diff --git a/private/pure-function.rkt b/private/pure-function.rkt
index 15cfe31..71736b7 100644
--- a/private/pure-function.rkt
+++ b/private/pure-function.rkt
@@ -9,6 +9,8 @@
          (prefix-in te: type-expander)
          phc-toolkit
          (for-syntax (rename-in racket/base [... …])
+                     racket/match
+                     syntax/modcollapse
                      racket/list
                      racket/syntax
                      racket/contract
@@ -62,13 +64,16 @@
 (define-for-syntax built-in-pure-functions-free-id-set
   (immutable-free-id-set
    (syntax->list
-    #'(+ - * / modulo add1 sub1;; …
+    #'(+ - * / modulo add1 sub1 =;; …
+         eq? eqv? equal? ;; TODO: equal? can still cause problems if the
+         ;; struct's prop:equal+hash is effectful.
          error
          format values
          promise/pure/maybe-stateful? promise/pure/stateless?
          ;; Does not have a type yet:
          ;; list*
-         cons car cdr list list? pair? length reverse ;; …
+         null cons car cdr list list? pair? null? length reverse ;; …
+         void
          vector-ref vector-immutable vector-length vector->list vector? ;; …
          hash-ref hash->list hash? ;; …
          set-member? set->list set? ;; …
@@ -80,7 +85,13 @@
          ))))
 
 (define-for-syntax (built-in-pure-function? id)
-  (free-id-set-member? built-in-pure-functions-free-id-set id))
+  (or (free-id-set-member? built-in-pure-functions-free-id-set id)
+      (match (identifier-binding id)
+        [(list (app collapse-module-path-index '(lib "racket/private/kw.rkt"))
+               'make-optional-keyword-procedure
+               _ _ _ _ _)
+         #t]
+        [_ #f])))
 
 (define-syntax (def-built-in-set stx)
   (syntax-case stx ()
@@ -299,18 +310,18 @@
 
     (define/with-syntax varref (datum->syntax self `(#%variable-reference)))
 
-    ;; Prevent the mutation of the cached copy, by making it a macro which
-    ;; rejects uses as the target of a set! .
     #`(let ()
         marked-as-unsafe ...
         (let ([free free] …)
+          ;; Prevent the mutation of the cached copy, by making it a macro which
+          ;; rejects uses as the target of a set! .
           (let-syntax ([free (make-no-set!-transformer #'free)] …)
             ;; The input should always be stateless
             (assert free (check-immutable/error varref 'stateless))
             …
             ;; The result must be pure too, otherwise it could (I
             ;; suppose) cause problems with occurrence typing, if a
-            ;; copy if mutated but not the other, and TR still
+            ;; copy is mutated but not the other, and TR still
             ;; expects them to be equal?
             ;; By construction, it should be immutable, except for functions
             ;; (which can hold internal state), but TR won't assume that when
@@ -368,8 +379,9 @@
      (quasisyntax/top-loc this-syntax
        (define name
          (declared-wrapper
-          (lam #,@(when-attr tvars #'(fa tvars)) args maybe-result-type …
-               (pure/? (let () body …))))))]))
+          (pure/?
+           (lam #,@(when-attr tvars #'(fa tvars)) args maybe-result-type …
+                (let () body …))))))]))
 
 (define-syntax define-pure/stateful (define-pure/impl 'stateful))
 (define-syntax define-pure/stateless (define-pure/impl 'stateless))
diff --git a/private/pure-safe.rkt b/private/pure-safe.rkt
index c42e884..bac87c2 100644
--- a/private/pure-safe.rkt
+++ b/private/pure-safe.rkt
@@ -17,8 +17,8 @@
  [make-promise/pure/stateful (∀ (a) (→ (→ a) (Promise a)))]
  [make-promise/pure/stateless (∀ (a) (→ (→ a) (Promise a)))])
 
-(define-syntax (delay/pure/stateful/unsafe stx)
-  (make-delayer stx #'make-promise/pure/stateful '()))
+(define-for-syntax (stx-e x)
+  (if (syntax? x) (syntax-e x) x))
 
 (define-syntax (delay/pure/stateless/unsafe stx)
   (make-delayer stx #'make-promise/pure/stateless '()))
@@ -27,10 +27,12 @@
   (syntax-parser
     [(_ e)
      (syntax/top-loc this-syntax
-       (delay/pure/stateful/unsafe (pure/stateful e)))]))
+       (make-promise/pure/stateful
+        (pure-thunk/stateful (λ () e))))]))
 
 (define-syntax delay/pure/stateless
   (syntax-parser
     [(_ e)
      (syntax/top-loc this-syntax
-       (delay/pure/stateless/unsafe (pure/stateless e)))]))
+       (make-promise/pure/stateless
+        (pure-thunk/stateless (λ () e))))]))
diff --git a/scribblings/delay-pure.scrbl b/scribblings/delay-pure.scrbl
index f71d135..138a8b8 100644
--- a/scribblings/delay-pure.scrbl
+++ b/scribblings/delay-pure.scrbl
@@ -46,6 +46,14 @@
  @racket[struct] accessors and predicates, and @racket[struct] constructors for
  immutable structures.
 
+ Note that the expressions can refer to variables mutated with @racket[set!]
+ by other code. Placing the expression in a lambda function and calling that
+ function twice may therefore yield different results, if other code mutates
+ some free variables between the two invocations. In order to produce a pure
+ thunk which caches its inputs (thereby shielding them from any mutation of the
+ external environment), use @racket[pure-thunk/stateless] and
+ @racket[pure-thunk/stateful] instead.
+
  The first form, @racket[pure/stateless], checks that once fully-expanded, the
  @racket[expression] does not contain uses of @racket[set!]. Since the free
  variables can never refer to stateful functions, this means that any function
diff --git a/test/test-external-mutation.rkt b/test/test-external-mutation.rkt
new file mode 100644
index 0000000..8207f9f
--- /dev/null
+++ b/test/test-external-mutation.rkt
@@ -0,0 +1,81 @@
+#lang typed/racket
+(require delay-pure
+         typed/rackunit)
+
+;; This file checks that externally mutating a free variable on which a pure
+;; function or promise depends does not affect the function's result.
+
+(check-equal? (let ([x 1])
+                (define d (delay/pure/stateful (add1 x)))
+                (list (begin (set! x -10) (force d))
+                      (begin (set! x -11) (force d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define d (delay/pure/stateless (add1 x)))
+                (list (begin (set! x -10) (force d))
+                      (begin (set! x -11) (force d))))
+              '(2 2))
+
+;; pure/stateless and pure/stateful do not protect the expression from
+;; external mutations, so we are not testing this case here.
+
+(check-equal? (let ([x 1])
+                (define d (pure-thunk/stateless (λ () (add1 x))))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define d (pure-thunk/stateless (λ () (add1 x)) #:check-result))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define d (pure-thunk/stateful (λ () (add1 x))))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define d (pure-thunk/stateful (λ () (add1 x)) #:check-result))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define-pure/stateless (d) (add1 x))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define-pure/stateless (d [opt : Number x]) (add1 opt))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define-pure/stateless (d #:kw [opt : Number x]) (add1 opt))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define-pure/stateful (d) (add1 x))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define-pure/stateful (d [opt : Number x]) (add1 opt))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
+
+(check-equal? (let ([x 1])
+                (define-pure/stateful (d #:kw [opt : Number x]) (add1 opt))
+                (list (begin (set! x -10) (d))
+                      (begin (set! x -11) (d))))
+              '(2 2))
\ No newline at end of file