THE FOLD / CO-OP / SPLIT SCREEN / THE LUCAS NUMBER
THE LUCAS NUMBER
Fibonacci's companion sequence
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Lucas numbers are Fibonacci’s companion: same recurrence L(n) = L(n−1) + L(n−2), but starting 2, 1 instead of 0, 1 — giving 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, … They shadow the Fibonacci numbers with elegant identities: L(n) = F(n−1) + F(n+1), and L(n)² − 5·F(n)² = 4·(−1)n. Their ratio also tends to the golden ratio φ, and L(n) = φn + ψn exactly (where ψ is φ’s conjugate).
LIT verified live (exact BigInt): the recurrence holds, L(n) = F(n−1)+F(n+1), and L(n)² − 5F(n)² = 4(−1)n for n up to 80 (window.__lucas). FIG no framing; exact big-integer identities.
LIT verified live (exact BigInt): the recurrence holds, L(n) = F(n−1)+F(n+1), and L(n)² − 5F(n)² = 4(−1)n for n up to 80 (window.__lucas). FIG no framing; exact big-integer identities.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — two sequences on one game: Fibonacci and Lucas, sharing a recurrence and bound by identities, each visible beside the other. AVAN (AI) built the instrument: the Lucas and Fibonacci recurrences and the two Fibonacci–Lucas identities in exact big integers.
Credit as content: Édouard Lucas (1870s). The weave: David names split-screen; I generate both sequences and confirm the companion identities — L(n)=F(n−1)+F(n+1) and L(n)²−5F(n)²=4(−1)n — hold exactly, Fibonacci and Lucas locked together.
Credit as content: Édouard Lucas (1870s). The weave: David names split-screen; I generate both sequences and confirm the companion identities — L(n)=F(n−1)+F(n+1) and L(n)²−5F(n)²=4(−1)n — hold exactly, Fibonacci and Lucas locked together.
3 ONE DIMENSION
L: 2, 1, 3, 4, 7, 11, 18, 29, … (same rule as Fibonacci, seeds 2,1). L(n) = F(n−1) + F(n+1). Ratio → φ. L(n)² − 5F(n)² = ±4.
4 TWO DIMENSIONS · INTERACTIVE
Lucas and Fibonacci side by side; the companion identities checked term by term.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: Fibonacci’s companion, locked by identity.
AVAN’s addition (the inverse-companion): pair the Fibonacci recurrence with a second seed (2, 1) to get a companion sequence bound to it by exact identities — L(n)=F(n−1)+F(n+1), L²−5F²=±4. The inverse of ‘Fibonacci alone from 0, 1’ is ‘its twin from 2, 1, tied to it forever.’ Magenta is Fibonacci by itself; green is the Lucas companion. Two sequences, one identity.
LIT Genuine Lucas numbers (Édouard Lucas, 1870s). Verified live with exact BigInt arithmetic: L(n)=L(n−1)+L(n−2) from seeds 2,1 (window.__lucas.recurrence), the identity L(n)=F(n−1)+F(n+1) (id1), and L(n)²−5F(n)²=4(−1)ⁿ (id2), all for n up to 80 (BigInt so the squared identity stays exact past 2⁵³).
FIG No framing: the Lucas and Fibonacci recurrences and the two companion identities run in-browser in exact big integers and agree. Honest note: the squared identity is checked in BigInt because F(n)² exceeds 2⁵³ for n past ~38. The AVAN inverse is honest — pairing the Fibonacci recurrence with the second seed (2,1) gives a companion sequence tied to it by exact identities; magenta is Fibonacci alone, green the Lucas companion. Two sequences, one identity.
FIG No framing: the Lucas and Fibonacci recurrences and the two companion identities run in-browser in exact big integers and agree. Honest note: the squared identity is checked in BigInt because F(n)² exceeds 2⁵³ for n past ~38. The AVAN inverse is honest — pairing the Fibonacci recurrence with the second seed (2,1) gives a companion sequence tied to it by exact identities; magenta is Fibonacci alone, green the Lucas companion. Two sequences, one identity.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN