fix(analytics): the funnel source read a referrer that never changes

The staging E2E gate has been red since #132 merged (run 1064, and
1066 after it): "a school reached from a location page is attributed
to it, not to direct" expects `place`, receives `direct`.

#132 fixed a real bug — `/schools/` had no case and fell through to
`direct` — but the mechanism underneath it never worked.
getNavigationSource read document.referrer, which the browser writes
only when a *document* loads. Every internal navigation here is an App
Router soft navigation: history.pushState, no new document, so
document.referrer goes on naming whatever opened the tab for the whole
session.

Verified on staging: load /schools/brentwood, click a school, the URL
becomes /school/… and document.referrer is still "".

So `from` reported `direct` for essentially every in-app journey, not
just the ones through the location layer — search, rankings, compare
and detail were all being counted as "typed the URL". The unit suite
passed throughout because every case set document.referrer directly,
which only happens on a full page load.

The fix is a module-level trail written by RouteTrail, a render-nothing
client component in the root layout. Its lifetime is exactly right: it
survives soft navigation, and it dies on a real document load — which
is precisely when document.referrer becomes meaningful again, so the
two cover each other with no overlap.

Reading it skips entries equal to the current path rather than taking
the second-to-last. That makes the answer independent of whether the
layout effect or the page effect ran first — React orders those by
tree position, which is not a contract worth resting a measurement on
— and it gives the right answer both when the user returns to a page
they came from and on a hard load of a school page.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01FuPUioHpxtaiDNagQvjxyM

nextjs-app/components/RouteTrail.tsx

@@ -0,0 +1,27 @@
+/**
+ * Writes the in-app navigation trail that funnel attribution reads.
+ *
+ * Renders nothing. It exists because document.referrer cannot answer "which
+ * page did they come from" in an App Router app: a soft navigation creates no
+ * document, so the browser never updates it. See the trail comment in
+ * lib/analytics.ts.
+ *
+ * Mounted once in the root layout, so every route is recorded — including the
+ * ones that fire no event of their own, which are still somebody else's
+ * previous page.
+ */
+'use client';
+
+import { useEffect } from 'react';
+import { usePathname } from 'next/navigation';
+import { recordVisitedPath } from '@/lib/analytics';
+
+export function RouteTrail() {
+  const pathname = usePathname();
+
+  useEffect(() => {
+    recordVisitedPath(pathname);
+  }, [pathname]);
+
+  return null;
+}