Unify agent startup with Manager pattern

Refactors all agent startup paths (witness, refinery, crew, polecat) to use
a consistent Manager interface with Start(), Stop(), IsRunning(), and
SessionName() methods.

Includes:
- Witness manager with GUPP propulsion nudge for startup
- Refinery manager for engineer sessions
- Crew manager for worker agents
- Session/polecat manager updates
- claude_settings_check doctor check for settings validation
- Settings management consolidated from rig/manager.go
- Settings location moved outside source repos to prevent conflicts

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

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

internal/templates/templates.go

@@ -215,6 +215,32 @@ func CommandNames() ([]string, error) {
 	return names, nil
 }
 
+// CreateMayorCLAUDEmd creates the Mayor's CLAUDE.md file at the specified directory.
+// This is used by both gt install and gt doctor --fix.
+func CreateMayorCLAUDEmd(mayorDir, townRoot, townName, mayorSession, deaconSession string) error {
+	tmpl, err := New()
+	if err != nil {
+		return err
+	}
+
+	data := RoleData{
+		Role:          "mayor",
+		TownRoot:      townRoot,
+		TownName:      townName,
+		WorkDir:       mayorDir,
+		MayorSession:  mayorSession,
+		DeaconSession: deaconSession,
+	}
+
+	content, err := tmpl.RenderRole("mayor", data)
+	if err != nil {
+		return err
+	}
+
+	claudePath := filepath.Join(mayorDir, "CLAUDE.md")
+	return os.WriteFile(claudePath, []byte(content), 0644)
+}
+
 // HasCommands checks if a workspace has the .claude/commands/ directory provisioned.
 func HasCommands(workspacePath string) bool {
 	commandsDir := filepath.Join(workspacePath, ".claude", "commands")