live: AMI-fed live state, /api/live WebSocket, dashboard and status dots

LiveState loads extension states and registered devices from Asterisk on
every AMI connect and follows ExtensionStatus/ContactStatus; the hub sends
a snapshot then deltas batched every 250 ms (devices for admins only),
pings every 30 s and closes unauthenticated sockets with 4401. The web UI
keeps one reconnecting socket in a zustand store, shows a Reconnecting
banner, a Dashboard (online, on a call, telephony engine) and status dots.

Fix: Asterisk 24 has no 'pjsip reload'; reload with
'module reload res_pjsip.so'. AMI errors now carry the CLI output.

Milestone check, automated: call-test.ts registers two baresip phones and
holds a 30 s call with RTP both ways (pnpm test:asterisk); the recorded
session drives the live-state tests. scripts/e2e/m2-live.mjs runs api +
Asterisk + phones + browser on a private Docker network: a new extension
is live in Asterisk in ~1.3 s, the dashboard shows 2/2 online and red dots
during the call.

Task: M2-T6

packages/shared/src/index.ts

@@ -6,3 +6,4 @@ export * from './settings.ts';
 export * from './auth.ts';
 export * from './users.ts';
 export * from './extensions.ts';
+export * from './live.ts';

packages/shared/src/live.ts

@@ -0,0 +1,40 @@
+// SPDX-License-Identifier: AGPL-3.0-or-later
+// Live state shapes sent over the /api/live WebSocket (04 §5.1, §5.2).
+
+export type ExtensionState =
+  'idle' | 'inuse' | 'busy' | 'unavailable' | 'ringing' | 'onhold' | 'ringinuse';
+
+export type ContactLive = {
+  endpoint: string; // "101" or "101-web"
+  uri: string;
+  userAgent?: string;
+  status: 'reachable' | 'unreachable' | 'unknown';
+  rttMs?: number;
+};
+
+export type ExtensionLive = {
+  number: string;
+  name: string;
+  state: ExtensionState;
+  /** Admins only; users see presence without devices. */
+  contacts?: ContactLive[];
+  /** Any registered device (desk phone, app or browser). */
+  online: boolean;
+  dnd: boolean;
+};
+
+export type LiveSnapshot = {
+  extensions: ExtensionLive[];
+  ami: { connected: boolean };
+};
+
+/** Server → client messages. */
+export type LiveMessage =
+  | { type: 'snapshot'; data: LiveSnapshot }
+  | { type: 'extension.updated'; data: ExtensionLive }
+  | { type: 'extension.removed'; data: { number: string } }
+  | { type: 'ami.status'; data: { connected: boolean } }
+  | {
+      type: 'system.configApplied';
+      data: { ok: boolean; error: string | null; restartRequired: boolean };
+    };