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/lib/analytics.ts

@@ -60,22 +60,77 @@ export function track(name: EventName, data?: Payload): void {
 export type NavigationSource =
   'search' | 'rankings' | 'compare' | 'detail' | 'place' | 'direct';
 
+/*
+ * The in-app trail.
+ *
+ * document.referrer is written by the browser 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 (usually nothing, or a search engine) for the whole
+ * session. Reading it to answer "which page did they come from" therefore
+ * returned 'direct' for essentially every in-app journey, including the one
+ * the location layer exists to produce.
+ *
+ * Verified on staging: /schools/brentwood, click a school, the URL becomes
+ * /school/… and document.referrer is still "".
+ *
+ * A module-level trail is the counterpart with exactly the right lifetime. 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.
+ */
+const TRAIL_LIMIT = 4;
+const trail: string[] = [];
+
+/** Record a path the user is now on. Called by RouteTrail on every route. */
+export function recordVisitedPath(path: string): void {
+  if (trail[trail.length - 1] === path) return;
+  trail.push(path);
+  if (trail.length > TRAIL_LIMIT) trail.shift();
+}
+
+/**
+ * The most recent path that is not the one being viewed.
+ *
+ * Skipping the current path rather than taking trail[length - 2] is what
+ * makes the answer independent of ordering: the trail is written by a
+ * layout-level effect and read by a page-level one, and React orders those by
+ * tree position — not a contract worth resting a measurement on. It also
+ * gives the right answer when the user goes back to a page they came from.
+ */
+function previousInAppPath(): string | null {
+  if (typeof window === 'undefined') return null;
+  const current = window.location.pathname;
+  for (let i = trail.length - 1; i >= 0; i -= 1) {
+    if (trail[i] !== current) return trail[i];
+  }
+  return null;
+}
+
+function classifyPath(p: string): NavigationSource {
+  if (p === '/' || p === '') return 'search';
+  if (p.startsWith('/rankings')) return 'rankings';
+  if (p.startsWith('/compare')) return 'compare';
+  // `/schools/` before `/school/`: they differ by one letter and mean
+  // different things — the location layer versus a single school. Checked
+  // first so the narrower-looking prefix cannot shadow it if either string
+  // is ever edited.
+  if (p.startsWith('/schools/')) return 'place';
+  if (p.startsWith('/school/')) return 'detail';
+  return 'direct';
+}
+
 export function getNavigationSource(): NavigationSource {
+  const internal = previousInAppPath();
+  if (internal) return classifyPath(internal);
+
+  // No trail means this is the first page of the document, so the referrer is
+  // the only witness — and an honest one.
   if (typeof window === 'undefined' || !document.referrer) return 'direct';
   try {
     const ref = new URL(document.referrer);
     if (ref.origin !== window.location.origin) return 'direct';
-    const p = ref.pathname;
-    if (p === '/' || p === '') return 'search';
-    if (p.startsWith('/rankings')) return 'rankings';
-    if (p.startsWith('/compare')) return 'compare';
-    // `/schools/` before `/school/`: they differ by one letter and mean
-    // different things — the location layer versus a single school. Checked
-    // first so the narrower-looking prefix cannot shadow it if either string
-    // is ever edited.
-    if (p.startsWith('/schools/')) return 'place';
-    if (p.startsWith('/school/')) return 'detail';
-    return 'direct';
+    return classifyPath(ref.pathname);
   } catch {
     return 'direct';
   }