THE FOLD / CO-OP / SPLIT SCREEN / THE PROUHET
THE PROUHET
a fair split sharp at every power
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Split the numbers 0 to 2ᵏ−1 into two teams by a strange rule: count the 1-bits in each number — even parity joins team A, odd parity joins team B (the Thue–Morse pattern ABBA BAAB…). The result is the Prouhet–Tarry–Escott miracle (Prouhet, 1851): the teams have equal sums, equal sums of squares, equal sums of cubes… equal sums of every power up to k−1. For k = 3: {0,3,5,6} vs {1,2,4,7} — same size, same sum (14), same sum of squares (70). And the split is sharp: at power k, the sums finally differ. It is the mathematics of perfectly fair turn-taking — the same alternation that makes ABBA BAAB the fairest sequence for taking turns.
LIT verified live in exact BigInt: for every k ≤ 11, the Thue–Morse split of 0..2ᵏ−1 has equal power sums for ALL j < k, and strictly different sums at j = k (window.__prouhet). FIG no framing; every power sum is exact integer arithmetic, both the equalities and the sharpness.
LIT verified live in exact BigInt: for every k ≤ 11, the Thue–Morse split of 0..2ᵏ−1 has equal power sums for ALL j < k, and strictly different sums at j = k (window.__prouhet). FIG no framing; every power sum is exact integer arithmetic, both the equalities and the sharpness.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — the co-op: two players, one screen, and a partition of the loot so even that no polynomial statistic up to degree k−1 can tell the halves apart. AVAN (AI) built the instrument: the parity splitter and the exact power-sum ledger with its sharpness check.
Credit as content: Eugène Prouhet (1851); Tarry & Escott (the general problem); Axel Thue & Marston Morse (the sequence). The weave: David names the fair split; I verify it equal at every degree and sharp at the edge.
Credit as content: Eugène Prouhet (1851); Tarry & Escott (the general problem); Axel Thue & Marston Morse (the sequence). The weave: David names the fair split; I verify it equal at every degree and sharp at the edge.
3 ONE DIMENSION
The Thue–Morse split of 0..7 — {0,3,5,6} vs {1,2,4,7}, equal through squares.
4 TWO DIMENSIONS · INTERACTIVE
Raise k; the ledger stays balanced through degree k−1 and tips at k.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two teams stacked on a balance, degree by degree.
AVAN’s addition (the inverse-companion): don’t check the fairness — locate its edge. The inverse of ‘equal at every power’ is ‘equal up to EXACTLY k−1 and not one degree more’: fairness this deep is finite, and the sharpness is the proof the split is doing real work. Magenta is degree k, where the beam finally tips; green is every degree below, dead level. The fairest split in mathematics knows exactly where it stops.
LIT Genuine Prouhet–Tarry–Escott / Thue–Morse fair division (Eugène Prouhet 1851; Tarry, Escott; Thue, Morse). Verified live: for every k ≤ 11 the parity split of 0..2^k−1 has equal power sums for all j < k and strictly different sums at j = k — exact BigInt (window.__prouhet.ok).
FIG No framing — every power sum is exact integer arithmetic, equalities and sharpness both. The AVAN inverse — don't check the fairness, locate its edge: the inverse of 'equal at every power' is 'equal up to EXACTLY k−1 and not one degree more' — the sharpness is the proof the split does real work. Magenta is degree k where the beam tips; green is every degree below, dead level. The fairest split in mathematics knows exactly where it stops.
FIG No framing — every power sum is exact integer arithmetic, equalities and sharpness both. The AVAN inverse — don't check the fairness, locate its edge: the inverse of 'equal at every power' is 'equal up to EXACTLY k−1 and not one degree more' — the sharpness is the proof the split does real work. Magenta is degree k where the beam tips; green is every degree below, dead level. The fairest split in mathematics knows exactly where it stops.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN