refactor(router): split route registration into v1 package categorizations

- Created `internal/router/v1` subpackage
- Split routes into `admin.go`, `user.go`, `public.go`, and `custom.go`
- Cleaned up imports and helper registration functions from `router.go`
- Removed obsolete `internal/router/custom.go`

internal/router/v1/public.go

@@ -0,0 +1,39 @@
+// Copyright 2026 Arctel.net
+// SPDX-License-Identifier: Apache-2.0
+
+// Package v1 contains router registrations for API V1
+package v1
+
+import (
+	capApp "github.com/Rain-kl/Wavelet/internal/apps/cap"
+	publicconfig "github.com/Rain-kl/Wavelet/internal/apps/config"
+	"github.com/Rain-kl/Wavelet/internal/apps/health"
+	"github.com/gin-gonic/gin"
+)
+
+// RegisterPublicRoutes registers all public routes (captcha, health, config).
+func RegisterPublicRoutes(apiV1Router *gin.RouterGroup, apiGroup *gin.RouterGroup) {
+	// CAPTCHA
+	registerCaptchaRoutes(apiGroup)
+
+	// Health
+	apiGroup.GET("/health", health.Health)
+
+	// Config (public)
+	registerConfigRoutes(apiV1Router)
+}
+
+func registerCaptchaRoutes(apiGroup *gin.RouterGroup) {
+	capGroup := apiGroup.Group("/cap")
+	{
+		capGroup.POST("/challenge", capApp.Challenge)
+		capGroup.POST("/redeem", capApp.Redeem)
+	}
+}
+
+func registerConfigRoutes(apiV1Router *gin.RouterGroup) {
+	configRouter := apiV1Router.Group("/config")
+	{
+		configRouter.GET("/public", publicconfig.GetPublicConfig)
+	}
+}