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/context/ComparisonContext.tsx

@@ -17,6 +17,8 @@ interface ComparisonContextType {
   addSchool: (school: School) => void;
   removeSchool: (urn: number) => void;
   replaceSchools: (schools: School[]) => void;
+  /** Fill in phases missing from stored entries; never overwrites one. */
+  backfillPhases: (phases: Record<number, string | null | undefined>) => void;
   clearAll: () => void;
   isSelected: (urn: number) => boolean;
   /** The comparison group with no room for this school, or null. */

nextjs-app/context/ComparisonProvider.tsx

@@ -72,6 +72,22 @@ export function ComparisonProvider({ children }: { children: React.ReactNode })
     setSelectedSchools(fitToGroupLimits(schools));
   }, []);
 
+  // Baskets saved before phases were recorded (or added from a path that
+  // lacked one) count against both groups. Fill the gaps once the compare
+  // page has fetched each school, so they stop holding a slot they don't need.
+  const backfillPhases = useCallback((phases: Record<number, string | null | undefined>) => {
+    setSelectedSchools((prev) => {
+      let changed = false;
+      const next = prev.map((s) => {
+        const phase = phases[s.urn];
+        if (s.phase || !phase) return s;
+        changed = true;
+        return { ...s, phase };
+      });
+      return changed ? next : prev;
+    });
+  }, []);
+
   const clearAll = useCallback(() => {
     setSelectedSchools([]);
   }, []);
@@ -99,6 +115,7 @@ export function ComparisonProvider({ children }: { children: React.ReactNode })
         addSchool,
         removeSchool,
         replaceSchools,
+        backfillPhases,
         clearAll,
         isSelected,
         fullGroupFor: fullGroupForSchool,