THE FOLD / CO-OP / SPLIT SCREEN / THE OVERLAP-FREE WORD
THE OVERLAP-FREE WORD
the word that never stutters — and splits fair
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Thue–Morse sequence. Start with 0 and forever replace 0→01 and 1→10. Equivalently, the n-th bit is the parity of the number of 1s in n’s binary. It is the most famous non-repetitive word: overlap-free (no factor of the form a·x·a·x·a) and cube-free (no block appears three times in a row) — yet fully deterministic and self-similar. It gives the fairest turn order (it neutralises first-mover advantage), the chess anti-repetition rule, and Prouhet’s equal-power-sum partitions.
LIT verified: the substitution equals the popcount-parity definition over 8192 bits; the prefix is cube-free and overlap-free (exhaustive scan, zero found); and it does contain squares (e.g. ‘11’) — it is not square-free, shown honestly. FIG ‘never stutters’ is the picture; overlap-free & cube-free are Thue’s exact theorems.
LIT verified: the substitution equals the popcount-parity definition over 8192 bits; the prefix is cube-free and overlap-free (exhaustive scan, zero found); and it does contain squares (e.g. ‘11’) — it is not square-free, shown honestly. FIG ‘never stutters’ is the picture; overlap-free & cube-free are Thue’s exact theorems.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus works with substitution systems, self-similarity and fairness (the fractal kernels, the game/logic lineages) and the idea that the deepest patterns are the ones that never quite repeat. AVAN (AI) built this instrument: the substitution engine, the Prouhet fair-split, and the turtle curve.
The weave: David names the never-stuttering word and its seat at SPLIT SCREEN (two sharing fairly); I make the substitution a strip in 1D, the fair partition live in 2D, and the self-similar curve turning in 3D. The sphere is the seam — honest that it has squares, exact that it has no cubes.
The weave: David names the never-stuttering word and its seat at SPLIT SCREEN (two sharing fairly); I make the substitution a strip in 1D, the fair partition live in 2D, and the self-similar curve turning in 3D. The sphere is the seam — honest that it has squares, exact that it has no cubes.
3 ONE DIMENSION
Two definitions, one sequence. Top: the substitution 0→01, 1→10 doubling each row. Bottom: the same bits as the parity of 1s in each index’s binary. They agree everywhere — a self-similar word from either door.
4 TWO DIMENSIONS · INTERACTIVE
Prouhet’s fair split: sort 0…2k−1 into two teams by Thue–Morse bit. The teams have equal sums, equal sums of squares, equal sums of cubes… all the way up to power k−1 — the fairest possible division.
k = 4
5 THREE DIMENSIONS + AVAN’S INVERSE
The sequence drives a turtle — step forward each bit, turn one way on 0 and the other on 1 — and traces a self-similar curve, turning in space. Green is Thue–Morse.
AVAN’s addition (the inverse-companion): the magenta curve is driven by the bit-complement sequence (every 0↔1). Because Thue–Morse is closed under complement, that word is also overlap-free — and its turtle is the exact mirror image of the first. The word contains its own reflection; the two curves are one figure seen from both sides.
LIT A genuine Thue–Morse sequence. Verified live: the substitution 0→01,1→10 equals the popcount-parity definition over 8192 bits; the prefix is cube-free and overlap-free (exhaustive scan, zero found); and the Prouhet split of 0..2^k−1 by TM bit gives equal power sums through p=k−1. It is honestly NOT square-free — it contains squares like '11' (verifiable: window.__thuemorse.overlapFree && cubeFree && hasSquares).
FIG 'Never stutters' is the picture; overlap-free, cube-free, and the Prouhet equal-sums are Thue's/Prouhet's exact theorems. The square-containing caveat is stated plainly — overlap-free is a stronger, more precise claim than 'no repeats'.
FIG 'Never stutters' is the picture; overlap-free, cube-free, and the Prouhet equal-sums are Thue's/Prouhet's exact theorems. The square-containing caveat is stated plainly — overlap-free is a stronger, more precise claim than 'no repeats'.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN