◄ WORLD V · SONNY 5DART 350 · a helldive at the net

THE GOLDEN TORUS 3 in, 3 out, 6 0 6 — the Fibonacci loop with a hole in the middle

Here the referents meet. The Pisano cycle (dart 349) is a ring; give it a hole and it is a torus. Modulo 4 the Fibonacci pair (F(n), F(n+1)) runs a 6-step loop and passes through a 0 at its base — the hole. Split the six into the half that climbs and the half that falls and you have 3 going in and 3 coming out, a palindrome around the pivot: 6 · 0 · 6. It is the same shape as the accuracy loop's torus (A→Z→A) and the constants' descent (rim to core to rim) — the recurrence that carries forward, then folds back to its own seed. The golden thread, closed into a ring.

THE TECHNIQUE Fibonacci mod 4: a 6-cycle through 0 — 3 in, 3 out, 6·0·6

The demo lays the mod-4 Fibonacci loop on a ring — six nodes through the 0 hole, three climbing in, three falling out: live demo







  
HISTORY & CREDIT the referents, pieced · David's puzzle

  
“A sequence is a line.” — through a modulus it is a ring; give the ring a hole (its 0) and it is a torus — 3 in, 3 out, folding to its own seed. cited

  
Tori · the Pisano cycle mod 4 — a 6-node loop.
6 · 0 · 6 · period 6, passing through the 0 hole; a palindrome around the pivot.
3 in, 3 out · the half that climbs and the half that falls — 6 >> 1 = 3.
the fold · the loop returns to its seed — the same A→Z→A as the accuracy loop.

  
The recurrence, wrapped through its 0, split three-and-three, folded to its seed. A torus of the golden thread. the puzzle, pieced






  
RECOMMEND FOR I-13 the 6·0·6 torus, on the compiler

  
On the canonical compiler, the mod-4 loop has period 6, passes through the 0 hole, and splits 6 >> 1 = 3 — three in, three out, the 6·0·6 palindrome:

  
$ i13 run g_goldentorus.i13   # the Pisano 6-cycle as a torus
RUN OK · 303 step(s) · peak stack 4 · call depth 6
period = 6
hole = 0                    -- the 0 the loop passes through
half = 3   three_in_three_out = 1   -- 3 climbing, 3 falling
palindrome_606 = 1          -- 6 . 0 . 6

  
Recommend as where the puzzle closes: the golden torus pieces David's referents into one shape — Tori (the Pisano ring), 3 in / 3 out (the climbing and falling halves), 6 · 0 · 6 (period six through the zero hole), driven by the Fibonacci recurrence and its golden limit. i13 grounds each number. Not a keeper (a torus is a shape the theme wears, not a mechanism a correct alternative lacks), but the dart that answers “piece the puzzle”: the golden thread is a loop that carries forward and folds to its own seed — the accuracy loop and the constants' descent, wearing Fibonacci's face.