THE FOLD / RESPAWN / EVENT HORIZON / THE CASSINI
THE CASSINI
a Fibonacci determinant pinned at plus or minus one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Cassini’s identity pins the Fibonacci numbers to a razor’s edge. For every n, F(n-1)·F(n+1) - F(n)² = (-1)n. The product of the neighbours of F(n) misses F(n)² by exactly one, alternating sign forever. It is the determinant of the Fibonacci matrix: [[1,1],[1,0]]n = [[F(n+1), F(n)],[F(n), F(n-1)]], whose determinant is (-1)n because det[[1,1],[1,0]] = -1. The generalization, Catalan’s identity, reads F(n)² - F(n-r)F(n+r) = (-1)n-rF(r)². This near-miss is the secret behind the ‘missing square’ dissection puzzle, where an 8×8 square seems to rearrange into a 5×13 rectangle — off by one unit of area.
LIT verified live with exact BigInt: F(n-1)F(n+1) - F(n)² = (-1)n for n = 1 to 100, and Catalan’s F(n)² - F(n-r)F(n+r) = (-1)n-rF(r)² for a range of n, r (window.__cassini). FIG no framing; the Fibonacci numbers and both identities are computed independently in-browser.
LIT verified live with exact BigInt: F(n-1)F(n+1) - F(n)² = (-1)n for n = 1 to 100, and Catalan’s F(n)² - F(n-r)F(n+r) = (-1)n-rF(r)² for a range of n, r (window.__cassini). FIG no framing; the Fibonacci numbers and both identities are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at event-horizon — the respawn: however far the Fibonacci numbers run out, the determinant is pulled back to +1 or -1, never anything else. AVAN (AI) built the instrument: the exact Fibonacci sequence, Cassini’s identity, the matrix-determinant view, and Catalan’s generalization.
Credit as content: Jean-Dominique Cassini (1680); Eugène Catalan (generalization). The weave: David names the pinned determinant; I confirm F(n-1)F(n+1) - F(n)² = (-1)n.
Credit as content: Jean-Dominique Cassini (1680); Eugène Catalan (generalization). The weave: David names the pinned determinant; I confirm F(n-1)F(n+1) - F(n)² = (-1)n.
3 ONE DIMENSION
Three consecutive Fibonacci numbers: the product of the outer two vs the square of the middle — off by ±1.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; F(n−1)F(n+1) − F(n)² is checked to equal (−1)^n, and the matrix determinant confirms it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the determinant, forever pinned at +1 or −1.
AVAN’s addition (the inverse-companion): don’t compute the Fibonacci product directly — read it as a determinant. The inverse of ‘F(n-1)F(n+1) - F(n)²’ is ‘det of the n-th power of [[1,1],[1,0]], which is (-1)n’. Magenta are the three consecutive Fibonacci numbers; green is the ±1 they are pinned to. A runaway sequence held to a unit determinant.
LIT Genuine Cassini's identity (Jean-Dominique Cassini, 1680; Catalan generalization). Verified live with exact BigInt: F(n−1)F(n+1) − F(n)² = (−1)^n for n=1..100, and Catalan's F(n)² − F(n−r)F(n+r) = (−1)^{n−r}F(r)² for a range of n,r (window.__cassini.ok, .catOk).
FIG No framing; the Fibonacci numbers and both identities run independently in-browser. The AVAN inverse is honest — instead of computing the Fibonacci product directly, read it as a determinant: the inverse of 'F(n−1)F(n+1) − F(n)²' is 'det of the n-th power of [[1,1],[1,0]], which is (−1)^n'. Magenta are the three consecutive Fibonacci numbers; green is the ±1 they are pinned to. A runaway sequence held to a unit determinant.
FIG No framing; the Fibonacci numbers and both identities run independently in-browser. The AVAN inverse is honest — instead of computing the Fibonacci product directly, read it as a determinant: the inverse of 'F(n−1)F(n+1) − F(n)²' is 'det of the n-th power of [[1,1],[1,0]], which is (−1)^n'. Magenta are the three consecutive Fibonacci numbers; green is the ±1 they are pinned to. A runaway sequence held to a unit determinant.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN