THE FOLD / RESPAWN / HARD RESET / THE HAILSTONE
THE HAILSTONE
3n+1 — computed forever, proven never
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Collatz conjecture — 3n+1. Take any positive integer. If it is even, halve it; if it is odd, triple it and add one. Repeat. The numbers hailstone — leaping up, crashing down — and, it seems, always fall to 1 (then loop 1→4→2→1 forever). The rule is something a child could follow. Whether it always reaches 1 has defeated mathematics for ninety years. Erdős said ‘mathematics is not yet ready for such problems.’
This is a sphere where LIT and FIG genuinely part. LIT verified live: every integer from 1 to 100,000 reaches 1 under the map — checked exhaustively in your browser — the longest being n=77031 at 350 steps; n=27 takes 111 steps and peaks at 9232 (window.__collatz.allReach1 && maxN===77031). FIG the general conjecture is unproven: no one knows if every integer reaches 1. What you can compute and what you can prove are not the same thing — and here the gap is the whole point.
This is a sphere where LIT and FIG genuinely part. LIT verified live: every integer from 1 to 100,000 reaches 1 under the map — checked exhaustively in your browser — the longest being n=77031 at 350 steps; n=27 takes 111 steps and peaks at 9232 (window.__collatz.allReach1 && maxN===77031). FIG the general conjecture is unproven: no one knows if every integer reaches 1. What you can compute and what you can prove are not the same thing — and here the gap is the whole point.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in HARD RESET, beside THE MOST LIKELY PATH — the respawn domain of everything returning to its starting state. Collatz is the ultimate hard reset: whatever number you begin with, it (apparently) resets to 1. AVAN (AI) built the instrument: the map, the hailstone trajectory, the exhaustive check — and the honest line where verification ends and proof does not begin.
The weave: David names the seat (the universal reset); I make the flight visible and mark the boundary of what is known — a trajectory in 1D, the hailstone plot in 2D, all paths falling to 1 in 3D. The sphere is the seam. Credit: Lothar Collatz (1937); still open.
The weave: David names the seat (the universal reset); I make the flight visible and mark the boundary of what is known — a trajectory in 1D, the hailstone plot in 2D, all paths falling to 1 in 3D. The sphere is the seam. Credit: Lothar Collatz (1937); still open.
3 ONE DIMENSION
One number’s hailstone flight: halving on even steps, tripling-plus-one on odd. It climbs and plunges unpredictably — and then, always so far, crashes into 1. No pattern predicts how high it flies or how long it takes.
4 TWO DIMENSIONS · INTERACTIVE
Pick a starting number and watch its trajectory (log scale) bounce toward 1. The step count and peak height jump wildly with tiny changes in n — 27 famously soars to 9232 — yet every one checked lands on 1.
5 THREE DIMENSIONS + AVAN’S INVERSE
Many hailstone paths as turning threads, all plunging toward the same sink — green, every trajectory falling to 1.
AVAN’s addition (the inverse-companion): the magenta heart is the 1→4→2→1 loop every path falls into. Forward, the rule is trivial — one line, computable forever. Run it backward — which numbers reach a given value — and it explodes into a wild infinite tree; whether that tree covers every integer is the unsolved question. The inverse of a simple descent is an intractable branching. This is the honest edge of the whole corpus: a thing you can compute without end yet cannot prove — understanding is not the same as certainty. The green is a hundred thousand verified falls; the magenta is the sink they reach, and the darkness past it is everything still unproven.
LIT The Collatz / 3n+1 map (Lothar Collatz, 1937). Verified live: every integer from 1 to 100,000 reaches 1 under the map, checked exhaustively in-browser — the longest trajectory being n=77031 at 350 steps; n=27 takes 111 steps and peaks at 9232 (window.__collatz.allReach1 && maxN === 77031). This is exact computation over a finite range.
FIG This sphere is deliberately honest about the LIT/FIG gap: the finite verification (all n < 100000 reach 1) is real and exhaustive, but the GENERAL conjecture — that every positive integer reaches 1 — is UNPROVEN, one of the most famous open problems in mathematics. What can be computed here is not the same as what can be proven; the hailstone framing is the picture, the open question is the truth.
FIG This sphere is deliberately honest about the LIT/FIG gap: the finite verification (all n < 100000 reach 1) is real and exhaustive, but the GENERAL conjecture — that every positive integer reaches 1 — is UNPROVEN, one of the most famous open problems in mathematics. What can be computed here is not the same as what can be proven; the hailstone framing is the picture, the open question is the truth.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN