fix(conversation_store,admin,cli): keep session when file lags db, sort pty by recency

src/beaver_gateway/frontends/admin/frontend.py

@@ -567,13 +567,19 @@ def _collect_pty_sessions(runtime: GatewayRuntime) -> list[dict[str, Any]]:
     (currently only ``ClaudeCodeBackendAdapter``). Other backend types
     are quietly skipped — the admin terminal viewer only makes sense for
     PTY-backed agents.
+
+    Sorted most-recently-used first. The underlying mapping is keyed by
+    conversation fingerprint and reshuffled on every pool hit, so its
+    iteration order carries no meaning a human could act on.
     """
+    now = time.time()
     out: list[dict[str, Any]] = []
     for agent_name, backend in runtime.backends.items():
         live = getattr(backend, "live_sessions", None)
         if not isinstance(live, dict):
             continue
         for session_id, pty in live.items():
+            last_used = getattr(pty, "last_activity_at", None) or now
             out.append(
                 {
                     "agent": agent_name,
@@ -582,11 +588,28 @@ def _collect_pty_sessions(runtime: GatewayRuntime) -> list[dict[str, Any]]:
                     "buffer_size": len(pty.captured_output())
                     if hasattr(pty, "captured_output")
                     else 0,
+                    "last_used": last_used,
+                    "idle": _humanize_duration(now - last_used),
+                    "age": _humanize_duration(
+                        now - (getattr(pty, "created_at", None) or now)
+                    ),
                 }
             )
+    out.sort(key=lambda s: s["last_used"], reverse=True)
     return out
 
 
+def _humanize_duration(seconds: float) -> str:
+    """Render a duration as the coarsest unit that still reads precisely."""
+    secs = max(0, int(seconds))
+    if secs < 60:
+        return f"{secs}s"
+    if secs < 3600:
+        return f"{secs // 60}m {secs % 60}s"
+    hours, rem = divmod(secs, 3600)
+    return f"{hours}h {rem // 60}m"
+
+
 def _find_pty(runtime: GatewayRuntime, session_id: str) -> Any:
     """Locate a live PTY by its claude session_id, across all backends."""
     for backend in runtime.backends.values():

src/beaver_gateway/frontends/admin/templates/pty_list.html

@@ -4,8 +4,9 @@
 {% block content %}
 <h2>Live PTY sessions</h2>
 <p class="muted">
-  Each row is a running <code>claude</code> subprocess. Open one to see what's
-  currently rendered on its TUI and (if needed) type into it directly.
+  Each row is a running <code>claude</code> subprocess, most recently used first.
+  Open one to see what's currently rendered on its TUI and (if needed) type into
+  it directly. Sessions idle past the agent's TTL are terminated automatically.
 </p>
 
 {% if sessions %}
@@ -15,6 +16,8 @@
       <tr>
         <th>Session ID</th>
         <th>Agent</th>
+        <th>Idle</th>
+        <th>Age</th>
         <th>PID</th>
         <th>Buffer</th>
         <th></th>
@@ -25,6 +28,8 @@
       <tr>
         <td><code>{{ s.session_id }}</code></td>
         <td>{{ s.agent }}</td>
+        <td>{{ s.idle }}</td>
+        <td class="muted">{{ s.age }}</td>
         <td><span class="pill">{{ s.pid or "?" }}</span></td>
         <td>{{ s.buffer_size }} bytes</td>
         <td>