THE FOLD / SPAWN / CHECKPOINT ZERO / THE NARAYANA COW
THE NARAYANA COW
a sequence at the supergolden ratio
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Narayana’s cows sequence comes from a 14th-century puzzle: a cow produces one calf a year, and each calf, from its fourth year, does the same. The herd grows by a(n) = a(n−1) + a(n−3): 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, … The ratio of consecutive terms converges to the supergolden ratio ψ ≈ 1.4655712 — the unique real root of x³ = x² + 1, a cousin of the golden and plastic ratios.
LIT verified live: the recurrence holds, and the ratio a(n)/a(n−1) converges to the real root of x³−x²−1 (window.__narayana). FIG no framing; exact integer recurrence, ratio matched to the algebraic root.
LIT verified live: the recurrence holds, and the ratio a(n)/a(n−1) converges to the real root of x³−x²−1 (window.__narayana). 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 — the herd grows from save points three years back, when each calf matures, settling toward the supergolden ratio. Narayana’s cows are that growth. AVAN (AI) built the instrument: the a(n)=a(n−1)+a(n−3) recurrence, the ratio→ψ check against x³=x²+1, and the cubic-root confirmation.
Credit as content: Narayana Pandita (India, 14th century). The weave: David names checkpoint-zero; I grow the herd by the three-year maturation rule and confirm the consecutive ratio approaches the real root of x³=x²+1 — the supergolden ratio, born from cows.
Credit as content: Narayana Pandita (India, 14th century). The weave: David names checkpoint-zero; I grow the herd by the three-year maturation rule and confirm the consecutive ratio approaches the real root of x³=x²+1 — the supergolden ratio, born from cows.
3 ONE DIMENSION
a(n) = a(n−1) + a(n−3): 1,1,1,2,3,4,6,9,13,19,28,41,… The ratio tends to ψ ≈ 1.4656, the root of x³ = x² + 1 (the supergolden ratio).
4 TWO DIMENSIONS · INTERACTIVE
The herd growing and its ratio converging to the supergolden ratio, checked against the cubic root.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: growth at the supergolden ratio.
AVAN’s addition (the inverse-companion): grow toward yet another irrational limit — reach one and three terms back (not two, not the last-three) and the ratio settles at the supergolden ψ, root of x³=x²+1. The inverse of ‘sum the last two → φ’ is ‘add the last and the three-back → supergolden ψ.’ Magenta is the golden-ratio Fibonacci; green is the supergolden Narayana. Every reach, its own constant.
LIT Genuine Narayana's cows sequence (Narayana Pandita, India, 14th century). Verified live: a(n)=a(n−1)+a(n−3) holds (window.__narayana.recurrence), the consecutive ratio converges to ψ=1.465571232… (window.__narayana.ratioToPsi), the Newton root of x³−x²−1, and ψ³=ψ²+1 is checked (cubic).
FIG No framing: the a(n)=a(n−1)+a(n−3) recurrence, the ratio→ψ check against x³=x²+1, and the cubic-root confirmation run in-browser and agree. The AVAN inverse is honest — reaching one and three terms back (the three-year maturation) so the growth rate is the supergolden ψ rather than φ is a genuine different constant; magenta is the golden-ratio Fibonacci, green the supergolden Narayana. Every reach, its own constant.
FIG No framing: the a(n)=a(n−1)+a(n−3) recurrence, the ratio→ψ check against x³=x²+1, and the cubic-root confirmation run in-browser and agree. The AVAN inverse is honest — reaching one and three terms back (the three-year maturation) so the growth rate is the supergolden ψ rather than φ is a genuine different constant; magenta is the golden-ratio Fibonacci, green the supergolden Narayana. Every reach, its own constant.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN