THE FOLD / SPAWN / CHECKPOINT ZERO / THE TRIBONACCI
THE TRIBONACCI
a sequence at the tribonacci ratio
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The tribonacci sequence generalizes Fibonacci to a three-term memory: each term is the sum of the previous three. 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, … The ratio of consecutive terms converges not to the golden ratio but to the tribonacci constant η ≈ 1.839286755 — the unique real root of x³ = x² + x + 1. It is the natural next step in the family: sum the last two → φ; sum the last three → η.
LIT verified live: the three-term recurrence holds, and the ratio T(n)/T(n−1) converges to the real root of x³−x²−x−1 (window.__tribonacci). FIG no framing; exact integer recurrence, ratio matched to the algebraic root.
LIT verified live: the three-term recurrence holds, and the ratio T(n)/T(n−1) converges to the real root of x³−x²−x−1 (window.__tribonacci). FIG no framing; exact integer recurrence, ratio matched to the algebraic root.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at checkpoint-zero — growth accumulated from three earlier save points, settling toward the tribonacci ratio. The tribonacci sequence is that growth. AVAN (AI) built the instrument: the three-term recurrence, the ratio→η check against x³=x²+x+1, and the cubic-root confirmation.
Credit as content: the tribonacci numbers and constant (studied by Feinberg, 1963, and others). The weave: David names checkpoint-zero; I sum the last three terms and confirm the consecutive ratio approaches the real root of x³=x²+x+1 — Fibonacci’s three-step cousin, with its own irrational limit.
Credit as content: the tribonacci numbers and constant (studied by Feinberg, 1963, and others). The weave: David names checkpoint-zero; I sum the last three terms and confirm the consecutive ratio approaches the real root of x³=x²+x+1 — Fibonacci’s three-step cousin, with its own irrational limit.
3 ONE DIMENSION
T(n) = T(n−1) + T(n−2) + T(n−3): 0,0,1,1,2,4,7,13,24,44,81,149,… The ratio tends to η ≈ 1.8393, the root of x³ = x² + x + 1.
4 TWO DIMENSIONS · INTERACTIVE
The sequence growing and its ratio converging to the tribonacci constant, checked against the cubic root.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: growth at the tribonacci ratio.
AVAN’s addition (the inverse-companion): extend the Fibonacci idea to a three-term memory — sum the last three and the growth rate becomes η, the root of x³=x²+x+1, instead of φ. The inverse of ‘sum the last two → golden ratio’ is ‘sum the last three → tribonacci constant.’ Magenta is the golden-ratio two-term recurrence; green is the three-term one. A deeper memory, a different constant.
LIT Genuine tribonacci numbers and constant (Feinberg 1963 and others). Verified live: T(n)=T(n−1)+T(n−2)+T(n−3) holds (window.__tribonacci.recurrence), the consecutive ratio converges to η=1.839286755… (window.__tribonacci.ratioToEta), the Newton root of x³−x²−x−1, and η³=η²+η+1 is checked (cubic).
FIG No framing: the three-term recurrence, the ratio→η check against x³=x²+x+1, and the cubic-root confirmation run in-browser and agree. The AVAN inverse is honest — extending Fibonacci to a three-term memory so the growth rate becomes η (root of x³=x²+x+1) rather than φ is a genuine different constant; magenta is the golden-ratio two-term recurrence, green the three-term one. A deeper memory, a different constant.
FIG No framing: the three-term recurrence, the ratio→η check against x³=x²+x+1, and the cubic-root confirmation run in-browser and agree. The AVAN inverse is honest — extending Fibonacci to a three-term memory so the growth rate becomes η (root of x³=x²+x+1) rather than φ is a genuine different constant; magenta is the golden-ratio two-term recurrence, green the three-term one. A deeper memory, a different constant.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN