Expose runtime activity in server diagnostics (#37486)

## What changed

- Add lifecycle-backed gauges for in-flight and queued app requests, pending server requests and mailbox messages, active turns, and live MCP connections.
- Register each gauge on first use and decrement it when the tracked work or resource is dropped.
- Document the new `server/diagnostics` gauge names and account for the diagnostics request itself in the response example and test.

GitOrigin-RevId: 4bb82a3918af0b502d149b5d761ad669b2083f39

codex-rs/core/src/tasks/mod.rs

@@ -8,6 +8,7 @@ use std::sync::Arc;
 use std::time::Duration;
 use std::time::Instant;
 
+use codex_diagnostics::Gauge;
 use codex_extension_api::ThreadIdleCause;
 use futures::future::BoxFuture;
 use tokio::select;
@@ -63,6 +64,7 @@ pub(crate) use user_shell::execute_user_shell_command;
 
 const GRACEFULL_INTERRUPTION_TIMEOUT_MS: u64 = 100;
 const TASK_COMPACT_METRIC: &str = "codex.task.compact";
+static ACTIVE_TURNS: Gauge = Gauge::new("core.turns.active");
 
 pub(crate) type SessionTaskResult = CodexResult<Option<String>>;
 
@@ -421,6 +423,7 @@ impl Session {
             cancellation_token,
             turn_context: Arc::clone(&turn_context),
             _agent_execution_guard: agent_execution_guard,
+            _diagnostics_guard: ACTIVE_TURNS.track(),
             _timer: timer,
         };
         turn.task = Some(running_task);