diff --git a/doc/execution-semantics.md b/doc/execution-semantics.md
index 9047d55637..5228e3a5eb 100644
--- a/doc/execution-semantics.md
+++ b/doc/execution-semantics.md
@@ -1767,8 +1767,13 @@ A native run can accept its result and then lose its server to a hard kill
 before finalization. Its result stays leased to the controller of the earlier
 boot, which cannot release the lease the way a graceful drain does. Startup
 native recovery releases that lease at once when the lease belongs to a
-different boot and its recorded controller process has exited or its PID now
-names another process. Only the executor's lease carries that controller
+different boot in the same known kernel and PID namespace, and its recorded
+controller process has exited or its PID now names another process. The
+executor saves that namespace with its claim. Linux uses its boot UUID and PID
+namespace; macOS uses its kernel boot-session UUID. An unreadable or unsupported
+identity, a different namespace, or an older claim without this proof keeps the
+original lease expiry and does not probe the recorded PID. Only the executor's
+lease carries that controller
 identity; a finalizer or cleanup lease is never released this way and expires
 on its own short lease. The stored result is then
 finalized without another
diff --git a/server/src/__tests__/heartbeat-native-restart-suspension.test.ts b/server/src/__tests__/heartbeat-native-restart-suspension.test.ts
index ec813739bb..4b868efbe2 100644
--- a/server/src/__tests__/heartbeat-native-restart-suspension.test.ts
+++ b/server/src/__tests__/heartbeat-native-restart-suspension.test.ts
@@ -128,8 +128,12 @@ describeEmbeddedPostgres("heartbeat native recovery interrupted by graceful shut
     // saved when it reaches the provider session.
     const savedAttempts: number[] = [];
     const reachedProvider = async () => {
-      const [owner] = await f.db.select({ attempt: nativeRunFinalizations.attempt }).from(nativeRunFinalizations)
+      const [owner] = await f.db.select({ attempt: nativeRunFinalizations.attempt,
+        controllerHostHash: nativeRunFinalizations.controllerHostHash }).from(nativeRunFinalizations)
         .where(eq(nativeRunFinalizations.runId, f.runId));
+      const actualProcess = await controllerProcess.currentControllerProcess();
+      expect(actualProcess.hostHash).not.toBe("unknown");
+      expect(owner.controllerHostHash).toBe(actualProcess.hostHash);
       savedAttempts.push(owner.attempt);
       throw new Error("stop at provider dispatch");
     };