fix(dashboard): deduplicate realtime log replay

tests/dashboard/live-console-model.test.ts

@@ -38,4 +38,19 @@ describe('live console state', () => {
     expect(state.entries.map(({ id }) => id)).toEqual(['3', '4']);
     expect(state.pendingEntries).toEqual([]);
   });
+
+  it('deduplicates the same log delivered by the initial snapshot and SSE replay', () => {
+    let state = createLiveConsoleState();
+    state = reduceLiveConsoleState(state, {
+      type: 'append',
+      entry: entry('initial-1', 'info', 'Database initialized'),
+    });
+    state = reduceLiveConsoleState(state, {
+      type: 'append',
+      entry: entry('realtime-1', 'info', 'Database initialized'),
+    });
+
+    expect(state.entries).toHaveLength(1);
+    expect(state.entries[0]?.id).toBe('initial-1');
+  });
 });

tests/log-broadcaster.test.ts

@@ -4,12 +4,21 @@ import { LogBroadcaster } from '../src/web/log-broadcaster.js';
 
 describe('LogBroadcaster realtime bridge', () => {
   it('publishes each log through the typed event hub', () => {
-    const hub = new RealtimeEventHub({ capacity: 10 });
+    const occurredAt = new Date('2026-08-08T15:00:00.000Z');
+    const hub = new RealtimeEventHub({
+      capacity: 10,
+      clock: { now: () => occurredAt },
+    });
     const received: RealtimeEvent[] = [];
+    let recent: unknown;
     hub.subscribe((event) => received.push(event));
     const broadcaster = new LogBroadcaster(hub);
 
     broadcaster.push('warn', 'Connection is stale', { attempts: 2 });
+    broadcaster.recent(
+      { query: {} } as never,
+      { json: (value: unknown) => { recent = value; } } as never,
+    );
 
     expect(received).toEqual([
       expect.objectContaining({
@@ -21,5 +30,8 @@ describe('LogBroadcaster realtime bridge', () => {
         },
       }),
     ]);
+    expect(recent).toEqual([
+      expect.objectContaining({ timestamp: occurredAt.getTime() }),
+    ]);
   });
 });