feat: add dag_engine app — CLI + web frontend for DAG execution (0007e)

Full DAG engine app with CLI subcommands (run, list, status, validate, server)
and React/Mantine web frontend. Uses net/http + embedded Vite build. SQLite
store for run history. Scheduler with cron_ticker for automated execution.
Compatible with existing dagu YAML format.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

apps/dag_engine/api.go

@@ -0,0 +1,47 @@
+package main
+
+import (
+	"io/fs"
+	"net/http"
+)
+
+// RegisterAPI sets up all HTTP routes on the given mux.
+func RegisterAPI(mux *http.ServeMux, executor *Executor, scheduler *Scheduler, frontendFS fs.FS) {
+	// API routes.
+	mux.HandleFunc("GET /api/dags", handleListDags(executor))
+	mux.HandleFunc("GET /api/dags/{name}", handleGetDag(executor))
+	mux.HandleFunc("POST /api/dags/{name}/run", handleRunDag(executor))
+
+	mux.HandleFunc("GET /api/runs", handleListRuns(executor))
+	mux.HandleFunc("GET /api/runs/{id}", handleGetRun(executor))
+
+	mux.HandleFunc("POST /api/scheduler/start", handleSchedulerStart(scheduler))
+	mux.HandleFunc("POST /api/scheduler/stop", handleSchedulerStop(scheduler))
+	mux.HandleFunc("GET /api/scheduler/status", handleSchedulerStatus(scheduler))
+
+	// Frontend SPA fallback.
+	if frontendFS != nil {
+		mux.Handle("/", spaHandler(frontendFS))
+	}
+}
+
+// spaHandler serves static files from the embedded FS, falling back to index.html
+// for unknown paths (SPA client-side routing).
+func spaHandler(fsys fs.FS) http.Handler {
+	fileServer := http.FileServer(http.FS(fsys))
+	return http.HandlerFunc(func(w http.ResponseWriter, r *http.Request) {
+		// Try to serve the file directly.
+		path := r.URL.Path
+		if path == "/" {
+			path = "index.html"
+		} else {
+			path = path[1:] // strip leading /
+		}
+
+		if _, err := fs.Stat(fsys, path); err != nil {
+			// File not found — serve index.html for SPA routing.
+			r.URL.Path = "/"
+		}
+		fileServer.ServeHTTP(w, r)
+	})
+}