heartbeat: fix backgrounded exec notification path

- resolveHeartbeatReasonKind: add `exec:` prefix match so `exec:<id>:exit`
  reasons emitted by maybeNotifyOnExit are classified as "exec-event"
  instead of "other", enabling shouldInspectPendingEvents = true

- isExecCompletionEvent: extend to also match "exec completed",
  "exec failed", and "exec killed" (maybeNotifyOnExit event text),
  not just "exec finished" (emitExecSystemEvent / gateway path)

Without both fixes, backgrounded allowlisted commands triggered a
heartbeat that routed to the correct session but fell through to a
regular heartbeat prompt instead of the exec-event prompt.

src/infra/heartbeat-events-filter.test.ts

@@ -91,3 +91,30 @@ describe("heartbeat event classification", () => {
     expect(isCronSystemEvent(value)).toBe(expected);
   });
 });
+
+describe("isExecCompletionEvent", () => {
+  it("matches emitExecSystemEvent (gateway/node approval path) events", () => {
+    expect(isExecCompletionEvent("Exec finished (gateway id=g1, session=s1, code 0)")).toBe(true);
+    expect(isExecCompletionEvent("exec finished (node=n1, code 1)\nsome output")).toBe(true);
+  });
+
+  it("matches maybeNotifyOnExit (backgrounded allowlisted commands) events", () => {
+    expect(isExecCompletionEvent("Exec completed (abc12345, code 0) :: some output")).toBe(true);
+    expect(isExecCompletionEvent("Exec completed (abc12345, code 0)")).toBe(true);
+    expect(isExecCompletionEvent("Exec failed (abc12345, code 1) :: error text")).toBe(true);
+    expect(isExecCompletionEvent("Exec failed (abc12345, signal SIGTERM)")).toBe(true);
+    expect(isExecCompletionEvent("Exec killed (abc12345, signal SIGKILL)")).toBe(true);
+  });
+
+  it("is case-insensitive", () => {
+    expect(isExecCompletionEvent("EXEC COMPLETED (abc12345, code 0)")).toBe(true);
+    expect(isExecCompletionEvent("exec failed (abc12345, code 2)")).toBe(true);
+  });
+
+  it("does not match non-exec events", () => {
+    expect(isExecCompletionEvent("Exec running (gateway id=g1, session=s1, >5s): ls")).toBe(false);
+    expect(isExecCompletionEvent("Exec denied (gateway id=g1, reason): rm -rf /")).toBe(false);
+    expect(isExecCompletionEvent("Heartbeat wake")).toBe(false);
+    expect(isExecCompletionEvent("")).toBe(false);
+  });
+});

src/infra/heartbeat-events-filter.ts

@@ -84,7 +84,15 @@ function isHeartbeatNoiseEvent(evt: string): boolean {
 }
 
 export function isExecCompletionEvent(evt: string): boolean {
-  return evt.toLowerCase().includes("exec finished");
+  const lower = evt.toLowerCase();
+  // "exec finished" — emitExecSystemEvent (gateway/node approval path)
+  // "exec completed/failed/killed" — maybeNotifyOnExit (backgrounded allowlisted commands)
+  return (
+    lower.includes("exec finished") ||
+    lower.includes("exec completed") ||
+    lower.includes("exec failed") ||
+    lower.includes("exec killed")
+  );
 }
 
 // Returns true when a system event should be treated as real cron reminder content.

src/infra/heartbeat-reason.test.ts

@@ -20,6 +20,8 @@ describe("heartbeat-reason", () => {
     { value: "interval", expected: "interval" },
     { value: "manual", expected: "manual" },
     { value: "exec-event", expected: "exec-event" },
+    // maybeNotifyOnExit backgrounded exec path: exec:<sessionId>:exit
+    { value: "exec:abc12345abc12345:exit", expected: "exec-event" },
     { value: "wake", expected: "wake" },
     { value: "acp:spawn:stream", expected: "wake" },
     { value: "acp:spawn:", expected: "wake" },
@@ -36,6 +38,7 @@ describe("heartbeat-reason", () => {
 
   it.each([
     { value: "exec-event", expected: true },
+    { value: "exec:abc123:exit", expected: true },
     { value: "cron:job-1", expected: true },
     { value: "wake", expected: true },
     { value: "acp:spawn:stream", expected: true },
@@ -50,6 +53,7 @@ describe("heartbeat-reason", () => {
   it.each([
     { value: "manual", expected: true },
     { value: "exec-event", expected: true },
+    { value: "exec:abc123:exit", expected: true },
     { value: "hook:wake", expected: true },
     { value: "interval", expected: false },
     { value: "cron:job-1", expected: false },

src/infra/heartbeat-reason.ts

@@ -31,6 +31,10 @@ export function resolveHeartbeatReasonKind(reason?: string): HeartbeatReasonKind
   if (trimmed === "exec-event") {
     return "exec-event";
   }
+  // exec:<sessionId>:exit — emitted by maybeNotifyOnExit for backgrounded commands
+  if (trimmed.startsWith("exec:")) {
+    return "exec-event";
+  }
   if (trimmed === "wake") {
     return "wake";
   }