diff --git a/isolated-contexts.bs b/isolated-contexts.bs
index 0e9365d..0e9afc2 100644
--- a/isolated-contexts.bs
+++ b/isolated-contexts.bs
@@ -388,20 +388,25 @@ the associated [=environment settings object/global object=].
 
 ### Integrity ### {#html-integrity}
 
-A [=browsing context group=] has an <dfn for="browsing context group" export>
-integrity origin</dfn>, which is an [=origin=] or `null`.
-
-A [=browsing context group=] has an <dfn for="browsing context group" export>
-integrity verification algorithm</dfn>, which is `null` or an
+An <dfn export>integrity verification algorithm</dfn> is an
 [=implementation-defined=] algorithm that accepts a [=request=] and a
-[=response=], and returns a [=boolean=]. A [=browsing context group=]'s
-[=integrity verification algorithm=] MUST be non-null if its
-[=integrity origin=] is non-null.
+[=response=], and returns a [=boolean=].
 
 Note: A typical [=integrity verification algorithm=] might verify that a
 response body hashes to an expected value, or that it originated from a known
 bundle of resources.
 
+A [=user agent=] holds an <dfn export>origin integrity verification map</dfn>,
+which is a [=map=] of [=tuple origins=] to
+[=integrity verification algorithms=].
+
+Note: How user agents populate the [=origin integrity verification map=] is
+outside the scope of this specification, which is focused on the properties
+needed to establish integrity and isolation.
+<a href="https://github.com/WICG/isolated-web-apps/">Isolated Web Apps</a>
+provide one possible implementation by basing this map on the set of installed
+Isolated Web Apps.
+
 ### Environment Settings Object properties ### {#html-environment-properties}
 
 <div algorithm="environment settings object mitigates injection">
@@ -427,20 +432,15 @@ these properties will not mutate during an environment's lifetime.
 <div algorithm="environment settings object is an isolated context">
 An [=environment settings object=] |environment| is an
 <dfn export>isolated context</dfn> if the following algorithm returns `true`:
-    1.  Let |browsing context group| be the [=browsing context group=] that 
-        |environment| belongs to.
     1.  If |environment| does not [=environment settings object/meaningfully
         mitigate injection attacks=], return `false`.
-    1.  If |environment|'s [=cross-origin isolated capability=] is not
-        [=concrete=], return `false`.
+    1.  If |environment|'s [=cross-origin isolated capability=] is
+        not [=concrete=], return `false`.
     1.  If |environment| does not [=environment settings object/mitigate UI
         Redressing attacks=], return `false`.
-    1.  If |browsing context group|'s [=browsing context group/integrity
-        origin=] is null, return `false`.
-    1.  Let |integrity origin| be |browsing context group|'s
-        [=browsing context group/integrity origin=].
-    1.  If |environment|'s [=origin=] is not [=same origin=] with |integrity
-        origin|, return `false`.
+    1.  Let |origin| be |environment|'s [=origin=].
+    1.  If the [=user agent=]'s [=origin integrity verification map=][|origin|]
+        does not [=map/exist=], return `false`.
     1.  Return `true`.
 </div>
 
@@ -459,24 +459,14 @@ and a [=response=] |response|, run these steps. Possible return values are
 <ol>
   <li>Let |client| be |request|'s [=request/client=].</li>
   <li>If |client| is `null`, return "`not applicable`".</li>
+  <li>Let |origin| be |request|'s [=request/origin=].</li>
   <li>
-    Let |browsing context group| be the [=browsing context group=] that
-    |client| belongs to.
+    If the [=user agent=]'s [=origin integrity verification map=][|origin|]
+    does not [=map/exist=], return "`not applicable`".
   </li>
   <li>
-    Let |integrity origin| be |browsing context group|'s [=integrity origin=].
-  </li>
-  <li>
-    Let |integrity verification algorithm| be |browsing context group|'s
-    [=integrity verification algorithm=].
-  </li>
-  <li>
-    If |integrity origin| or |integrity verification algorithm| are `null`,
-    return "`not applicable`".
-  </li>
-  <li>
-    If |request|'s [=request/origin=] is not [=same origin=] with |integrity
-    origin|, return "`not applicable`".
+    Let |integrity verification algorithm| be the [=user agent=]'s
+    [=origin integrity verification map=][|origin|].
   </li>
   <li>
     If |response|'s [=response/body=] is `null`, return "`invalid`".
@@ -612,8 +602,10 @@ after similarly handling [{{CrossOriginIsolated}}] (step 4 below).
 ## Storage ## {#monkey-storage}
 
 The [=obtain a storage key for non-storage purposes=] algorithm is extended to
-require double-keying on all storage within a [=browsing context group=]
-containing [=Isolated Contexts=].
+require double-keying on all storage belonging to an
+<a href="https://html.spec.whatwg.org/multipage/webappapis.html#environment">
+environment</a> with a [=top-level origin=] known by the [=user agent=] to have
+an [=integrity verification algorithm=].
 
 <div algorithm="obtain a storage key for non-storage purposes isolated context">
 To obtain a storage key for non-storage purposes, given an
@@ -628,13 +620,12 @@ environment</a> |environment|, run these steps:
     </li>
 
     <li><ins>
-      Let |integrity origin| be the [=browsing context group/integrity origin=]
-      of the [=browsing context group=] that |environment| belongs to.
+      Let |top-level origin| be |environment|'s [=top-level origin=].
     </ins></li>
-
     <li><ins>
-      If |integrity origin| is non-null, return a [=tuple=] consisting of
-      |integrity origin| and |origin|.
+      If the [=user agent=]'s [=origin integrity verification map=]
+      [|top-level origin|] [=map/exists=], return a [=tuple=] consisting of
+      |top-level origin| and |origin|.
     </ins></li>
 
     <li>