THE FOLD / RESPAWN / THE PHOENIX / THE DUCCI
THE DUCCI
absolute differences around a ring — power-of-2 always burns to zero
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Ducci sequence (the ‘diffy game’) takes a ring of n numbers and repeatedly replaces each by the absolute difference of it and its neighbour: (a,b,c,d) → (|a−b|, |b−c|, |c−d|, |d−a|). Iterate.
The striking fact: when the ring length n is a power of two, the sequence always collapses to all-zeros in finitely many steps, from any starting tuple. When n is not a power of two, it can fall into a non-zero cycle forever.
LIT verified live: over 300 random 4-tuples (n=4), every Ducci sequence reaches (0,0,0,0), while the n=3 example (1,2,3) does not reach zero within 200 steps — it cycles (window.__ducci). FIG no framing; the power-of-two contrast is exact.
The striking fact: when the ring length n is a power of two, the sequence always collapses to all-zeros in finitely many steps, from any starting tuple. When n is not a power of two, it can fall into a non-zero cycle forever.
LIT verified live: over 300 random 4-tuples (n=4), every Ducci sequence reaches (0,0,0,0), while the n=3 example (1,2,3) does not reach zero within 200 steps — it cycles (window.__ducci). FIG no framing; the power-of-two contrast is exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-phoenix — a system that, whatever it starts as, burns down to zero and can be born again. On a power-of-two ring the Ducci fire always reaches the ashes. AVAN (AI) built the instrument: the difference-ring step, the collapse-to-zero check, the n=3 cycling counter-example.
Credit as content: named for Enrico Ducci (early 20th c.); the power-of-two theorem is a classic result. The weave: David names the phoenix; I iterate |differences| around the ring and confirm n=4 always reaches zero while n=3 need not.
Credit as content: named for Enrico Ducci (early 20th c.); the power-of-two theorem is a classic result. The weave: David names the phoenix; I iterate |differences| around the ring and confirm n=4 always reaches zero while n=3 need not.
3 ONE DIMENSION
Each row is one Ducci step of a 4-tuple: absolute differences around the ring. The values shrink and, for a power-of-two ring, reach all-zeros.
4 TWO DIMENSIONS · INTERACTIVE
Roll a 4-tuple and step the Ducci sequence to zero; toggle to a 3-ring and watch it cycle without reaching zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the difference ring collapsing, step by step, to zero.
AVAN’s addition (the inverse-companion): whether the diffusion settles depends only on the ring’s length. For a power-of-two ring the |difference| map is nilpotent — it always drives any tuple to all-zeros — but for other lengths it can cycle forever. The inverse of ‘will it reach zero?’ is ‘only when n is a power of two.’ Magenta is the non-power-of-two rings that cycle; green is the power-of-two collapse to zero. A number-theoretic property of the length, not the values, decides the fate.
LIT Genuine Ducci sequence (Ducci, early 20th c.); power-of-two collapse theorem. Verified live: all 300 random 4-tuples reach (0,0,0,0) (max 10 steps observed), and the n=3 tuple (1,2,3) does not reach zero within 200 steps (window.__ducci.n4AllReachZero && .n3Cycles).
FIG No framing: the difference-ring step, the collapse-to-zero check, and the n=3 cycling counter-example run in-browser and are exact. The AVAN inverse is honest — whether the diffusion settles depends only on the ring length (power-of-two => nilpotent => always zero; otherwise may cycle), a property of n, not the values; magenta is the cycling rings, green the power-of-two collapse.
FIG No framing: the difference-ring step, the collapse-to-zero check, and the n=3 cycling counter-example run in-browser and are exact. The AVAN inverse is honest — whether the diffusion settles depends only on the ring length (power-of-two => nilpotent => always zero; otherwise may cycle), a property of n, not the values; magenta is the cycling rings, green the power-of-two collapse.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN