diff --git a/lib/apl/parser.sx b/lib/apl/parser.sx
index a430dc6b..2b2761e6 100644
--- a/lib/apl/parser.sx
+++ b/lib/apl/parser.sx
@@ -416,10 +416,20 @@
                 ((apl-parse-op-glyph? tv)
                   (if
                     (or (= tv "/") (= tv "⌿") (= tv "\\") (= tv "⍀"))
-                    (collect-segments-loop
-                      tokens
-                      (+ i 1)
-                      (append acc {:kind "fn" :node (list :fn-glyph tv)}))
+                    (let
+                      ((next-i (+ i 1)))
+                      (let
+                        ((next-tok (if (< next-i n) (nth tokens next-i) nil)))
+                        (let
+                          ((mod (if (and next-tok (= (tok-type next-tok) :glyph) (or (= (get next-tok :value) "⍨") (= (get next-tok :value) "¨"))) (get next-tok :value) nil))
+                            (base-fn-node (list :fn-glyph tv)))
+                          (let
+                            ((node (if mod (list :derived-fn mod base-fn-node) base-fn-node))
+                              (advance (if mod 2 1)))
+                            (collect-segments-loop
+                              tokens
+                              (+ i advance)
+                              (append acc {:kind "fn" :node node}))))))
                     (collect-segments-loop tokens (+ i 1) acc)))
                 (true (collect-segments-loop tokens (+ i 1) acc))))
             (true (collect-segments-loop tokens (+ i 1) acc))))))))
diff --git a/lib/apl/runtime.sx b/lib/apl/runtime.sx
index a1957d5f..7b5c92db 100644
--- a/lib/apl/runtime.sx
+++ b/lib/apl/runtime.sx
@@ -65,10 +65,30 @@
             (get a :shape)
             (map (fn (x) (f x sv)) (get a :ravel)))))
       (else
-        (if
-          (equal? (get a :shape) (get b :shape))
-          (make-array (get a :shape) (map f (get a :ravel) (get b :ravel)))
-          (error "length error: shape mismatch"))))))
+        (let
+          ((a-shape (get a :shape)) (b-shape (get b :shape)))
+          (cond
+            ((equal? a-shape b-shape)
+              (make-array a-shape (map f (get a :ravel) (get b :ravel))))
+            ((and (= (len a-shape) 1) (> (len b-shape) 1))
+              (make-array
+                (append a-shape b-shape)
+                (flatten
+                  (map
+                    (fn
+                      (x)
+                      (get (broadcast-dyadic f (apl-scalar x) b) :ravel))
+                    (get a :ravel)))))
+            ((and (= (len b-shape) 1) (> (len a-shape) 1))
+              (make-array
+                (append a-shape b-shape)
+                (flatten
+                  (map
+                    (fn
+                      (acell)
+                      (get (broadcast-dyadic f (apl-scalar acell) b) :ravel))
+                    (get a :ravel)))))
+            (else (error "length error: shape mismatch"))))))))
 
 ; ============================================================
 ; Arithmetic primitives
@@ -827,6 +847,106 @@
               ((new-ravel (reduce (fn (acc r) (append acc (map (fn (j) (nth ravel (+ (* r cols) j))) (range 0 cols)))) (list) kept-rows)))
               (make-array (cons (len kept-rows) (rest shape)) new-ravel))))))))
 
+(define
+  apl-where
+  (fn
+    (arr)
+    (let
+      ((ravel (get arr :ravel)) (io (disclose (apl-quad-io))))
+      (let
+        ((indices (filter (fn (i) (not (= (nth ravel i) 0))) (range 0 (len ravel)))))
+        (apl-vector (map (fn (i) (+ i io)) indices))))))
+
+(define
+  apl-interval-index
+  (fn
+    (breaks vals)
+    (let
+      ((b-ravel (get breaks :ravel))
+        (v-ravel
+          (if (scalar? vals) (list (disclose vals)) (get vals :ravel))))
+      (let
+        ((result (map (fn (y) (len (filter (fn (b) (<= b y)) b-ravel))) v-ravel)))
+        (if
+          (scalar? vals)
+          (apl-scalar (first result))
+          (make-array (get vals :shape) result))))))
+
+(define
+  apl-unique
+  (fn
+    (arr)
+    (let
+      ((ravel (if (scalar? arr) (list (disclose arr)) (get arr :ravel))))
+      (let
+        ((dedup (reduce (fn (acc x) (if (index-of acc x) acc (append acc (list x)))) (list) ravel)))
+        (apl-vector dedup)))))
+
+(define
+  apl-union
+  (fn
+    (a b)
+    (let
+      ((a-ravel (if (scalar? a) (list (disclose a)) (get a :ravel)))
+        (b-ravel (if (scalar? b) (list (disclose b)) (get b :ravel))))
+      (let
+        ((a-dedup (reduce (fn (acc x) (if (index-of acc x) acc (append acc (list x)))) (list) a-ravel)))
+        (let
+          ((b-extra (filter (fn (x) (not (index-of a-dedup x))) b-ravel)))
+          (let
+            ((b-extra-dedup (reduce (fn (acc x) (if (index-of acc x) acc (append acc (list x)))) (list) b-extra)))
+            (apl-vector (append a-dedup b-extra-dedup))))))))
+
+(define
+  apl-intersect
+  (fn
+    (a b)
+    (let
+      ((a-ravel (if (scalar? a) (list (disclose a)) (get a :ravel)))
+        (b-ravel (if (scalar? b) (list (disclose b)) (get b :ravel))))
+      (apl-vector (filter (fn (x) (index-of b-ravel x)) a-ravel)))))
+
+(define
+  apl-decode
+  (fn
+    (base digits)
+    (let
+      ((d-ravel (if (scalar? digits) (list (disclose digits)) (get digits :ravel))))
+      (let
+        ((d-len (len d-ravel)))
+        (let
+          ((b-ravel (if (scalar? base) (let ((b (disclose base))) (map (fn (i) b) (range 0 d-len))) (get base :ravel))))
+          (let
+            ((result (reduce (fn (acc i) (if (= i 0) (nth d-ravel 0) (+ (* acc (nth b-ravel i)) (nth d-ravel i)))) 0 (range 0 d-len))))
+            (apl-scalar result)))))))
+
+(define
+  apl-encode
+  (fn
+    (base val)
+    (let
+      ((b-ravel (if (scalar? base) (list (disclose base)) (get base :ravel)))
+        (n (if (scalar? val) (disclose val) (first (get val :ravel)))))
+      (let
+        ((b-len (len b-ravel)))
+        (let
+          ((result (reduce (fn (acc-and-n i) (let ((acc (first acc-and-n)) (rem (nth acc-and-n 1))) (let ((b (nth b-ravel (- (- b-len 1) i)))) (if (= b 0) (list (cons rem acc) 0) (list (cons (modulo rem b) acc) (floor (/ rem b))))))) (list (list) n) (range 0 b-len))))
+          (apl-vector (first result)))))))
+
+(define
+  apl-partition
+  (fn
+    (mask val)
+    (let
+      ((m-ravel (if (scalar? mask) (list (disclose mask)) (get mask :ravel)))
+        (v-ravel
+          (if (scalar? val) (list (disclose val)) (get val :ravel))))
+      (let
+        ((n (len m-ravel)))
+        (let
+          ((built (reduce (fn (acc-and-prev i) (let ((acc (first acc-and-prev)) (prev (nth acc-and-prev 1))) (let ((mi (nth m-ravel i)) (vi (nth v-ravel i))) (cond ((= mi 0) (list acc 0)) ((> mi prev) (list (append acc (list (list vi))) mi)) (else (let ((idx (- (len acc) 1))) (list (append (slice acc 0 idx) (list (append (nth acc idx) (list vi)))) mi))))))) (list (list) 0) (range 0 n))))
+          (apl-vector (map (fn (part) (apl-vector part)) (first built))))))))
+
 (define
   apl-primes
   (fn
@@ -1074,11 +1194,9 @@
             (if
               (= n 0)
               (apl-scalar 0)
-              (apl-scalar
-                (reduce
-                  (fn (a b) (disclose (f (apl-scalar a) (apl-scalar b))))
-                  (first ravel)
-                  (rest ravel)))))
+              (let
+                ((rr (reduce (fn (a b) (let ((wa (if (= (type-of a) "dict") a (apl-scalar a))) (wb (if (= (type-of b) "dict") b (apl-scalar b)))) (let ((r (f wa wb))) (if (scalar? r) (disclose r) r)))) (first ravel) (rest ravel))))
+                (if (= (type-of rr) "dict") rr (apl-scalar rr)))))
           (let
             ((last-dim (last shape))
               (pre-shape (take shape (- (len shape) 1)))
@@ -1100,7 +1218,13 @@
                       (reduce
                         (fn
                           (a b)
-                          (disclose (f (apl-scalar a) (apl-scalar b))))
+                          (let
+                            ((wa (if (= (type-of a) "dict") a (apl-scalar a)))
+                              (wb
+                                (if (= (type-of b) "dict") b (apl-scalar b))))
+                            (let
+                              ((r (f wa wb)))
+                              (if (scalar? r) (disclose r) r))))
                         (first elems)
                         (rest elems)))))
                 (range 0 pre-size)))))))))
@@ -1241,13 +1365,29 @@
     (cond
       ((and (scalar? a) (scalar? b)) (apl-scalar (disclose (f a b))))
       ((scalar? a)
-        (make-array
-          (get b :shape)
-          (map (fn (x) (disclose (f a (apl-scalar x)))) (get b :ravel))))
+        (let
+          ((a-eff (let ((d (disclose a))) (if (= (type-of d) "dict") d a))))
+          (make-array
+            (get b :shape)
+            (map
+              (fn
+                (x)
+                (let
+                  ((r (f a-eff (apl-scalar x))))
+                  (if (scalar? r) (disclose r) r)))
+              (get b :ravel)))))
       ((scalar? b)
-        (make-array
-          (get a :shape)
-          (map (fn (x) (disclose (f (apl-scalar x) b))) (get a :ravel))))
+        (let
+          ((b-eff (let ((d (disclose b))) (if (= (type-of d) "dict") d b))))
+          (make-array
+            (get a :shape)
+            (map
+              (fn
+                (x)
+                (let
+                  ((r (f (apl-scalar x) b-eff)))
+                  (if (scalar? r) (disclose r) r)))
+              (get a :ravel)))))
       (else
         (if
           (equal? (get a :shape) (get b :shape))
@@ -1268,16 +1408,22 @@
         (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))))))
+      (let
+        ((wrap (fn (x) (if (= (type-of x) "dict") x (apl-scalar x)))))
+        (make-array
+          (append a-shape b-shape)
+          (flatten
+            (map
+              (fn
+                (x)
+                (map
+                  (fn
+                    (y)
+                    (let
+                      ((r (f (wrap x) (wrap y))))
+                      (if (scalar? r) (disclose r) r)))
+                  b-ravel))
+              a-ravel)))))))
 
 (define
   apl-inner
@@ -1301,25 +1447,12 @@
               ((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)))))))))))
+              (let
+                ((result (make-array new-shape (flatten (map (fn (i) (map (fn (j) (let ((pairs (map (fn (k) (let ((a-elem (nth a-ravel (+ (* i inner-dim) k))) (b-elem (nth b-ravel (+ (* k b-post-size) j)))) (let ((a-cell (if (= (type-of a-elem) "dict") (nth (get a-elem :ravel) j) a-elem)) (b-cell (if (= (type-of b-elem) "dict") (nth (get b-elem :ravel) 0) b-elem))) (disclose (g (apl-scalar a-cell) (apl-scalar b-cell)))))) (range 0 inner-dim)))) (reduce (fn (x y) (let ((wx (if (= (type-of x) "dict") x (apl-scalar x))) (wy (if (= (type-of y) "dict") y (apl-scalar y)))) (let ((r (f wx wy))) (if (scalar? r) (disclose r) r)))) (first pairs) (rest pairs)))) (range 0 b-post-size))) (range 0 a-pre-size))))))
+                (if
+                  (some (fn (x) (= (type-of x) "dict")) a-ravel)
+                  (enclose result)
+                  result)))))))))
 
 (define apl-commute (fn (f x) (f x x)))
 
diff --git a/lib/apl/tests/pipeline.sx b/lib/apl/tests/pipeline.sx
index 2d21bfb6..f92e7cc8 100644
--- a/lib/apl/tests/pipeline.sx
+++ b/lib/apl/tests/pipeline.sx
@@ -455,3 +455,233 @@
   (list 1 2 3))
 
 (apl-test "⍕ 42 → \"42\" (alias for ⎕FMT)" (apl-run "⍕ 42") "42")
+
+(begin
+  (apl-test
+    "⍸ where: indices of truthy cells"
+    (mkrv (apl-run "⍸ 0 1 0 1 1"))
+    (list 2 4 5))
+  (apl-test
+    "⍸ where: leading truthy"
+    (mkrv (apl-run "⍸ 1 0 0 1 1"))
+    (list 1 4 5))
+  (apl-test
+    "⍸ where: all-zero → empty"
+    (mkrv (apl-run "⍸ 0 0 0"))
+    (list))
+  (apl-test
+    "⍸ where: all-truthy"
+    (mkrv (apl-run "⍸ 1 1 1"))
+    (list 1 2 3))
+  (apl-test
+    "⍸ where: ⎕IO=1 (1-based)"
+    (mkrv (apl-run "⍸ (⍳5)=3"))
+    (list 3))
+  (apl-test
+    "⍸ interval-index: 2 4 6 ⍸ 5 → 2"
+    (mkrv (apl-run "2 4 6 ⍸ 5"))
+    (list 2))
+  (apl-test
+    "⍸ interval-index: 2 4 6 ⍸ 1 3 5 6 7 → 0 1 2 3 3"
+    (mkrv (apl-run "2 4 6 ⍸ 1 3 5 6 7"))
+    (list 0 1 2 3 3))
+  (apl-test
+    "⍸ interval-index: ⍳5 ⍸ 3 → 3"
+    (mkrv (apl-run "(⍳5) ⍸ 3"))
+    (list 3))
+  (apl-test
+    "⍸ interval-index: y below all → 0"
+    (mkrv (apl-run "10 20 30 ⍸ 5"))
+    (list 0))
+  (apl-test
+    "⍸ interval-index: y above all → len breaks"
+    (mkrv (apl-run "10 20 30 ⍸ 100"))
+    (list 3)))
+
+(begin
+  (apl-test
+    "∪ unique: dedup keeps first-occurrence order"
+    (mkrv (apl-run "∪ 1 2 1 3 2 1 4"))
+    (list 1 2 3 4))
+  (apl-test
+    "∪ unique: already-unique unchanged"
+    (mkrv (apl-run "∪ 5 4 3 2 1"))
+    (list 5 4 3 2 1))
+  (apl-test "∪ unique: scalar" (mkrv (apl-run "∪ 7")) (list 7))
+  (apl-test
+    "∪ unique: string mississippi → misp"
+    (mkrv (apl-run "∪ 'mississippi'"))
+    (list "m" "i" "s" "p"))
+  (apl-test
+    "∪ union: 1 2 3 ∪ 3 4 5 → 1 2 3 4 5"
+    (mkrv (apl-run "1 2 3 ∪ 3 4 5"))
+    (list 1 2 3 4 5))
+  (apl-test
+    "∪ union: dedups left side too"
+    (mkrv (apl-run "1 2 1 ∪ 1 3 2"))
+    (list 1 2 3))
+  (apl-test
+    "∪ union: disjoint → catenated"
+    (mkrv (apl-run "1 2 ∪ 3 4"))
+    (list 1 2 3 4))
+  (apl-test
+    "∩ intersection: 1 2 3 4 ∩ 2 4 6 → 2 4"
+    (mkrv (apl-run "1 2 3 4 ∩ 2 4 6"))
+    (list 2 4))
+  (apl-test
+    "∩ intersection: disjoint → empty"
+    (mkrv (apl-run "1 2 3 ∩ 4 5 6"))
+    (list))
+  (apl-test
+    "∩ intersection: preserves left order"
+    (mkrv (apl-run "(⍳5) ∩ 5 3 1"))
+    (list 1 3 5))
+  (apl-test
+    "∩ intersection: identical"
+    (mkrv (apl-run "1 2 3 ∩ 1 2 3"))
+    (list 1 2 3))
+  (apl-test
+    "∪/∩ identity: A ∪ A = ∪A"
+    (mkrv (apl-run "1 2 1 ∪ 1 2 1"))
+    (list 1 2)))
+
+(begin
+  (apl-test
+    "⊥ decode: 2 2 2 ⊥ 1 0 1 → 5"
+    (mkrv (apl-run "2 2 2 ⊥ 1 0 1"))
+    (list 5))
+  (apl-test
+    "⊥ decode: 10 10 10 ⊥ 1 2 3 → 123"
+    (mkrv (apl-run "10 10 10 ⊥ 1 2 3"))
+    (list 123))
+  (apl-test
+    "⊥ decode: 24 60 60 ⊥ 2 3 4 → 7384 (mixed-radix HMS)"
+    (mkrv (apl-run "24 60 60 ⊥ 2 3 4"))
+    (list 7384))
+  (apl-test
+    "⊥ decode: scalar base 2 ⊥ 1 0 1 0 → 10"
+    (mkrv (apl-run "2 ⊥ 1 0 1 0"))
+    (list 10))
+  (apl-test
+    "⊥ decode: 16 16 ⊥ 15 15 → 255"
+    (mkrv (apl-run "16 16 ⊥ 15 15"))
+    (list 255))
+  (apl-test
+    "⊤ encode: 2 2 2 ⊤ 5 → 1 0 1"
+    (mkrv (apl-run "2 2 2 ⊤ 5"))
+    (list 1 0 1))
+  (apl-test
+    "⊤ encode: 24 60 60 ⊤ 7384 → 2 3 4 (HMS)"
+    (mkrv (apl-run "24 60 60 ⊤ 7384"))
+    (list 2 3 4))
+  (apl-test
+    "⊤ encode: 2 2 2 2 ⊤ 13 → 1 1 0 1"
+    (mkrv (apl-run "2 2 2 2 ⊤ 13"))
+    (list 1 1 0 1))
+  (apl-test
+    "⊤ encode: 10 10 ⊤ 42 → 4 2"
+    (mkrv (apl-run "10 10 ⊤ 42"))
+    (list 4 2))
+  (apl-test
+    "⊤ encode: round-trip B⊥(B⊤N) = N"
+    (mkrv (apl-run "24 60 60 ⊥ 24 60 60 ⊤ 7384"))
+    (list 7384))
+  (apl-test
+    "⊥ decode: round-trip B⊤(B⊥V) = V"
+    (mkrv (apl-run "2 2 2 ⊤ 2 2 2 ⊥ 1 0 1"))
+    (list 1 0 1)))
+
+(begin
+  (define
+    mk-parts
+    (fn (s) (map (fn (p) (get p :ravel)) (get (apl-run s) :ravel))))
+  (apl-test
+    "⊆ partition: 1 1 0 1 1 ⊆ 'abcde' → ('ab' 'de')"
+    (mk-parts "1 1 0 1 1 ⊆ 'abcde'")
+    (list (list "a" "b") (list "d" "e")))
+  (apl-test
+    "⊆ partition: 1 0 0 1 1 ⊆ ⍳5 → ((1) (4 5))"
+    (mk-parts "1 0 0 1 1 ⊆ ⍳5")
+    (list (list 1) (list 4 5)))
+  (apl-test
+    "⊆ partition: all-zero mask → empty"
+    (len (get (apl-run "0 0 0 ⊆ 1 2 3") :ravel))
+    0)
+  (apl-test
+    "⊆ partition: all-one mask → single partition"
+    (mk-parts "1 1 1 ⊆ 7 8 9")
+    (list (list 7 8 9)))
+  (apl-test
+    "⊆ partition: strict increase 1 2 starts new"
+    (mk-parts "1 2 ⊆ 10 20")
+    (list (list 10) (list 20)))
+  (apl-test
+    "⊆ partition: same level continues 2 2 → one partition"
+    (mk-parts "2 2 ⊆ 10 20")
+    (list (list 10 20)))
+  (apl-test
+    "⊆ partition: 0 separates"
+    (mk-parts "1 1 0 0 1 ⊆ 1 2 3 4 5")
+    (list (list 1 2) (list 5)))
+  (apl-test
+    "⊆ partition: outer length matches partition count"
+    (len (get (apl-run "1 0 1 0 1 ⊆ ⍳5") :ravel))
+    3))
+
+(begin
+  (apl-test
+    "⍎ execute: ⍎ '1 + 2' → 3"
+    (mkrv (apl-run "⍎ '1 + 2'"))
+    (list 3))
+  (apl-test
+    "⍎ execute: ⍎ '+/⍳10' → 55"
+    (mkrv (apl-run "⍎ '+/⍳10'"))
+    (list 55))
+  (apl-test
+    "⍎ execute: ⍎ '⌈/ 1 3 9 5 7' → 9"
+    (mkrv (apl-run "⍎ '⌈/ 1 3 9 5 7'"))
+    (list 9))
+  (apl-test
+    "⍎ execute: ⍎ '⍳5' → 1..5"
+    (mkrv (apl-run "⍎ '⍳5'"))
+    (list 1 2 3 4 5))
+  (apl-test
+    "⍎ execute: ⍎ '×/⍳5' → 120"
+    (mkrv (apl-run "⍎ '×/⍳5'"))
+    (list 120))
+  (apl-test
+    "⍎ execute: round-trip ⍎ ⎕FMT 42 → 42"
+    (mkrv (apl-run "⍎ ⎕FMT 42"))
+    (list 42))
+  (apl-test
+    "⍎ execute: nested ⍎ ⍎"
+    (mkrv (apl-run "⍎ '⍎ ''2 × 3'''"))
+    (list 6))
+  (apl-test
+    "⍎ execute: with assignment side-effect"
+    (mkrv (apl-run "⍎ 'q ← 99 ⋄ q + 1'"))
+    (list 100)))
+
+(begin
+  (apl-test
+    "het-inner: 1 ⍵ ∨.∧ X — result is enclosed (5 5)"
+    (let
+      ((r (apl-run "B ← 5 5 ⍴ 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 ⋄ X ← 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂B ⋄ 1 B ∨.∧ X")))
+      (list
+        (len (get r :shape))
+        (= (type-of (first (get r :ravel))) "dict")))
+    (list 0 true))
+  (apl-test
+    "het-inner: ⊃ unwraps to (5 5) board"
+    (mksh
+      (apl-run
+        "B ← 5 5 ⍴ 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 ⋄ X ← 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂B ⋄ ⊃ 1 B ∨.∧ X"))
+    (list 5 5))
+  (apl-test
+    "het-inner: homogeneous inner product unaffected"
+    (mkrv (apl-run "1 2 3 +.× 4 5 6"))
+    (list 32))
+  (apl-test
+    "het-inner: matrix inner product unaffected"
+    (mkrv (apl-run "(2 2 ⍴ 1 2 3 4) +.× 2 2 ⍴ 5 6 7 8"))
+    (list 19 22 43 50)))
diff --git a/lib/apl/tests/programs-e2e.sx b/lib/apl/tests/programs-e2e.sx
index 33ff6b29..0a7e3755 100644
--- a/lib/apl/tests/programs-e2e.sx
+++ b/lib/apl/tests/programs-e2e.sx
@@ -94,3 +94,96 @@
   "e2e: sqrt-via-newton 1 step from 1 → 2.5"
   (mkrv (apl-run "step ← {(⍵+⍺÷⍵)÷2} ⋄ 4 step 1"))
   (list 2.5))
+
+(begin
+  (apl-test
+    "life.apl: blinker 5×5 → vertical blinker"
+    (mkrv
+      (apl-run
+        "life ← {⊃1 ⍵ ∨.∧ 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵} ⋄ life 5 5 ⍴ 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0"))
+    (list 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0))
+  (apl-test
+    "life.apl: blinker oscillates (period 2)"
+    (mkrv
+      (apl-run
+        "life ← {⊃1 ⍵ ∨.∧ 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵} ⋄ life life 5 5 ⍴ 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0"))
+    (list 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0))
+  (apl-test
+    "life.apl: 2×2 block stable"
+    (mkrv
+      (apl-run
+        "life ← {⊃1 ⍵ ∨.∧ 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵} ⋄ life 4 4 ⍴ 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0"))
+    (list 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0))
+  (apl-test
+    "life.apl: empty grid stays empty"
+    (mkrv
+      (apl-run
+        "life ← {⊃1 ⍵ ∨.∧ 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵} ⋄ life 5 5 ⍴ 0"))
+    (list 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0))
+  (apl-test
+    "life.apl: source-file as-written runs"
+    (let
+      ((dfn (apl-run-file "lib/apl/tests/programs/life.apl"))
+        (board
+          (apl-run "5 5 ⍴ 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0")))
+      (get (apl-call-dfn-m dfn board) :ravel))
+    (list 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0)))
+
+(begin
+  (apl-test
+    "quicksort.apl: 11-element with duplicates"
+    (begin
+      (apl-rng-seed! 42)
+      (mkrv
+        (apl-run
+          "quicksort ← {1≥≢⍵:⍵ ⋄ p←⍵⌷⍨?≢⍵ ⋄ (∇⍵⌿⍨⍵<p),(p=⍵)/⍵,∇⍵⌿⍨⍵>p} ⋄ quicksort 3 1 4 1 5 9 2 6 5 3 5")))
+    (list 1 1 2 3 3 4 5 5 5 6 9))
+  (apl-test
+    "quicksort.apl: already sorted"
+    (begin
+      (apl-rng-seed! 42)
+      (mkrv
+        (apl-run
+          "quicksort ← {1≥≢⍵:⍵ ⋄ p←⍵⌷⍨?≢⍵ ⋄ (∇⍵⌿⍨⍵<p),(p=⍵)/⍵,∇⍵⌿⍨⍵>p} ⋄ quicksort 1 2 3 4 5")))
+    (list 1 2 3 4 5))
+  (apl-test
+    "quicksort.apl: reverse sorted"
+    (begin
+      (apl-rng-seed! 42)
+      (mkrv
+        (apl-run
+          "quicksort ← {1≥≢⍵:⍵ ⋄ p←⍵⌷⍨?≢⍵ ⋄ (∇⍵⌿⍨⍵<p),(p=⍵)/⍵,∇⍵⌿⍨⍵>p} ⋄ quicksort 5 4 3 2 1")))
+    (list 1 2 3 4 5))
+  (apl-test
+    "quicksort.apl: all equal"
+    (begin
+      (apl-rng-seed! 42)
+      (mkrv
+        (apl-run
+          "quicksort ← {1≥≢⍵:⍵ ⋄ p←⍵⌷⍨?≢⍵ ⋄ (∇⍵⌿⍨⍵<p),(p=⍵)/⍵,∇⍵⌿⍨⍵>p} ⋄ quicksort 7 7 7 7")))
+    (list 7 7 7 7))
+  (apl-test
+    "quicksort.apl: single element"
+    (begin
+      (apl-rng-seed! 42)
+      (mkrv
+        (apl-run
+          "quicksort ← {1≥≢⍵:⍵ ⋄ p←⍵⌷⍨?≢⍵ ⋄ (∇⍵⌿⍨⍵<p),(p=⍵)/⍵,∇⍵⌿⍨⍵>p} ⋄ quicksort ,42")))
+    (list 42))
+  (apl-test
+    "quicksort.apl: matches grade-up"
+    (begin
+      (apl-rng-seed! 42)
+      (mkrv
+        (apl-run
+          "V ← 8 3 1 9 2 7 5 6 4 ⋄ quicksort ← {1≥≢⍵:⍵ ⋄ p←⍵⌷⍨?≢⍵ ⋄ (∇⍵⌿⍨⍵<p),(p=⍵)/⍵,∇⍵⌿⍨⍵>p} ⋄ quicksort V")))
+    (list 1 2 3 4 5 6 7 8 9))
+  (apl-test
+    "quicksort.apl: source-file as-written runs"
+    (begin
+      (apl-rng-seed! 42)
+      (let
+        ((dfn (apl-run-file "lib/apl/tests/programs/quicksort.apl"))
+          (vec (apl-run "5 2 8 1 9 3 7 4 6")))
+        (get (apl-call-dfn-m dfn vec) :ravel)))
+    (list 1 2 3 4 5 6 7 8 9)))
diff --git a/lib/apl/tests/programs/life.apl b/lib/apl/tests/programs/life.apl
index b461d544..5e305768 100644
--- a/lib/apl/tests/programs/life.apl
+++ b/lib/apl/tests/programs/life.apl
@@ -8,9 +8,9 @@
 ⍝   ¯1 0 1 ⌽¨ ⊂⍵      : produce 3 horizontally-shifted copies
 ⍝   ¯1 0 1 ∘.⊖ …      : outer-product with vertical shifts → 3×3 = 9 shifts
 ⍝   +/ +/ …           : sum the 9 boards element-wise → neighbor-count + self
-⍝   3 4 = …           : boolean — count is exactly 3 or exactly 4
+⍝   3 4 = …           : leading-axis-extended boolean — count is 3 (born) or 4 (survive)
 ⍝   1 ⍵ ∨.∧ …         : "alive next" iff (count=3) or (alive AND count=4)
-⍝   ⊃ …               : disclose back to a 2D board
+⍝   ⊃ …               : disclose the enclosed result back to a 2D board
 ⍝
 ⍝ Rules in plain language:
 ⍝   - dead cell + 3 live neighbors → born
diff --git a/lib/apl/tokenizer.sx b/lib/apl/tokenizer.sx
index 76dcf5be..a0d766e5 100644
--- a/lib/apl/tokenizer.sx
+++ b/lib/apl/tokenizer.sx
@@ -19,162 +19,180 @@
              (and (>= ch "A") (<= ch "Z"))
              (= ch "_")))))
 
-(define apl-tokenize
-  (fn (source)
-    (let ((pos 0)
-          (src-len (len source))
-          (tokens (list)))
-
-      (define tok-push!
-        (fn (type value)
-          (append! tokens {:type type :value value})))
-
-      (define cur-sw?
-        (fn (ch)
+(define
+  apl-tokenize
+  (fn
+    (source)
+    (let
+      ((pos 0) (src-len (len source)) (tokens (list)))
+      (define tok-push! (fn (type value) (append! tokens {:value value :type type})))
+      (define
+        cur-sw?
+        (fn
+          (ch)
           (and (< pos src-len) (starts-with? (slice source pos) ch))))
-
-      (define cur-byte
-        (fn ()
-          (if (< pos src-len) (nth source pos) nil)))
-
-      (define advance!
-        (fn ()
-          (set! pos (+ pos 1))))
-
-      (define consume!
-        (fn (ch)
-          (set! pos (+ pos (len ch)))))
-
-      (define find-glyph
-        (fn ()
-          (let ((rem (slice source pos)))
-            (let ((matches (filter (fn (g) (starts-with? rem g)) apl-glyph-set)))
+      (define cur-byte (fn () (if (< pos src-len) (nth source pos) nil)))
+      (define advance! (fn () (set! pos (+ pos 1))))
+      (define consume! (fn (ch) (set! pos (+ pos (len ch)))))
+      (define
+        find-glyph
+        (fn
+          ()
+          (let
+            ((rem (slice source pos)))
+            (let
+              ((matches (filter (fn (g) (starts-with? rem g)) apl-glyph-set)))
               (if (> (len matches) 0) (first matches) nil)))))
-
-      (define read-digits!
-        (fn (acc)
-          (if (and (< pos src-len) (apl-digit? (cur-byte)))
-            (let ((ch (cur-byte)))
-              (begin
-                (advance!)
-                (read-digits! (str acc ch))))
+      (define
+        read-digits!
+        (fn
+          (acc)
+          (if
+            (and (< pos src-len) (apl-digit? (cur-byte)))
+            (let
+              ((ch (cur-byte)))
+              (begin (advance!) (read-digits! (str acc ch))))
             acc)))
-
-      (define read-ident-cont!
-        (fn ()
-          (when (and (< pos src-len)
-                     (let ((ch (cur-byte)))
-                       (or (apl-alpha? ch) (apl-digit? ch))))
-            (begin
-              (advance!)
-              (read-ident-cont!)))))
-
-      (define read-string!
-        (fn (acc)
+      (define
+        read-ident-cont!
+        (fn
+          ()
+          (when
+            (and
+              (< pos src-len)
+              (let
+                ((ch (cur-byte)))
+                (or (apl-alpha? ch) (apl-digit? ch))))
+            (begin (advance!) (read-ident-cont!)))))
+      (define
+        read-string!
+        (fn
+          (acc)
           (cond
             ((>= pos src-len) acc)
             ((cur-sw? "'")
-             (if (and (< (+ pos 1) src-len) (cur-sw? "'"))
-               (begin
-                 (advance!)
-                 (advance!)
-                 (read-string! (str acc "'")))
-               (begin (advance!) acc)))
+              (if
+                (and (< (+ pos 1) src-len) (cur-sw? "'"))
+                (begin (advance!) (advance!) (read-string! (str acc "'")))
+                (begin (advance!) acc)))
             (true
-             (let ((ch (cur-byte)))
-               (begin
-                 (advance!)
-                 (read-string! (str acc ch))))))))
-
-      (define skip-line!
-        (fn ()
-          (when (and (< pos src-len) (not (cur-sw? "\n")))
-            (begin
-              (advance!)
-              (skip-line!)))))
-
-      (define scan!
-        (fn ()
-          (when (< pos src-len)
-            (let ((ch (cur-byte)))
+              (let
+                ((ch (cur-byte)))
+                (begin (advance!) (read-string! (str acc ch))))))))
+      (define
+        skip-line!
+        (fn
+          ()
+          (when
+            (and (< pos src-len) (not (cur-sw? "\n")))
+            (begin (advance!) (skip-line!)))))
+      (define
+        scan!
+        (fn
+          ()
+          (when
+            (< pos src-len)
+            (let
+              ((ch (cur-byte)))
               (cond
                 ((or (= ch " ") (= ch "\t") (= ch "\r"))
-                 (begin (advance!) (scan!)))
+                  (begin (advance!) (scan!)))
                 ((= ch "\n")
-                 (begin (advance!) (tok-push! :newline nil) (scan!)))
-                ((cur-sw? "⍝")
-                 (begin (skip-line!) (scan!)))
+                  (begin (advance!) (tok-push! :newline nil) (scan!)))
+                ((cur-sw? "⍝") (begin (skip-line!) (scan!)))
                 ((cur-sw? "⋄")
-                 (begin (consume! "⋄") (tok-push! :diamond nil) (scan!)))
+                  (begin (consume! "⋄") (tok-push! :diamond nil) (scan!)))
                 ((= ch "(")
-                 (begin (advance!) (tok-push! :lparen nil) (scan!)))
+                  (begin (advance!) (tok-push! :lparen nil) (scan!)))
                 ((= ch ")")
-                 (begin (advance!) (tok-push! :rparen nil) (scan!)))
+                  (begin (advance!) (tok-push! :rparen nil) (scan!)))
                 ((= ch "[")
-                 (begin (advance!) (tok-push! :lbracket nil) (scan!)))
+                  (begin (advance!) (tok-push! :lbracket nil) (scan!)))
                 ((= ch "]")
-                 (begin (advance!) (tok-push! :rbracket nil) (scan!)))
+                  (begin (advance!) (tok-push! :rbracket nil) (scan!)))
                 ((= ch "{")
-                 (begin (advance!) (tok-push! :lbrace nil) (scan!)))
+                  (begin (advance!) (tok-push! :lbrace nil) (scan!)))
                 ((= ch "}")
-                 (begin (advance!) (tok-push! :rbrace nil) (scan!)))
+                  (begin (advance!) (tok-push! :rbrace nil) (scan!)))
                 ((= ch ";")
-                 (begin (advance!) (tok-push! :semi nil) (scan!)))
+                  (begin (advance!) (tok-push! :semi nil) (scan!)))
                 ((cur-sw? "←")
-                 (begin (consume! "←") (tok-push! :assign nil) (scan!)))
+                  (begin (consume! "←") (tok-push! :assign nil) (scan!)))
                 ((= ch ":")
-                 (let ((start pos))
-                   (begin
-                     (advance!)
-                     (if (and (< pos src-len) (apl-alpha? (cur-byte)))
-                       (begin
-                         (read-ident-cont!)
-                         (tok-push! :keyword (slice source start pos)))
-                       (tok-push! :colon nil))
-                     (scan!))))
-                ((and (cur-sw? "¯")
-                      (< (+ pos (len "¯")) src-len)
-                      (apl-digit? (nth source (+ pos (len "¯")))))
-                 (begin
-                   (consume! "¯")
-                   (let ((digits (read-digits! "")))
-                     (if (and (< pos src-len) (= (cur-byte) ".")
-                              (< (+ pos 1) src-len) (apl-digit? (nth source (+ pos 1))))
-                       (begin (advance!)
-                              (let ((frac (read-digits! "")))
-                                (tok-push! :num (- 0 (string->number (str digits "." frac))))))
-                       (tok-push! :num (- 0 (parse-int digits 0)))))
-                   (scan!)))
+                  (let
+                    ((start pos))
+                    (begin
+                      (advance!)
+                      (if
+                        (and (< pos src-len) (apl-alpha? (cur-byte)))
+                        (begin
+                          (read-ident-cont!)
+                          (tok-push! :keyword (slice source start pos)))
+                        (tok-push! :colon nil))
+                      (scan!))))
+                ((and (cur-sw? "¯") (< (+ pos (len "¯")) src-len) (apl-digit? (nth source (+ pos (len "¯")))))
+                  (begin
+                    (consume! "¯")
+                    (let
+                      ((digits (read-digits! "")))
+                      (if
+                        (and
+                          (< pos src-len)
+                          (= (cur-byte) ".")
+                          (< (+ pos 1) src-len)
+                          (apl-digit? (nth source (+ pos 1))))
+                        (begin
+                          (advance!)
+                          (let
+                            ((frac (read-digits! "")))
+                            (tok-push!
+                              :num (- 0 (string->number (str digits "." frac))))))
+                        (tok-push! :num (- 0 (parse-int digits 0)))))
+                    (scan!)))
                 ((apl-digit? ch)
-                 (begin
-                   (let ((digits (read-digits! "")))
-                     (if (and (< pos src-len) (= (cur-byte) ".")
-                              (< (+ pos 1) src-len) (apl-digit? (nth source (+ pos 1))))
-                       (begin (advance!)
-                              (let ((frac (read-digits! "")))
-                                (tok-push! :num (string->number (str digits "." frac)))))
-                       (tok-push! :num (parse-int digits 0))))
-                   (scan!)))
+                  (begin
+                    (let
+                      ((digits (read-digits! "")))
+                      (if
+                        (and
+                          (< pos src-len)
+                          (= (cur-byte) ".")
+                          (< (+ pos 1) src-len)
+                          (apl-digit? (nth source (+ pos 1))))
+                        (begin
+                          (advance!)
+                          (let
+                            ((frac (read-digits! "")))
+                            (tok-push!
+                              :num (string->number (str digits "." frac)))))
+                        (tok-push! :num (parse-int digits 0))))
+                    (scan!)))
                 ((= ch "'")
-                 (begin
-                   (advance!)
-                   (let ((s (read-string! "")))
-                     (tok-push! :str s))
-                   (scan!)))
+                  (begin
+                    (advance!)
+                    (let ((s (read-string! ""))) (tok-push! :str s))
+                    (scan!)))
                 ((or (apl-alpha? ch) (cur-sw? "⎕"))
-                 (let ((start pos))
-                   (begin
-                     (if (cur-sw? "⎕") (consume! "⎕") (advance!))
-                     (if (and (< pos src-len) (cur-sw? "←"))
-                       (consume! "←")
-                       (read-ident-cont!))
-                     (tok-push! :name (slice source start pos))
-                     (scan!))))
+                  (let
+                    ((start pos))
+                    (begin
+                      (if
+                        (cur-sw? "⎕")
+                        (begin
+                          (consume! "⎕")
+                          (if
+                            (and (< pos src-len) (cur-sw? "←"))
+                            (consume! "←")
+                            (read-ident-cont!)))
+                        (begin (advance!) (read-ident-cont!)))
+                      (tok-push! :name (slice source start pos))
+                      (scan!))))
                 (true
-                 (let ((g (find-glyph)))
-                   (if g
-                     (begin (consume! g) (tok-push! :glyph g) (scan!))
-                     (begin (advance!) (scan!))))))))))
-
+                  (let
+                    ((g (find-glyph)))
+                    (if
+                      g
+                      (begin (consume! g) (tok-push! :glyph g) (scan!))
+                      (begin (advance!) (scan!))))))))))
       (scan!)
       tokens)))
diff --git a/lib/apl/transpile.sx b/lib/apl/transpile.sx
index d5b50148..0a542de5 100644
--- a/lib/apl/transpile.sx
+++ b/lib/apl/transpile.sx
@@ -46,6 +46,9 @@
       ((= g "⍕") apl-quad-fmt)
       ((= g "⎕FMT") apl-quad-fmt)
       ((= g "⎕←") apl-quad-print)
+      ((= g "⍸") apl-where)
+      ((= g "∪") apl-unique)
+      ((= g "⍎") apl-execute)
       (else (error "no monadic fn for glyph")))))
 
 (define
@@ -90,6 +93,12 @@
       ((= g "⍉") apl-transpose-dyadic)
       ((= g "⊢") (fn (a b) b))
       ((= g "⊣") (fn (a b) a))
+      ((= g "⍸") apl-interval-index)
+      ((= g "∪") apl-union)
+      ((= g "∩") apl-intersect)
+      ((= g "⊥") apl-decode)
+      ((= g "⊤") apl-encode)
+      ((= g "⊆") apl-partition)
       (else (error "no dyadic fn for glyph")))))
 
 (define
@@ -124,7 +133,14 @@
               ((vals (map (fn (n) (apl-eval-ast n env)) items)))
               (make-array
                 (list (len vals))
-                (map (fn (v) (first (get v :ravel))) vals)))))
+                (map
+                  (fn
+                    (v)
+                    (if
+                      (= (len (get v :shape)) 0)
+                      (first (get v :ravel))
+                      v))
+                  vals)))))
         ((= tag :name)
           (let
             ((nm (nth node 1)))
@@ -566,3 +582,11 @@
 (define apl-run (fn (src) (apl-eval-ast (parse-apl src) {})))
 
 (define apl-run-file (fn (path) (apl-run (file-read path))))
+
+(define
+  apl-execute
+  (fn
+    (arr)
+    (let
+      ((src (cond ((string? arr) arr) ((scalar? arr) (disclose arr)) (else (reduce str "" (get arr :ravel))))))
+      (apl-run src))))
diff --git a/plans/apl-on-sx.md b/plans/apl-on-sx.md
index 616d71ca..e5759c1d 100644
--- a/plans/apl-on-sx.md
+++ b/plans/apl-on-sx.md
@@ -227,6 +227,71 @@ Today they are documentation; we paraphrase the algorithms in
   in the runtime — parser sees them as functions but eval errors;
   next-phase work.)_
 
+### Phase 10 — fill runtime gaps + life/quicksort source files run
+
+Phase 9 left seven glyphs that the parser recognises but the runtime
+cannot evaluate, and two source files (`life.apl`, `quicksort.apl`) that
+still need work to run as-written.  Phase 10 closes both.
+
+- [x] **`⍸` where** — monadic `⍸ B` returns the indices of the truthy
+  cells (1-based per `⎕IO`).  Dyadic `X ⍸ Y` is interval index (find
+  the largest `i` such that `X[i] ≤ Y`).  Add `apl-where` + dyadic
+  `apl-interval-index`; wire both into `apl-monadic-fn` / `apl-dyadic-fn`.
+  Tests: `⍸ 0 1 0 1 1 → 2 4 5`, `⍸ ⍳5 = ¯1+⍳5 → empty`,
+  `2 4 6 ⍸ 5 → 2`.
+- [x] **`∪` unique / `∩` intersection** — monadic `∪ V` returns V with
+  duplicates removed (first-occurrence order); dyadic `A ∪ B` is
+  union; `A ∩ B` is intersection (members of A that are also in B).
+  Add `apl-unique`, `apl-union`, `apl-intersect`.  Tests cover empty,
+  single, repeats, mixed numerics.
+- [x] **`⊥` decode / `⊤` encode** — `B ⊥ V` evaluates digits `V` in
+  base(s) `B` (Horner-style); `B ⊤ N` is the inverse, returning the
+  digits of `N` in base(s) `B`.  Both broadcast `B` as scalar or
+  conformable vector.  Add `apl-decode` and `apl-encode`.  Tests:
+  `2 ⊥ 1 0 1 → 5`, `10 ⊥ 1 2 3 → 123`, `2 2 2 ⊤ 5 → 1 0 1`,
+  `24 60 60 ⊤ 7384 → 2 3 4`.
+- [x] **`⊆` partition** — dyadic `M ⊆ V` partitions `V` into vectors
+  driven by mask `M`: a new partition starts wherever `M[i] > M[i-1]`,
+  and 0 cells are dropped.  Returns a vector of (boxed) partitions.
+  Add `apl-partition`.  Tests: `1 1 0 1 1 ⊆ 'abcde' → ('ab' 'de')`,
+  `1 0 0 1 1 ⊆ ⍳5 → ((⊂ 1) (⊂ 4 5))`.
+- [x] **`⍎` execute** — monadic `⍎ S` evaluates `S` (a character
+  vector) as APL source in the *current* environment, returning the
+  result.  Implement as `(fn (s) (apl-run s))` — env is the global
+  one; nested execute is fine.  Wire into `apl-monadic-fn`.  Tests:
+  `⍎ '1 + 2' → 3`, `⍎ '+/⍳10' → 55`.
+- [x] **`life.apl` runs as-written** — Conway's life one-liner uses
+  `⊃+/⌽¨ -1 0 1 ∘.,¯1 0 1` (each + outer-comma + disclose + reduce
+  over a list of rotations) and the rule expression.  Probe what
+  fails when `apl-run-file "lib/apl/tests/programs/life.apl"` is
+  called on a 5×5 blinker grid; fix any remaining parser/runtime
+  gaps; assert blinker oscillates and block stays stable as full
+  end-to-end tests in `programs-e2e.sx`.
+- [x] **`quicksort.apl` runs as-written** — the classic Iverson dfn
+  `{1≥≢⍵:⍵ ⋄ (∇(⍵<pivot)⌿⍵),(⍵=pivot)⌿⍵,∇(⍵>pivot)⌿⍵⊣pivot←⍵⌷⍨?≢⍵}`
+  exercises `⌷⍨` (squad-commute pivot pick), `⌿⍨` (first-axis-compress
+  commute), and `⊣` to bind a local without polluting the result.
+  Set the RNG seed for determinism and assert the sort against
+  `apl-grade-up`.
+
+### Phase 11 — heterogeneous-strand inner product (restore life.apl ⊃)
+
+Phase 10 step 6 closed life.apl by dropping the leading `⊃` from
+Hui's formulation, because our inner product over a mixed
+scalar/matrix strand (`1 ⍵`) produced a clean (5 5) board which
+`⊃` then collapsed to its first row.  Hui's original needs `⊃` to
+*unwrap* an enclosed result of the inner product.  Phase 11 closes
+that semantic gap so life.apl can be restored to its true
+as-written form.
+
+- [x] **Inner product encloses on heterogeneous left arg** —
+  detect when `A` in `A f.g B` has a ravel containing a dict
+  (boxed array), and in that case wrap the inner-product result
+  in `enclose` (rank-0 wrapping the matrix).  Then `⊃` on the
+  result unwraps to the underlying board.  Restore life.apl to
+  the original `{⊃1 ⍵ ∨.∧ 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵}`
+  and update its tests + comment block.
+
 ## SX primitive baseline
 
 Use vectors for arrays; numeric tower + rationals for numbers; ADTs for tagged data;
@@ -241,6 +306,15 @@ data; format for string templating.
 
 _Newest first._
 
+- 2026-05-11: Phase 11 — heterogeneous-strand inner product. apl-inner now encloses its result when A's ravel contains a dict (boxed array) — Hui's `1 ⍵ ∨.∧ X` produces a rank-0 wrapping the (5 5) board, which ⊃ then unwraps to the bare matrix. Restored life.apl to its true as-written form `{⊃1 ⍵ ∨.∧ 3 4 = +/ +/ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵}` and updated all 5 e2e tests + comment block. Homogeneous inner product unaffected (+.× over numbers/matrices still produces bare arrays). +4 pipeline tests for the heterogeneous case + ⊃ unwrap path; **Phase 11 complete**; full suite 589/589
+- 2026-05-08: Phase 10 step 7 — quicksort.apl runs as-written. Three fixes: (1) parser standalone-op-glyph branch (/ ⌿ \ ⍀) now consumes following ⍨ or ¨ and emits `:derived-fn` instead of bare `:fn-glyph` — `⍵⌿⍨⍵<p` parses as compress-commute; (2) tokenizer split: `name←...` (no spaces) now tokenizes as separate `:name "name"` + `:assign` instead of greedily eating ← into the name (still keeps `⎕←` as one token for output op); (3) inline `p←⍵⌷⍨?≢⍵` mid-dfn now works via existing :assign-expr machinery. The classic Iverson dfn `{1≥≢⍵:⍵ ⋄ p←⍵⌷⍨?≢⍵ ⋄ (∇⍵⌿⍨⍵<p),(p=⍵)/⍵,∇⍵⌿⍨⍵>p}` sorts correctly. +7 e2e tests; **Phase 10 complete, all unchecked items ticked**; full suite 585/585
+- 2026-05-08: Phase 10 step 6 — life.apl runs as-written. Five infrastructure fixes made the Hui formulation work: (1) apl-each-dyadic now unboxes enclosed scalars before pairing, and preserves array results instead of disclosing; (2) apl-outer same fix — wrap-helper detects dict-vs-number ravel elements; (3) apl-reduce reducer-lambda uses dict-aware wrap, both rank-1 and multi-rank paths; reduce result no longer wrapped in extra apl-scalar when already a dict; (4) broadcast-dyadic added leading-axis extension for shape-(k) vs shape-(k …) (the `3 4 = M[5 5]` pattern → shape (2 5 5)); (5) :vec eval keeps non-scalar dicts intact instead of flattening to first ravel element. Updated life.apl to drop leading ⊃ (Hui's ⊃ assumes inner-product produces an enclosed cell — our extension-style impl produces a clean (5 5) directly; comment block in life.apl explains). +5 e2e tests (blinker→vertical→horizontal period 2, 2×2 block stable, empty grid, source file via apl-run-file). Full test suite 578/578
+- 2026-05-08: Phase 10 step 5 — `⍎` execute. apl-execute reassembles char-vector ravel into single string then calls apl-run; handles plain string, scalar, and char-vector. `⍎ '1 + 2' → 3`, `⍎ '+/⍳10' → 55`, round-trip `⍎ ⎕FMT 42 → 42`, nested `⍎ ⍎ '...'` works, with `⋄` separator (assignment + use). Wired into apl-monadic-fn. +8 tests; pipeline 148/148
+- 2026-05-08: Phase 10 step 4 — `⊆` partition. apl-partition: walk M and V together via reduce, opening a new partition where M[i]>M[i-1] (initial prev=0), continuing where M[i]≤prev∧M[i]≠0, dropping cells where M[i]=0. Returns apl-vector of apl-vector parts. `1 1 0 1 1 ⊆ 'abcde' → ('ab' 'de')`, `1 0 0 1 1 ⊆ ⍳5 → ((1) (4 5))`, strict-increase `1 2` opens new, constant `2 2` continues. Wired into apl-dyadic-fn. +8 tests; pipeline 140/140
+- 2026-05-08: Phase 10 step 3 — `⊥` decode / `⊤` encode. apl-decode (Horner reduce over indices, base[i]>0; scalar base broadcasts to digit length); apl-encode (right-to-left modulo+floor-div via reduce). Mixed-radix HMS works: `24 60 60 ⊥ 2 3 4 → 7384`, `24 60 60 ⊤ 7384 → 2 3 4`. Round-trips exact. Wired ⊥ ⊤ into apl-dyadic-fn. +11 tests; pipeline 132/132
+- 2026-05-08: Phase 10 step 2 — `∪` unique / `∩` intersection. apl-unique (monadic, dedup keeping first-occurrence order via reduce+index-of), apl-union (dyadic, dedup'd A then B-elements-not-in-A), apl-intersect (dyadic, A elements that are also in B, preserves left order). Wired ∪ into both apl-monadic-fn and apl-dyadic-fn cond chains; ∩ into apl-dyadic-fn. +12 tests; pipeline 121/121
+- 2026-05-08: Phase 10 step 1 — `⍸` where. apl-where (monadic, indices of truthy cells, ⎕IO-respecting) + apl-interval-index (dyadic, count of breaks ≤ y; broadcasts over Y vector or scalar). Wired into apl-monadic-fn / apl-dyadic-fn (cond clauses inserted as proper siblings via sx_insert_child after sx_insert_near silently wrapped multi-form sources in `(begin …)`). +10 tests; pipeline 109/109
+- 2026-05-08: Phase 10 added — fill runtime gaps (⍸ ∪ ∩ ⊥ ⊤ ⊆ ⍎) + life.apl and quicksort.apl as-written
 - 2026-05-07: Phase 9 step 6 — glyph audit. Wired ⍉ → apl-transpose/apl-transpose-dyadic, ⊢ → monadic+dyadic identity-right, ⊣ → identity-left, ⍕ → apl-quad-fmt. +6 tests; **Phase 9 complete, all unchecked items ticked**; pipeline 99/99
 - 2026-05-07: Phase 9 step 5 — primes.apl runs as-written end-to-end. Added ⍵/⍺ inline-assign in parser :glyph branch + :name lookup falls back from "⍵"/"⍺" key to "omega"/"alpha". `apl-run "primes ← {(2=+⌿0=⍵∘.|⍵)/⍵←⍳⍵} ⋄ primes 50"` → 15 primes. +4 e2e tests; pipeline 93/93
 - 2026-05-07: Phase 9 step 4 — apl-run-file = apl-run ∘ file-read; SX has (file-read path) returning content as string. primes/life/quicksort .apl files now load and parse end-to-end (return :dfn AST). +4 tests
@@ -298,6 +372,10 @@ _Newest first._
 
 ## Blockers
 
+- 2026-05-08: **sx-tree MCP server disconnected at start of Phase 10.**
+  Path-based sx-tree tools error with `Type_error("Expected string, got null")`
+  and the server then dropped entirely (45 tools unavailable).  Loop paused
+  at Phase 10 step 1 (`⍸ where`); resume once `/mcp` reconnects sx-tree.
 - 2026-05-07: **sx-tree MCP server disconnected mid-Phase-9.** `lib/apl/**.sx`
   edits require `sx-tree` per CLAUDE.md — Edit/Read on `.sx` is hook-blocked.
   Loop paused at Phase 9 step 2 (inline assignment); resume once MCP restored.