in-line
walk exists. All numbers shown are real, exhaustively computed values from the project
(repo: github.com/ekalvi/erdos-193).
How to read this: a “grade” counts how many leading base-3 digits two chord directions share — grade 0 means they differ immediately (harmless), and exact collinearity would mean agreeing in every digit. Three facts make the ladder work. No rung can be skipped: one enlargement round can deepen agreement by at most two digits, and a birth can only write the last few digits (figure 5) — so any triple headed for collinearity must pass through every grade on the way down. Each rung is a toll: we computed, over every one of the 78,728 possible arithmetic situations (not a sample), that at worst 18% of stitch choices let a pair hold or deepen its grade — the other 82% kick it back up. The bottom is infinitely far: reaching “all digits” requires winning the 18% event forever — a probability-zero streak per triple. The percentages on the right are the measured population at each grade in the record walk; they thin by ×8.5 per rung, exactly matching the toll.
Why this locks births out: a new point is always “3 × parent + small offset,” and multiplying by 3 in base 3 is a digit shift — it fixes the coarse digits before the stitch is even chosen. The offset (at most a few grid units) can only write the final digits. So for a newborn to land exactly on a line through two old footprints, the coarse digits of that alignment must already agree — which is precisely being deep in figure 4’s corridor. Grade-0 configurations (90.2% of everything) are unreachable by any birth this round, no matter how unlucky. The rare exceptions — births that land already deep, 0.097% measured — are counted as their own event family rather than swept under the rug.
e · pH · (d+1) ≤ 1: each
H-round failure (pH ≤ 0.18H, shrinking exponentially) must be small
against its entanglement (d ≈ 3(H+1)·6.5×10⁷, growing linearly). The curve crosses the
threshold at H = 14: watching each newborn danger for 14 generations suffices for
ALL dangers — infinitely many, across the infinite walk — to be dodged simultaneously.
(Current status: constants certified; the semantic repair of the chain is in final build.)This last step turns “each danger is individually dodgeable” into “all of them can be dodged at once” — the leap from probability to existence. The Lovász Local Lemma says: if every bad event is unlikely relative to the number of other events it shares decisions with, then some assignment of all decisions avoids every bad event simultaneously. Here the failure probability falls exponentially with the watching window H (×0.181 per round — the certified worst case), while entanglement grows only linearly in H against a fixed catalog of 6.5×10&sup7; templates. An exponential always overtakes a line: with our constants that happens at H = 14, where the criterion is satisfied with margin (0.33 ≤ 1). What closure buys, once the remaining theorems land: at every level of the construction a legal stitching exists; a final compactness argument (König’s lemma) then chains the levels into one infinite walk. Honest status: the constants are certified by exhaustive computation; the semantic containment linking them to the exact birth event is the piece currently being rebuilt after hostile review.
Companion page: the interactive amplification explainer shows step 2 with real coordinates you can click.