diff --git a/lib/apl/runtime.sx b/lib/apl/runtime.sx
index e26dcf1f..8d26c5a9 100644
--- a/lib/apl/runtime.sx
+++ b/lib/apl/runtime.sx
@@ -996,3 +996,23 @@
               (get a :ravel)
               (get b :ravel)))
           (error "length error: shape mismatch"))))))
+
+(define
+  apl-outer
+  (fn
+    (f a b)
+    (let
+      ((a-shape (get a :shape))
+        (b-shape (get b :shape))
+        (a-ravel (get a :ravel))
+        (b-ravel (get b :ravel)))
+      (make-array
+        (append a-shape b-shape)
+        (flatten
+          (map
+            (fn
+              (x)
+              (map
+                (fn (y) (disclose (f (apl-scalar x) (apl-scalar y))))
+                b-ravel))
+            a-ravel))))))
diff --git a/lib/apl/tests/operators.sx b/lib/apl/tests/operators.sx
index 9c17911b..fbe07e1a 100644
--- a/lib/apl/tests/operators.sx
+++ b/lib/apl/tests/operators.sx
@@ -244,4 +244,95 @@
       apl-mul
       (make-array (list 2 2) (list 1 2 3 4))
       (make-array (list 2 2) (list 5 6 7 8))))
-  (list 5 12 21 32))
\ No newline at end of file
+  (list 5 12 21 32))
+
+(apl-test
+  "outer product mult table values"
+  (rv
+    (apl-outer
+      apl-mul
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3) (list 1 2 3))))
+  (list 1 2 3 2 4 6 3 6 9))
+
+(apl-test
+  "outer product mult table shape"
+  (sh
+    (apl-outer
+      apl-mul
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3) (list 1 2 3))))
+  (list 3 3))
+
+(apl-test
+  "outer product add table values"
+  (rv
+    (apl-outer
+      apl-add
+      (make-array (list 2) (list 1 2))
+      (make-array (list 3) (list 10 20 30))))
+  (list 11 21 31 12 22 32))
+
+(apl-test
+  "outer product add table shape"
+  (sh
+    (apl-outer
+      apl-add
+      (make-array (list 2) (list 1 2))
+      (make-array (list 3) (list 10 20 30))))
+  (list 2 3))
+
+(apl-test
+  "outer product scalar+vector shape"
+  (sh
+    (apl-outer apl-mul (apl-scalar 5) (make-array (list 3) (list 1 2 3))))
+  (list 3))
+
+(apl-test
+  "outer product scalar+vector values"
+  (rv
+    (apl-outer apl-mul (apl-scalar 5) (make-array (list 3) (list 1 2 3))))
+  (list 5 10 15))
+
+(apl-test
+  "outer product vector+scalar shape"
+  (sh
+    (apl-outer apl-mul (make-array (list 3) (list 1 2 3)) (apl-scalar 10)))
+  (list 3))
+
+(apl-test
+  "outer product scalar+scalar"
+  (rv (apl-outer apl-mul (apl-scalar 6) (apl-scalar 7)))
+  (list 42))
+
+(apl-test
+  "outer product scalar+scalar shape"
+  (sh (apl-outer apl-mul (apl-scalar 6) (apl-scalar 7)))
+  (list))
+
+(apl-test
+  "outer product equality identity matrix values"
+  (rv
+    (apl-outer
+      apl-eq
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3) (list 1 2 3))))
+  (list 1 0 0 0 1 0 0 0 1))
+
+(apl-test
+  "outer product matrix+vector rank doubling shape"
+  (sh
+    (apl-outer
+      apl-add
+      (make-array (list 2 2) (list 1 2 3 4))
+      (make-array (list 3) (list 10 20 30))))
+  (list 2 2 3))
+
+(apl-test
+  "outer product matrix+vector rank doubling values"
+  (rv
+    (apl-outer
+      apl-add
+      (make-array (list 2 2) (list 1 2 3 4))
+      (make-array (list 3) (list 10 20 30))))
+  (list 11 21 31 12 22 32 13 23 33 14 24 34))
\ No newline at end of file
diff --git a/plans/apl-on-sx.md b/plans/apl-on-sx.md
index 1140acef..291be996 100644
--- a/plans/apl-on-sx.md
+++ b/plans/apl-on-sx.md
@@ -76,7 +76,7 @@ Core mapping:
 - [x] Reduce `f/` (last axis), `f⌿` (first axis) — including `∧/`, `∨/`, `+/`, `×/`, `⌈/`, `⌊/`
 - [x] Scan `f\`, `f⍀`
 - [x] Each `f¨` — applies `f` to each scalar/element
-- [ ] Outer product `∘.f` — `1 2 3 ∘.× 1 2 3` ↦ multiplication table
+- [x] Outer product `∘.f` — `1 2 3 ∘.× 1 2 3` ↦ multiplication table
 - [ ] Inner product `f.g` — `+.×` is matrix multiply
 - [ ] Commute `f⍨` — `f⍨ x` ↔ `x f x`, `x f⍨ y` ↔ `y f x`
 - [ ] Compose `f∘g` — applies `g` first then `f`
@@ -118,6 +118,7 @@ data; format for string templating.
 
 _Newest first._
 
+- 2026-05-06: Phase 4 step 4 — outer product ∘.f (apl-outer); rank-doubling result shape = a-shape++b-shape; 151/151 tests
 - 2026-05-06: Phase 4 step 3 — each f¨ (monadic apl-each + dyadic apl-each-dyadic); scalar broadcast both sides; 139/139 tests
 - 2026-05-06: Phase 4 step 2 — scan f\ (last axis) + f⍀ (first axis); apl-scan/apl-scan-first; 125/125 tests
 - 2026-05-06: Phase 4 step 1 — reduce f/ (last axis) + f⌿ (first axis); apl-reduce/apl-reduce-first; 110/110 tests