fix(places): phase links must stay in their own namespace

Every place page built its phase links as /schools/[slug]/[phase], the
shape that belongs to towns alone.

On an authority page that pointed into the town namespace. For 87 of the
151 authorities the target does not exist and the link 404s; for the
other 64 it resolves to the town of the same name — a different set of
schools, which is precisely the near-duplicate the two namespaces were
introduced to prevent. On an outcode page it 404s outright.

Two causes behind it, both a rule written twice and inherited by only
one of the places that needed it.

The authority phase route was in the spec and dropped by the plan, which
built the three bare routes and no fourth. The sitemap is generated from
the place registry, which was right about them all along, so 302
authority phase URLs have been submitted to Google and every one 404s.
Adding the route makes the sitemap true and serves a real query —
admissions are authority-run, so "primary schools in Kent" is how a
parent searches before they have settled on a town.

The outcode variants were the opposite: the registry computed phases for
outcodes although the spec gives them no route, and the sitemap knew to
skip them while the API did not. The registry now decides alone, and the
sitemap's duplicate of that rule is gone.

Also: an authority under the five-school threshold has no page, so the
API sends a null slug for it and the page names it without linking.
Two English authorities are in that position. It was unreachable in
today's data — verified across the EC and TR outcodes — but the thin
place redirect would have sent a reader to a 404 the year it isn't.

The e2e journey now walks every /schools link a page of each family
emits and requires a 200, which is the check that was missing.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_015mWQnpye9F299NVRCCSRvj

backend/app.py

@@ -222,10 +222,10 @@ def _place_sitemap_rows(kinds: tuple[str, ...]) -> list[str]:
         if p.kind not in kinds:
             continue
         rows.append(_url_element(BASE_URL + _place_url(p)))
-        # Outcodes carry no phase variants: nobody searches "primary schools
-        # in SW11", so the routes do not exist to submit.
-        if p.kind == "outcode":
-            continue
+        # Which phases a place publishes is the registry's decision alone —
+        # outcodes report none, because the spec gives them no phase route.
+        # Repeating that rule here was how the page and the sitemap came to
+        # disagree about which URLs exist.
         for phase in ("primary", "secondary"):
             if p.publishes_phase(phase):
                 rows.append(_url_element(f"{BASE_URL}{_place_url(p)}/{phase}"))
@@ -1200,7 +1200,8 @@ async def get_place(request: Request, kind: str, slug: str,
     if kind not in VALID_PLACE_KINDS:
         raise HTTPException(status_code=404, detail="No such place")
 
-    place = get_place_registry().get(f"{kind}:{slug}")
+    registry = get_place_registry()
+    place = registry.get(f"{kind}:{slug}")
     if place is None:
         raise HTTPException(status_code=404, detail="No such place")
 
@@ -1240,8 +1241,17 @@ async def get_place(request: Request, kind: str, slug: str,
                   # Every authority the place meaningfully sits in. SW19 is
                   # mostly Merton but partly Wandsworth; naming one asserts
                   # something false.
+                  #
+                  # The slug is null where that authority has no page of its
+                  # own: City of London and the Isles of Scilly hold fewer
+                  # schools than the threshold. Naming them is still right;
+                  # linking them would be a 404.
                   "authorities": [
-                      {"name": name, "slug": _slugify(name), "count": n}
+                      {"name": name,
+                       "slug": (_slugify(name)
+                                if f"authority:{_slugify(name)}" in registry
+                                else None),
+                       "count": n}
                       for name, n in place.authorities
                   ],
                   # Only phases that clear the threshold, so the page links