feat(nav): the live YouTube search and Home's budget are named in the address

A YouTube search is `?yt=<term>` (+ `ytn` for a non-default count), so a reload, a link
or Back/Forward reopens it. It runs by itself only where it costs no quota: the free page
scrape, a search asked for in this document, or results still cached. An API search
reopened any other way shows its term and waits for "Search YouTube again".

Home's time budget is `hb`; the question's draw stays on the history entry.

An e2e spec opens every page's bookmark in a fresh browser context; CLAUDE.md and
docs/development.md name the module URL codec as the place for view state.

frontend/src/modules/home/module.ts

@@ -1,6 +1,7 @@
 import { Home } from "lucide-react";
 import { lazyWithPreload } from "../../lib/lazyWithPreload";
 import type { ModuleDef } from "../../lib/moduleDef";
+import { HOME_KEYS } from "../../lib/homeUrlState";
 
 // This page as the shell and the navigation know it (lib/moduleDef). Its page loads lazily.
 export const homeModule: ModuleDef = {
@@ -11,8 +12,8 @@ export const homeModule: ModuleDef = {
   icon: Home,
   Page: lazyWithPreload(() => import("./Home")),
   maxWidth: "80rem",
-  // Nothing of its own: the question is an answer, not a setting — the same question draws other
-  // cards once what you have watched moves on.
-  urlKeys: [],
+  // Its time budget, a setting. Not the question's draw: the same question draws other cards once
+  // what you have watched moves on (lib/homeUrlState).
+  urlKeys: HOME_KEYS,
   playsVideos: true,
 };