fix(compare): give every basket entry a real phase

Review of the per-phase limit found entries reaching the basket with no
phase, and a phase-less entry holds a slot in both groups:

- Nearby-school cards added without one. The API now returns each
  peer's own phase (its pool is a phase group, so an all-through school
  can sit beside a primary); the button passes it through, and an older
  API simply leaves the conservative both-groups count in place.
- Baskets saved before this change were never migrated. The compare
  page now backfills missing phases from the data it already fetches,
  and never overwrites a phase the basket has.
- "16 plus" counted against both groups; it is secondary, as the API's
  PHASE_GROUPS files it.

Also rewraps the HomeView doc comment the previous commit left awkward.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

nextjs-app/components/ComparisonView.tsx

@@ -53,7 +53,8 @@ export function ComparisonView({
   const router = useRouter();
   const pathname = usePathname();
   const searchParams = useSearchParams();
-  const { selectedSchools, removeSchool, replaceSchools, isInitialized } = useComparison();
+  const { selectedSchools, removeSchool, replaceSchools, backfillPhases, isInitialized } =
+    useComparison();
 
   const [selectedMetric, setSelectedMetric] = useState(initialMetric);
   const [isModalOpen, setIsModalOpen] = useState(false);
@@ -157,6 +158,17 @@ export function ComparisonView({
     };
   }, [urnKey, isInitialized]);
 
+  useEffect(() => {
+    if (!comparisonData) return;
+    backfillPhases(
+      Object.fromEntries(
+        Object.values(comparisonData)
+          .filter((d) => d?.school_info)
+          .map((d) => [d.school_info.urn, d.school_info.phase]),
+      ),
+    );
+  }, [comparisonData, backfillPhases]);
+
   const primarySchools = selectedSchools.filter((school) => {
     const info = comparisonData?.[school.urn]?.school_info;
     const hasPrimaryData =