diff --git a/lib/apl/runtime.sx b/lib/apl/runtime.sx
index 8d26c5a9..19db9f8d 100644
--- a/lib/apl/runtime.sx
+++ b/lib/apl/runtime.sx
@@ -1016,3 +1016,45 @@
                 (fn (y) (disclose (f (apl-scalar x) (apl-scalar y))))
                 b-ravel))
             a-ravel))))))
+
+(define
+  apl-inner
+  (fn
+    (f g a b)
+    (let
+      ((a-shape (get a :shape))
+        (b-shape (get b :shape))
+        (a-ravel (get a :ravel))
+        (b-ravel (get b :ravel)))
+      (let
+        ((a-rank (len a-shape)) (b-rank (len b-shape)))
+        (if
+          (and (= a-rank 0) (= b-rank 0))
+          (apl-scalar (disclose (g a b)))
+          (let
+            ((inner-dim (last a-shape))
+              (a-pre (take a-shape (- a-rank 1)))
+              (b-post (rest b-shape)))
+            (let
+              ((a-pre-size (reduce * 1 a-pre))
+                (b-post-size (reduce * 1 b-post))
+                (new-shape (append a-pre b-post)))
+              (make-array
+                new-shape
+                (flatten
+                  (map
+                    (fn
+                      (i)
+                      (map
+                        (fn
+                          (j)
+                          (let
+                            ((pairs (map (fn (k) (disclose (g (apl-scalar (nth a-ravel (+ (* i inner-dim) k))) (apl-scalar (nth b-ravel (+ (* k b-post-size) j)))))) (range 0 inner-dim))))
+                            (reduce
+                              (fn
+                                (x y)
+                                (disclose (f (apl-scalar x) (apl-scalar y))))
+                              (first pairs)
+                              (rest pairs))))
+                        (range 0 b-post-size)))
+                    (range 0 a-pre-size)))))))))))
diff --git a/lib/apl/tests/operators.sx b/lib/apl/tests/operators.sx
index fbe07e1a..2e507811 100644
--- a/lib/apl/tests/operators.sx
+++ b/lib/apl/tests/operators.sx
@@ -335,4 +335,124 @@
       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
+  (list 11 21 31 12 22 32 13 23 33 14 24 34))
+
+(apl-test
+  "inner +.× dot product"
+  (rv
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3) (list 4 5 6))))
+  (list 32))
+
+(apl-test
+  "inner +.× dot product shape is scalar"
+  (sh
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3) (list 4 5 6))))
+  (list))
+
+(apl-test
+  "inner +.× matrix multiply 2x3 * 3x2 shape"
+  (sh
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 2 3) (list 1 2 3 4 5 6))
+      (make-array (list 3 2) (list 7 8 9 10 11 12))))
+  (list 2 2))
+
+(apl-test
+  "inner +.× matrix multiply 2x3 * 3x2 values"
+  (rv
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 2 3) (list 1 2 3 4 5 6))
+      (make-array (list 3 2) (list 7 8 9 10 11 12))))
+  (list 58 64 139 154))
+
+(apl-test
+  "inner +.× identity matrix 2x2"
+  (rv
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 2 2) (list 1 0 0 1))
+      (make-array (list 2 2) (list 5 6 7 8))))
+  (list 5 6 7 8))
+
+(apl-test
+  "inner ∧.= equal vectors"
+  (rv
+    (apl-inner
+      apl-and
+      apl-eq
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3) (list 1 2 3))))
+  (list 1))
+
+(apl-test
+  "inner ∧.= unequal vectors"
+  (rv
+    (apl-inner
+      apl-and
+      apl-eq
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3) (list 1 9 3))))
+  (list 0))
+
+(apl-test
+  "inner +.× matrix * vector shape"
+  (sh
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 2 3) (list 1 2 3 4 5 6))
+      (make-array (list 3) (list 7 8 9))))
+  (list 2))
+
+(apl-test
+  "inner +.× matrix * vector values"
+  (rv
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 2 3) (list 1 2 3 4 5 6))
+      (make-array (list 3) (list 7 8 9))))
+  (list 50 122))
+
+(apl-test
+  "inner +.× vector * matrix shape"
+  (sh
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3 2) (list 4 5 6 7 8 9))))
+  (list 2))
+
+(apl-test
+  "inner +.× vector * matrix values"
+  (rv
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 3) (list 1 2 3))
+      (make-array (list 3 2) (list 4 5 6 7 8 9))))
+  (list 40 46))
+
+(apl-test
+  "inner +.× single-element vectors"
+  (rv
+    (apl-inner
+      apl-add
+      apl-mul
+      (make-array (list 1) (list 6))
+      (make-array (list 1) (list 7))))
+  (list 42))
\ No newline at end of file
diff --git a/plans/apl-on-sx.md b/plans/apl-on-sx.md
index 291be996..08e68318 100644
--- a/plans/apl-on-sx.md
+++ b/plans/apl-on-sx.md
@@ -77,7 +77,7 @@ Core mapping:
 - [x] Scan `f\`, `f⍀`
 - [x] Each `f¨` — applies `f` to each scalar/element
 - [x] Outer product `∘.f` — `1 2 3 ∘.× 1 2 3` ↦ multiplication table
-- [ ] Inner product `f.g` — `+.×` is matrix multiply
+- [x] 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`
 - [ ] Power `f⍣n` — apply f n times; `f⍣≡` until fixed point
@@ -118,6 +118,7 @@ data; format for string templating.
 
 _Newest first._
 
+- 2026-05-06: Phase 4 step 5 — inner product f.g (apl-inner); +.× matrix multiply, ∧.= equal-vectors; 163/163 tests
 - 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