feat(queue): server-side PTY-inject + idle-drained follow-up queue (W2)

The walk-away primitive: queue follow-up prompts and let a session advance itself
while you're gone — "run the tests" drains, and when Claude next goes idle "open a
PR" drains. Injection reuses writeInput (byte-identical to a keystroke, broadcasts
to all mirrored devices); the byte-shuttle is untouched.

- POST/GET/DELETE /live-sessions/:id/queue — POST/DELETE behind requireAllowedOrigin
  + a per-IP rate limit (QUEUE_RATE_MAX=20/min) + SESSION_ID_RE; text non-empty,
  byte-capped (QUEUE_ITEM_MAX_BYTES=4096, 16kb body), count-capped (QUEUE_MAX_ITEMS=10).
- Bounded per-session queue in manager (enqueueFollowup/drainOne/clearQueue);
  new optional queueLength on LiveSessionInfo.
- Idle drain: the Stop/SessionEnd /hook branch scheduleDrain()s a debounced timer
  (QUEUE_SETTLE_MS=1500). It drains exactly one entry only if the session still
  exists, hasn't exited, produced no output during settle, and is idle — three
  guards (debounced single timer + pop-one + settle re-check) → one entry per idle.
- New additive ServerMessage {type:'queue',length} broadcasts the count to mirrors
  (⧗N badge); FE enqueue via the quick-reply editor + TabApp.enqueueToActive.

Injected bytes go verbatim to the PTY (never built into a shell command). Verified
independently: typecheck + build:web clean, 1737 tests pass (real-PTY integration
covers idle-drain / one-per-idle / settle-guard). Foundation for templated launches,
auto-continue, and issue-intake (docs/ROADMAP.md).

public/terminal-session.ts

@@ -115,6 +115,9 @@ export interface TerminalSessionOpts {
   /** Optional: fired when new statusLine telemetry arrives (B2). Single source of
    *  truth — T-tabs reads session.telemetry via the getter, not its own copy. */
   onTelemetry?: (telemetry: StatusTelemetry) => void
+  /** Optional: fired when the pending inject-queue depth changes (W2), so the tab
+   *  can render an "N queued" badge. Broadcast to every mirrored device. */
+  onQueue?: (length: number) => void
   /** Optional: fired on a pointerdown anywhere in this pane (split-grid focus).
    *  Lets TabApp move the focused-pane (activeIndex) to the clicked quadrant. */
   onFocus?: () => void
@@ -137,6 +140,7 @@ export class TerminalSession {
   private readonly onStatus: ((status: SessionStatus) => void) | undefined
   private readonly onClaudeStatus: ((status: ClaudeStatus, detail?: string) => void) | undefined
   private readonly onTelemetry: ((telemetry: StatusTelemetry) => void) | undefined
+  private readonly onQueue: ((length: number) => void) | undefined
   private readonly onFocus: (() => void) | undefined
   private readonly spawnCwd: string | undefined
   private readonly initialInput: string | undefined
@@ -147,6 +151,7 @@ export class TerminalSession {
   private pendingApprovalValue = false
   private pendingToolValue: string | undefined = undefined
   private telemetryValue: StatusTelemetry | null = null
+  private queueLengthValue = 0
   private pendingGateValue: PermissionGate | null = null
   // W1: bounded command/diff preview of the held tool, from the last pending
   // status frame. Null when no approval is held or the tool wasn't previewable.
@@ -179,6 +184,7 @@ export class TerminalSession {
     this.onStatus = opts.onStatus
     this.onClaudeStatus = opts.onClaudeStatus
     this.onTelemetry = opts.onTelemetry
+    this.onQueue = opts.onQueue
     this.onFocus = opts.onFocus
     this.spawnCwd = opts.cwd
     this.initialInput = opts.initialInput
@@ -255,6 +261,12 @@ export class TerminalSession {
     return this.telemetryValue
   }
 
+  /** W2: current pending inject-queue depth (0 when empty), from the last
+   *  `queue` frame. Every mirrored device sees the same value (broadcast). */
+  get queueLength(): number {
+    return this.queueLengthValue
+  }
+
   /** The gate kind from the last pending status frame (B4): 'plan' or 'tool'.
    *  Null when no approval is held or the last status had no gate. */
   get pendingGate(): PermissionGate | null {
@@ -393,6 +405,13 @@ export class TerminalSession {
         this.onTelemetry?.(msg.telemetry)
         break
       }
+      case 'queue': {
+        // W2: pending inject-queue depth changed — update the badge on every
+        // mirrored device (shared session).
+        this.queueLengthValue = msg.length
+        this.onQueue?.(msg.length)
+        break
+      }
       case 'exit': {
         this.setStatus('exited')
         const reason = msg.reason ? ` (${msg.reason})` : ''