ocaml: phase 5.1 fib_doubling.ml baseline (Fibonacci by doubling, fib(40) = 102334155)

Uses the identities:
  F(2k)   = F(k) * (2 * F(k+1) - F(k))
  F(2k+1) = F(k)^2 + F(k+1)^2

to compute Fibonacci in O(log n) recursive depth instead of O(n).

  let rec fib_pair n =
    if n = 0 then (0, 1)
    else
      let (a, b) = fib_pair (n / 2) in
      let c = a * (2 * b - a) in
      let d = a * a + b * b in
      if n mod 2 = 0 then (c, d)
      else (d, c + d)

Each call returns the pair (F(n), F(n+1)). fib(40) = 102334155 fits
in JS safe-int (< 2^53). Tests tuple returns with let-tuple
destructuring + recursion on n / 2.

86 baseline programs total.

plans/ocaml-on-sx.md

@@ -407,6 +407,12 @@ _Newest first._
   binary search tree (`type 'a tree = Leaf | Node of 'a * 'a tree *
   'a tree`) with insert + in-order traversal. Tests parametric ADT,
   recursive match, List.append, List.fold_left.
+- 2026-05-09 Phase 5.1 — fib_doubling.ml baseline (Fibonacci by
+  doubling, fib(40) = 102334155). Uses the identity F(2k) = F(k) *
+  (2*F(k+1) - F(k)) and F(2k+1) = F(k)^2 + F(k+1)^2 to compute fib
+  in O(log n) recursive depth. Returns a tuple (F(n), F(n+1)) at
+  each step. fib(40) = 102334155 fits in JS safe-int (< 2^53). 86
+  baseline programs total.
 - 2026-05-09 Phase 5.1 — merge_two.ml baseline (merge two sorted
   lists, length*sum = 9*49 = 441). Standard two-finger merge with
   nested match-in-match. Used as a building block in merge_sort.ml