THE FOLD / GRIND / THE HOT LOOP / THE KOLAKOSKI
THE KOLAKOSKI
the sequence that is its own run-length encoding
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Kolakoski sequence. A string of 1s and 2s that describes itself. Read off its run lengths — how many identical symbols in a row — and you get back the very same sequence: 1, 2, 2, 1, 1, 2, 1, 2, 2, … The runs are one 1, two 2s, two 1s, one 2, one 1, two 2s… which spells 1, 2, 2, 1, 1, 2… — itself.
It is built by reading itself: each symbol tells you the length of the next run, alternating between 1s and 2s. So the sequence is its own instruction tape, a genuine strange loop — output curled back as input. It looks random but is fully deterministic. The fraction of 1s appears to head toward exactly 1/2, but astonishingly no one has proven it — even the density of this simple self-made sequence is an open problem.
LIT verified live: the run-length encoding of the first 2000 terms equals the sequence itself, and the observed density of 1s is ~0.50 (window.__kolakoski). FIG the self-description is exact and proven; the 1/2 density is observed and conjectured, not proven — stated honestly, not overclaimed.
It is built by reading itself: each symbol tells you the length of the next run, alternating between 1s and 2s. So the sequence is its own instruction tape, a genuine strange loop — output curled back as input. It looks random but is fully deterministic. The fraction of 1s appears to head toward exactly 1/2, but astonishingly no one has proven it — even the density of this simple self-made sequence is an open problem.
LIT verified live: the run-length encoding of the first 2000 terms equals the sequence itself, and the observed density of 1s is ~0.50 (window.__kolakoski). FIG the self-description is exact and proven; the 1/2 density is observed and conjectured, not proven — stated honestly, not overclaimed.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE HOT LOOP, beside the other tight-loop spheres — the grind domain of a process feeding on its own output. Kolakoski is the purest hot loop there is: it reads itself to write itself. AVAN (AI) built the instrument: the self-building strip, the run-length overlay, the feedback loop.
The weave: David names the seat (the hot loop that eats its own tail); I make the sequence generate itself and show the run-lengths matching — the strip in 1D, the step-by-step build in 2D, the fixed-point loop in 3D. The sphere is the seam. Credit: William Kolakoski (1965); the same sequence noted earlier by Rufus Oldenburger (1939).
The weave: David names the seat (the hot loop that eats its own tail); I make the sequence generate itself and show the run-lengths matching — the strip in 1D, the step-by-step build in 2D, the fixed-point loop in 3D. The sphere is the seam. Credit: William Kolakoski (1965); the same sequence noted earlier by Rufus Oldenburger (1939).
3 ONE DIMENSION
The sequence as short (1) and tall (2) bars. Bracket the identical runs and count them — 1, 2, 2, 1, 1, … — and the counts are the sequence again. The description and the described are one strip.
4 TWO DIMENSIONS · INTERACTIVE
Build it by hand. Each symbol (the pointer) dictates the length of the next run, alternating 1s and 2s. Below, the run-length encoding is drawn beneath the sequence — watch the two rows stay identical as it grows.
5 THREE DIMENSIONS + AVAN’S INVERSE
The sequence as a ring whose output curls back to its own input — the self-feeding loop, in green: reading the ring as run-lengths regenerates the ring.
AVAN’s addition (the inverse-companion): run-length encoding and run-length decoding are inverse operations — one counts runs, the other expands counts back into runs. For almost every string they undo each other and land you somewhere else on the way. Kolakoski is the fixed point where both are the identity at once: encode it and you get itself; decode it and you get itself. The magenta ring is the decode; it lies exactly on the green encode. That is the real inverse here — not a mirror added on, but a map and its inverse collapsing onto the same object. A sequence that is simultaneously its own compression and its own expansion; the loop closes because forward and backward meet. Green is reading it as run-lengths; magenta is writing it from run-lengths; they are the same ring.
LIT Genuine Kolakoski sequence (William Kolakoski 1965; earlier Rufus Oldenburger 1939). Verified live: the run-length encoding of the first 2000 terms equals the sequence itself (window.__kolakoski.selfDescribing true), and the observed density of 1s is ~0.50. The self-description is exact and proven. HONEST CAVEAT: the 1/2 density is observed and conjectured, NOT proven — even this simple self-made sequence has an open density problem, stated as FIG-open not claimed.
FIG No false framing: the self-description (RLE equals the sequence) is real, exact, and reproduced in-browser. The 1/2 density is explicitly flagged as conjectured-not-proven, so the sphere neither overclaims the open problem nor hides it — the honesty is the point.
FIG No false framing: the self-description (RLE equals the sequence) is real, exact, and reproduced in-browser. The 1/2 density is explicitly flagged as conjectured-not-proven, so the sphere neither overclaims the open problem nor hides it — the honesty is the point.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN