6.4 /health reports MQTT status

Inject MqttPublisher into HealthController and surface IsConnected as
mqttConnected, replacing the hardcoded false.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

IMPLEMENTATION.md

@@ -164,7 +164,7 @@ Each type lives in `src/FrameProcessor/Domain/`. Tests in `tests/FrameProcessor.
 ### [x] 6.3 Wire into `FramesController`
 - After successful save, call `PublishAsync`; set `mqttPublished` accordingly in the response.
 
-### [ ] 6.4 `/health` reports MQTT status
+### [x] 6.4 `/health` reports MQTT status
 - Replace hardcoded `mqttConnected` with `MqttPublisher.IsConnected`.
 
 ### [ ] 6.5 Background retry queue

src/FrameProcessor/Controllers/HealthController.cs

@@ -1,3 +1,4 @@
+using FrameProcessor.Mqtt;
 using Microsoft.AspNetCore.Mvc;
 
 namespace FrameProcessor.Controllers;
@@ -6,6 +7,10 @@ namespace FrameProcessor.Controllers;
 [Route("health")]
 public class HealthController : ControllerBase
 {
+    private readonly MqttPublisher _mqtt;
+
+    public HealthController(MqttPublisher mqtt) => _mqtt = mqtt;
+
     [HttpGet]
-    public IActionResult Get() => Ok(new { status = "Healthy", mqttConnected = false });
+    public IActionResult Get() => Ok(new { status = "Healthy", mqttConnected = _mqtt.IsConnected });
 }