ocaml: phase 5.1 polygon_area.ml baseline (pentagon 2x area = 32)

Shoelace formula on a pentagon with integer vertices:

  pts = [(0,0); (4,0); (4,3); (2,5); (0,3)]

  2 * area = | Σ (x_i * y_{i+1} - x_{i+1} * y_i) |
           = | 0*0 - 4*0 + 4*3 - 4*0 + 4*5 - 2*3 + 2*3 - 0*5
                + 0*0 - 0*3 |
           = 32

Returns the doubled form (32) to stay integral.

Tests:
  - let (x1, y1) = arr.(i) in   -- tuple destructure from array
  - arr.((i + 1) mod n)         -- modular wrap-around index
  - if a < 0 then - a else a    -- prefix - negation

162 baseline programs total.

plans/ocaml-on-sx.md

@@ -407,6 +407,15 @@ _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.
+- 2026-05-10 Phase 5.1 — polygon_area.ml baseline (shoelace formula
+  on pentagon, returns 2× area = 32). Vertices (0,0), (4,0), (4,3),
+  (2,5), (0,3); shoelace sum |Σ(x_i·y_{i+1} − x_{i+1}·y_i)| = 32 so
+  the area is 16, and we return the doubled form to stay integral.
+  Tests `let (x1, y1) = arr.(i) in` tuple destructuring from array
+  access (previously suspected broken in bfs_grid but works here —
+  the iter-242 bug was actually `&&` short-circuit, not destructure),
+  modular wrap-around `arr.((i+1) mod n)`, prefix-`-` negation.
+  162 baseline programs total.
 - 2026-05-10 Phase 5.1 — min_cost_path.ml baseline (min-cost path
   through 4×4 cost grid, top-left to bottom-right with moves
   right/down only, optimal = 12). Standard 2D DP: dp[i][j] = min