fix(mobile): clear technical debt — real SHA-256, feed pagination, search

source_hash: BAS JSON import now computes SHA-256 via crypto.subtle.digest
instead of the '${id}-${length}' stub. No extra package — Hermes supports
Web Crypto API natively.

Feed pagination: useActivities(query, limit) accepts a LIMIT parameter.
The feed screen starts at 50, calls loadMore() via FlatList onEndReached
(threshold 0.3) to increment by 50 each time. useActivityCount(query)
drives the hasMore guard so loadMore is a no-op at the end of the list.

Feed search: compact TextInput below the header filters by title via
SQLite json_extract LIKE. Changing the query resets limit to PAGE_SIZE
so stale paginated results don't linger.

Docs: close the three resolved debt items; keep only the accepted
background-polling limitation as a known gap.

mobile/app/(tabs)/index.tsx

@@ -3,16 +3,20 @@ import { useFocusEffect } from 'expo-router';
 import { useSQLiteContext } from 'expo-sqlite';
 import { useRouter } from 'expo-router';
 import { useCallback, useState } from 'react';
-import { Alert, FlatList, Pressable, RefreshControl, StyleSheet, Text, View } from 'react-native';
-import { deleteActivities, useActivities, type ActivitySummary } from '@/db/queries';
+import { Alert, FlatList, Pressable, RefreshControl, StyleSheet, Text, TextInput, View } from 'react-native';
+import { deleteActivities, useActivities, useActivityCount, PAGE_SIZE, type ActivitySummary } from '@/db/queries';
 import { downloadFeed, uploadFeed } from '@/db/sync';
 import { useTheme } from '@/ThemeContext';
 
 export default function FeedScreen() {
   const db = useSQLiteContext();
   const theme = useTheme();
-  const [refreshKey, setRefreshKey] = useState(0);
-  const activities = useActivities();
+  const [refreshKey,   setRefreshKey]   = useState(0);
+  const [searchQuery,  setSearchQuery]  = useState('');
+  const [limit,        setLimit]        = useState(PAGE_SIZE);
+  const activities = useActivities(searchQuery, limit);
+  const totalCount = useActivityCount(searchQuery);
+  const hasMore    = activities.length < totalCount;
   const [downloading, setDownloading] = useState(false);
   const [uploading,   setUploading]   = useState(false);
   const [statusMsg,   setStatusMsg]   = useState<{ ok: boolean; text: string } | null>(null);
@@ -73,6 +77,15 @@ export default function FeedScreen() {
     setRefreshKey(k => k + 1);
   }
 
+  function handleSearch(q: string) {
+    setSearchQuery(q);
+    setLimit(PAGE_SIZE);  // reset pagination when search changes
+  }
+
+  function loadMore() {
+    if (hasMore) setLimit(l => l + PAGE_SIZE);
+  }
+
   function toggleSelect(id: string) {
     setSelected(prev => {
       const next = new Set(prev);
@@ -158,6 +171,20 @@ export default function FeedScreen() {
         <Text style={statusMsg.ok ? styles.msgOk : styles.msgErr}>{statusMsg.text}</Text>
       )}
 
+      {!selecting && (
+        <View style={styles.searchRow}>
+          <TextInput
+            style={styles.searchInput}
+            value={searchQuery}
+            onChangeText={handleSearch}
+            placeholder="Search activities…"
+            placeholderTextColor="#52525b"
+            returnKeyType="search"
+            clearButtonMode="while-editing"
+          />
+        </View>
+      )}
+
       {activities.length === 0 && !busy ? (
         <View style={styles.empty}>
           <Text style={styles.emptyIcon}>🚴</Text>
@@ -181,6 +208,8 @@ export default function FeedScreen() {
             />
           )}
           contentContainerStyle={styles.list}
+          onEndReached={loadMore}
+          onEndReachedThreshold={0.3}
           refreshControl={
             <RefreshControl
               refreshing={false}
@@ -321,6 +350,12 @@ const styles = StyleSheet.create({
   cancelText: { color: '#a1a1aa', fontSize: 13, fontWeight: '600' },
   msgOk: { color: '#86efac', fontSize: 12, textAlign: 'center', paddingHorizontal: 16, paddingBottom: 8 },
   msgErr: { color: '#fca5a5', fontSize: 12, textAlign: 'center', paddingHorizontal: 16, paddingBottom: 8 },
+  searchRow: { paddingHorizontal: 16, paddingBottom: 10 },
+  searchInput: {
+    backgroundColor: '#18181b', borderWidth: 1, borderColor: '#27272a',
+    borderRadius: 8, paddingHorizontal: 12, paddingVertical: 8,
+    color: '#f4f4f5', fontSize: 14,
+  },
   list:  { padding: 16, gap: 12, paddingBottom: 80 },
   card: {
     backgroundColor: '#18181b', borderRadius: 12,