refactor: store selected week in a cookie, not the URL

The home page no longer reads ?week=YYYY-MM-DD from the URL. Selected week
lives in the tcWeek cookie, set via a server action that revalidates the
home page so the next render reflects it. The URL stays at "/" regardless
of which week the user is viewing.

WeekNav prev/next/today buttons (and the arrow-key bindings) call the
server action directly — no router.refresh dance, no client-side cookie
write. BackToCalendarLink drops its localStorage-based href reconstruction
and just links to "/" since the cookie already remembers the right week
across navigations.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

app/page.tsx

@@ -1,15 +1,14 @@
+import { cookies } from "next/headers";
 import { HomePageClient } from "@/components/HomePageClient";
 import { getTodayLocal, formatDateLocal } from "@/lib/env";
 
 const BACKEND_URL = process.env.BACKEND_URL ?? "http://localhost:3001";
 
-interface PageProps {
-  searchParams: Promise<{ week?: string }>;
-}
+const WEEK_COOKIE = "tcWeek";
 
-function getWeekStart(param: string | undefined): string {
-  if (param && /^\d{4}-\d{2}-\d{2}$/.test(param)) {
-    const d = new Date(param + "T00:00:00");
+function getWeekStart(saved: string | undefined): string {
+  if (saved && /^\d{4}-\d{2}-\d{2}$/.test(saved)) {
+    const d = new Date(saved + "T00:00:00");
     if (!isNaN(d.getTime())) {
       d.setDate(d.getDate() - d.getDay());
       return formatDateLocal(d);
@@ -21,9 +20,9 @@ function getWeekStart(param: string | undefined): string {
   return formatDateLocal(d);
 }
 
-export default async function HomePage({ searchParams }: PageProps) {
-  const params = await searchParams;
-  const weekStart = getWeekStart(params.week);
+export default async function HomePage() {
+  const saved = (await cookies()).get(WEEK_COOKIE)?.value;
+  const weekStart = getWeekStart(saved);
 
   const data = await fetch(
     `${BACKEND_URL}/api/calendar/week?start=${weekStart}`,