fix(auth): harden unified diagnostic execution

tools/tht/internal/service/service.go

@@ -97,6 +97,20 @@ func CoreRunning(value string) (bool, error) {
 	return false, nil
 }
 
+// CoreHealthy reports whether core is both running and certified healthy by Compose.
+func CoreHealthy(value string) (bool, error) {
+	statuses, err := parseStatuses(value)
+	if err != nil {
+		return false, err
+	}
+	for _, item := range statuses {
+		if item.Service == "core" {
+			return strings.EqualFold(item.State, "running") && strings.EqualFold(item.Health, "healthy"), nil
+		}
+	}
+	return false, nil
+}
+
 // Healthy verifies the full expected Compose service set, including the one-shot model initializer.
 func Healthy(value string) error {
 	statuses, err := parseStatuses(value)