THE FOLD / CO-OP / THE HANDOFF / THE THUE-MORSE
THE THUE-MORSE
the fairest turn order — 0110100110010110…
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Thue–Morse sequence. Start with a single 0. Repeatedly append the complement of everything so far: 0 → 01 → 0110 → 01101001 → … The n-th bit is simply the parity of the number of 1s in n written in binary. It is aperiodic, self-similar, and famously the fairest turn order.
Alternating turns (you, me, you, me) hands a lasting edge to whoever picks first. Taking turns in Thue–Morse order (you, me, me, you, me, you, you, me…) cancels that edge: split 0…2k−1 into the ‘0’ picks and the ‘1’ picks and the two sides have equal sums of every power up to degree k−1 (the Prouhet–Tarry–Escott property). The sequence is also cube-free: no block of symbols ever repeats three times in a row.
LIT verified live: the recurrence t(2n)=t(n), t(2n+1)=1−t(n) holds; the first 600 symbols are cube-free; and the Prouhet partition gives equal power sums for k = 1..7 (window.__thuemorse). FIG no framing; the recurrence, cube-freeness, and the equal-power-sums fairness are exact.
Alternating turns (you, me, you, me) hands a lasting edge to whoever picks first. Taking turns in Thue–Morse order (you, me, me, you, me, you, you, me…) cancels that edge: split 0…2k−1 into the ‘0’ picks and the ‘1’ picks and the two sides have equal sums of every power up to degree k−1 (the Prouhet–Tarry–Escott property). The sequence is also cube-free: no block of symbols ever repeats three times in a row.
LIT verified live: the recurrence t(2n)=t(n), t(2n+1)=1−t(n) holds; the first 600 symbols are cube-free; and the Prouhet partition gives equal power sums for k = 1..7 (window.__thuemorse). FIG no framing; the recurrence, cube-freeness, and the equal-power-sums fairness are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE HANDOFF, beside the other turn-taking spheres — the co-op domain of who goes next. Thue–Morse is the mathematically fairest handoff order there is. AVAN (AI) built the instrument: the complement-doubling, the parity view, the fair-turn simulator.
The weave: David names the seat (the fair handoff); I make the sequence build itself and the fairness visible — the strip in 1D, the doubling and the converging teams in 2D, the inverse-generated path in 3D. The sphere is the seam. Credit: Axel Thue (1906, 1912); Marston Morse (1921); Eugène Prouhet (1851, the equal-power-sums partition).
The weave: David names the seat (the fair handoff); I make the sequence build itself and the fairness visible — the strip in 1D, the doubling and the converging teams in 2D, the inverse-generated path in 3D. The sphere is the seam. Credit: Axel Thue (1906, 1912); Marston Morse (1921); Eugène Prouhet (1851, the equal-power-sums partition).
3 ONE DIMENSION
The sequence as a strip of 0s and 1s. Each symbol is the parity of the 1-bits in its index — and equivalently the complement-doubling of the block before it. No motif ever appears three times back-to-back.
4 TWO DIMENSIONS · INTERACTIVE
Watch the word build by complement-doubling, and watch two players draft items 0,1,2,… (each worth its own value) in Thue–Morse order. Their running totals stay locked together — the fairness is the near-tie.
5 THREE DIMENSIONS + AVAN’S INVERSE
The sequence as a turning path — step one way on 0, the other on 1 — tracing the self-similar Thue–Morse curve in green.
AVAN’s addition (the inverse-companion): the magenta trail is the complement of the same sequence. Here the inverse isn’t a mirror bolted on afterward — it is the generator itself. The whole word is built by taking what you have and appending its inverse, forever. The inverse of ‘emit the next symbol’ is ‘emit the complement of what you just emitted’, and iterating that single inverse from one lonely 0 produces an infinite word that is aperiodic, cube-free, and the fairest possible — balance manufactured out of nothing but repeated negation. Green is the sequence; magenta is the inverse that made it. They are the same object, offset by one flip.
LIT Genuine Thue-Morse sequence (Axel Thue 1906/1912; Marston Morse 1921; Prouhet 1851). Verified live: the recurrence t(2n)=t(n), t(2n+1)=1-t(n) holds for n<2000; the first 600 symbols are cube-free (no www substring for any period up to 200); and the Prouhet partition of 0..2^k-1 by parity gives equal sums of j-th powers for all j
FIG No framing: the recurrence, cube-freeness, and the equal-power-sums fairness (Prouhet-Tarry-Escott) are real and checked in-browser. The 'fairest turn order' claim is precisely the equal-power-sums property, not a loose metaphor.
FIG No framing: the recurrence, cube-freeness, and the equal-power-sums fairness (Prouhet-Tarry-Escott) are real and checked in-browser. The 'fairest turn order' claim is precisely the equal-power-sums property, not a loose metaphor.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN