fix(search): start postcode searches at half a mile

New postcode searches, the near-me button and URLs without a radius now
use 0.5 miles. A postcode URL with no radius used to show "1 mile" in the
Distance control while the API applied its own 5-mile default; the page
and the map fetch now send the same default the control displays.

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

nextjs-app/components/HomeView.tsx

@@ -17,6 +17,7 @@ import { useComparisonContext } from '@/context/ComparisonContext';
 import { fetchSchools, fetchLAaverages, fetchNationalAverages } from '@/lib/api';
 import type { SchoolsResponse, Filters, School } from '@/lib/types';
 import { track } from '@/lib/analytics';
+import { DEFAULT_RADIUS_MILES } from '@/lib/utils';
 import styles from './HomeView.module.css';
 
 interface HomeViewProps {
@@ -340,6 +341,7 @@ export function HomeView({ initialSchools, filters, totalSchools, howItWorks, ed
     setIsLoadingMap(true);
     const params: Record<string, any> = {};
     searchParams.forEach((value, key) => { params[key] = value; });
+    params.radius ??= DEFAULT_RADIUS_MILES;
     params.page = 1;
     params.page_size = 500;
     fetchSchools(params, { cache: 'no-store', signal: controller.signal })
@@ -419,7 +421,7 @@ export function HomeView({ initialSchools, filters, totalSchools, howItWorks, ed
             setGeoState('idle');
             track('near_me_used', { outcome: 'granted' });
             track('search_submitted', { query: postcode, via: 'near_me', has_postcode: true, filters_active: '', filters_count: 0 });
-            router.push(`/?postcode=${encodeURIComponent(postcode)}&radius=1`);
+            router.push(`/?postcode=${encodeURIComponent(postcode)}&radius=${DEFAULT_RADIUS_MILES}`);
           } else {
             track('near_me_used', { outcome: 'no_postcode' });
             setGeoState('error');