The Fibonacci sequence \(F_1=1, F_2=1, F_{{n}}=F_{{n-1}}+F_{{n-2}}\) produces ratios \(F_{{n+1}}/F_n\) that converge to \(\varphi\) but never equal it — \(\varphi\) is irrational, every ratio of integers is rational, so exact equality is structurally impossible. What is worth studying closely is how the approach happens:
This alternation is a direct consequence of the continued-fraction expansion of \(\varphi\): \(\varphi = [1;1,1,1,\ldots]\), all partial quotients equal to 1, which is the slowest-converging continued fraction possible for an irrational number (this is the formal sense in which \(\varphi\) is “the most irrational number”). Compare this to \(\eta\) (Tribonacci, Week 15 preview) where the analogous ratio converges faster, with a different error-decay profile.
Embedding this in contact geometry (rather than treating it as a pure number-theory fact): the \(\eta^{{-k}}\) amplitude-weighting scheme from Week 4 generalizes to any n-bonacci rung, including \(n=2\). The Fibonacci case gives weights \(w(k) = \varphi^{{-k}}\) — a slower decay per step than the \(\eta^{{-k}}\) weights used for the (critical, \(n=3\)) Tribonacci case, consistent with \(\varphi\) sitting further from the \(\tau=2\) threshold on the ladder than \(\eta\) does.
AXLE_v6.lean — theorem names and Lean identifiers above point at the actual source files.-- dm³ 101 · Week 10 · Fibonacci convergence rate def fib : ℕ → ℕ | 0 => 0 | 1 => 1 | (n+2) => fib n + fib (n+1) -- Ratios fib(n+1)/fib(n) as a real-valued sequence converge to φ, -- alternating above/below — a consequence of φ's continued fraction -- [1;1,1,1,...] having the slowest possible convergence among -- irrationals (Hurwitz's theorem territory). No AXLE Lean proof of -- the convergence rate itself yet — numeric verification only, -- shown in the table above. #eval (List.range 10).map (fun n => (fib (n+1), fib n))