Expose app-server diagnostics through the experimental API (#37470)

## What changed

- Add the experimental `server/diagnostics` request, returning the app-server process ID, resident memory, platform-dependent physical footprint, and registered diagnostic gauges.
- Export the new protocol types in the generated JSON and TypeScript schemas and document the request.

## Testing

- Verify the response includes process measurements and the registered live-thread gauge.
- Verify the request is rejected unless `capabilities.experimentalApi` is enabled.

GitOrigin-RevId: 099d60a9c421a74ff1a72fe524e0dfeb89cf1057

codex-rs/app-server-protocol/schema/json/codex_app_server_protocol.v2.schemas.json

@@ -14076,6 +14076,52 @@
       "title": "SendAddCreditsNudgeEmailResponse",
       "type": "object"
     },
+    "ServerDiagnosticsGauge": {
+      "properties": {
+        "name": {
+          "type": "string"
+        },
+        "value": {
+          "format": "uint64",
+          "minimum": 0.0,
+          "type": "integer"
+        }
+      },
+      "required": [
+        "name",
+        "value"
+      ],
+      "type": "object"
+    },
+    "ServerDiagnosticsProcess": {
+      "properties": {
+        "id": {
+          "format": "uint32",
+          "minimum": 0.0,
+          "type": "integer"
+        },
+        "physicalFootprintBytes": {
+          "format": "uint64",
+          "minimum": 0.0,
+          "type": [
+            "integer",
+            "null"
+          ]
+        },
+        "residentMemoryBytes": {
+          "format": "uint64",
+          "minimum": 0.0,
+          "type": [
+            "integer",
+            "null"
+          ]
+        }
+      },
+      "required": [
+        "id"
+      ],
+      "type": "object"
+    },
     "ServerNotification": {
       "$schema": "http://json-schema.org/draft-07/schema#",
       "description": "Notification sent from the server to the client.",