fix(doctor): detect missing git repo and improve daemon startup message (#890)

When not in a git repository:
- Daemon startup now shows clear message immediately instead of waiting
  5 seconds: "Note: No git repository initialized — running without
  background sync"
- Added new doctor check "Git Sync Setup" that explains the situation

Doctor check now shows three states:
1. No git repo → Warning with fix: "Run 'git init' to enable background sync"
2. Git repo, no sync-branch → OK with hint about team collaboration benefits
3. Git repo + sync-branch → OK, fully configured

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com>

cmd/bd/doctor/daemon_test.go

@@ -2,8 +2,11 @@ package doctor
 
 import (
 	"os"
+	"os/exec"
 	"path/filepath"
 	"testing"
+
+	"github.com/steveyegge/beads/internal/git"
 )
 
 func TestCheckDaemonStatus(t *testing.T) {
@@ -33,3 +36,71 @@ func TestCheckDaemonStatus(t *testing.T) {
 		}
 	})
 }
+
+func TestCheckGitSyncSetup(t *testing.T) {
+	t.Run("not in git repository", func(t *testing.T) {
+		tmpDir := t.TempDir()
+		oldDir, err := os.Getwd()
+		if err != nil {
+			t.Fatal(err)
+		}
+		defer func() {
+			_ = os.Chdir(oldDir)
+			git.ResetCaches()
+		}()
+
+		if err := os.Chdir(tmpDir); err != nil {
+			t.Fatal(err)
+		}
+		git.ResetCaches()
+
+		check := CheckGitSyncSetup(tmpDir)
+
+		if check.Status != StatusWarning {
+			t.Errorf("Status = %q, want %q", check.Status, StatusWarning)
+		}
+		if check.Name != "Git Sync Setup" {
+			t.Errorf("Name = %q, want %q", check.Name, "Git Sync Setup")
+		}
+		if check.Fix == "" {
+			t.Error("Expected Fix to contain instructions")
+		}
+	})
+
+	t.Run("in git repository without sync-branch", func(t *testing.T) {
+		tmpDir := t.TempDir()
+		oldDir, err := os.Getwd()
+		if err != nil {
+			t.Fatal(err)
+		}
+		defer func() {
+			_ = os.Chdir(oldDir)
+			git.ResetCaches()
+		}()
+
+		// Initialize git repo
+		cmd := exec.Command("git", "init")
+		cmd.Dir = tmpDir
+		if err := cmd.Run(); err != nil {
+			t.Fatalf("Failed to init git repo: %v", err)
+		}
+
+		if err := os.Chdir(tmpDir); err != nil {
+			t.Fatal(err)
+		}
+		git.ResetCaches()
+
+		check := CheckGitSyncSetup(tmpDir)
+
+		if check.Status != StatusOK {
+			t.Errorf("Status = %q, want %q", check.Status, StatusOK)
+		}
+		if check.Name != "Git Sync Setup" {
+			t.Errorf("Name = %q, want %q", check.Name, "Git Sync Setup")
+		}
+		// Should mention sync-branch not configured
+		if check.Detail == "" {
+			t.Error("Expected Detail to contain sync-branch hint")
+		}
+	})
+}