feat(app-server): add remote control pairing status RPC (#26450)

## What

Exposes the pairing status transport as experimental app-server v2 RPC
`remoteControl/pairing/status`.

- Adds request/response protocol types for exactly one lookup key:
`pairingCode` or `manualPairingCode`, returning `{ claimed }`.
- Registers the RPC with `global_shared_read("remote-control-pairing")`.
- Wires the method through `MessageProcessor` and
`RemoteControlRequestProcessor`.
- Validates missing/conflicting pairing-code params as invalid requests.
- Documents the RPC in `app-server/README.md`.
- Adds processor, protocol export, and JSON-RPC integration coverage for
both code paths.

## Why

This is the app-server surface the desktop app can poll while the
QR/manual pairing modal is active.

Depends on https://github.com/openai/codex/pull/26449
Related backend change: https://github.com/openai/openai/pull/990244

## Verification

- `cargo test --manifest-path app-server-protocol/Cargo.toml
remote_control`
- `cargo test --manifest-path app-server/Cargo.toml remote_control`
- `cargo fmt --all --check`
- `git diff --check`

codex-rs/app-server-protocol/src/export.rs

@@ -2963,11 +2963,14 @@ permissionProfile?: string | null};
 
         let client_request_json = fs::read_to_string(output_dir.join("ClientRequest.json"))?;
         assert!(client_request_json.contains("remoteControl/pairing/start"));
+        assert!(client_request_json.contains("remoteControl/pairing/status"));
         assert!(client_request_json.contains("remoteControl/client/list"));
         assert!(client_request_json.contains("remoteControl/client/revoke"));
         for schema in [
             "RemoteControlPairingStartParams.json",
             "RemoteControlPairingStartResponse.json",
+            "RemoteControlPairingStatusParams.json",
+            "RemoteControlPairingStatusResponse.json",
             "RemoteControlClientsListParams.json",
             "RemoteControlClientsListResponse.json",
             "RemoteControlClientsRevokeParams.json",

codex-rs/app-server-protocol/src/protocol/common.rs

@@ -849,6 +849,12 @@ client_request_definitions! {
         serialization: global("remote-control-pairing"),
         response: v2::RemoteControlPairingStartResponse,
     },
+    #[experimental("remoteControl/pairing/status")]
+    RemoteControlPairingStatus => "remoteControl/pairing/status" {
+        params: v2::RemoteControlPairingStatusParams,
+        serialization: global_shared_read("remote-control-pairing"),
+        response: v2::RemoteControlPairingStatusResponse,
+    },
     #[experimental("remoteControl/client/list")]
     RemoteControlClientsList => "remoteControl/client/list" {
         params: v2::RemoteControlClientsListParams,
@@ -2014,6 +2020,19 @@ mod tests {
                 "remote-control-pairing"
             ))
         );
+        let remote_control_pairing_status = ClientRequest::RemoteControlPairingStatus {
+            request_id: request_id(),
+            params: v2::RemoteControlPairingStatusParams {
+                pairing_code: Some("pairing-code".to_string()),
+                manual_pairing_code: None,
+            },
+        };
+        assert_eq!(
+            remote_control_pairing_status.serialization_scope(),
+            Some(ClientRequestSerializationScope::GlobalSharedRead(
+                "remote-control-pairing"
+            ))
+        );
         let remote_control_clients_list = ClientRequest::RemoteControlClientsList {
             request_id: request_id(),
             params: v2::RemoteControlClientsListParams::default(),