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THE FIBONACCI MATRIX
Fibonacci as a matrix power
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Fibonacci Q-matrix turns the Fibonacci recurrence into a single matrix. Because Fn+1 = Fn + Fn-1, one step is multiplication by Q = [[1,1],[1,0]], so Qn = [[Fn+1, Fn], [Fn, Fn-1]]. That single fact gives Fibonacci numbers in O(log n) time by fast matrix exponentiation (repeated squaring), and it hands you identities for free: taking determinants of both sides gives Cassini’s identity, Fn-1Fn+1 - Fn2 = (-1)n, because det Q = -1 and determinants multiply.
LIT verified live (exact BigInt): for n up to 200, Qn by repeated squaring has exactly Fn and Fn+1 in the right entries, matching the direct recurrence, and its determinant equals (-1)n — Cassini’s identity (window.__fibmatrix). FIG no framing; the matrix power, the direct Fibonacci, and the determinant all run in-browser with arbitrary-precision integers.
LIT verified live (exact BigInt): for n up to 200, Qn by repeated squaring has exactly Fn and Fn+1 in the right entries, matching the direct recurrence, and its determinant equals (-1)n — Cassini’s identity (window.__fibmatrix). FIG no framing; the matrix power, the direct Fibonacci, and the determinant all run in-browser with arbitrary-precision integers.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — heavy exact-integer matrix arithmetic, the Fibonacci recurrence folded into one 2×2 whose powers are computed by repeated squaring, Fn for huge n in a few big multiplications. AVAN (AI) built the instrument: the BigInt matrix power, the direct Fibonacci, and the determinant/Cassini check.
Credit as content: the Q-matrix identity (folklore; Cassini 1680). The weave: David names the mainframe; I confirm Qn carries the Fibonacci numbers and its determinant is Cassini’s identity.
Credit as content: the Q-matrix identity (folklore; Cassini 1680). The weave: David names the mainframe; I confirm Qn carries the Fibonacci numbers and its determinant is Cassini’s identity.
3 ONE DIMENSION
Powers of Q = [[1,1],[1,0]] — each entry is a Fibonacci number; Qⁿ holds F_{n+1}, F_n, F_n, F_{n-1}.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; Qⁿ (by repeated squaring) is compared to the direct Fibonacci, and its determinant to (−1)ⁿ (Cassini).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Fibonacci number F_n living in the matrix power.
AVAN’s addition (the inverse-companion): don’t add n times — square log n times. The inverse of ‘compute Fn by stepping the recurrence’ is ‘raise Q to the n by repeated squaring’, giving Fn in O(log n) and Cassini’s identity from det Q = -1. Magenta is the matrix Q and its squarings; green is the Fibonacci number that falls out. A recurrence made a power.
LIT Genuine Fibonacci Q-matrix identity (folklore; Cassini's identity, 1680). Verified live with exact BigInt: for n≤200, Qⁿ=[[1,1],[1,0]]ⁿ by repeated squaring has F_n and F_{n+1} in the correct entries (matching the direct recurrence), and det Qⁿ=F_{n−1}F_{n+1}−F_n²=(−1)ⁿ, Cassini's identity; F_100=354224848179261915075 (window.__fibmatrix.matchOk, .cassiniOk, .f100).
FIG No framing; the matrix power, the direct Fibonacci, and the determinant all run in-browser with arbitrary-precision integers. The AVAN inverse is honest — instead of adding n times, square log n times: raise Q to the n by repeated squaring for F_n in O(log n), and Cassini's identity falls out of det Q=−1. Magenta is the matrix Q and its squarings; green is the Fibonacci number that falls out. A recurrence made a power.
FIG No framing; the matrix power, the direct Fibonacci, and the determinant all run in-browser with arbitrary-precision integers. The AVAN inverse is honest — instead of adding n times, square log n times: raise Q to the n by repeated squaring for F_n in O(log n), and Cassini's identity falls out of det Q=−1. Magenta is the matrix Q and its squarings; green is the Fibonacci number that falls out. A recurrence made a power.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN