Restructure step 0: Connection extracted from the coordinator

Connection owns the socket lifecycle, connect/dial with epochs,
reconnect, the clock offset, diagnostics upload and a sealed ConnState
signal; the hello is built by an injectable builder, the resume token
stays private, and the reconnect delay and silence limit are parameters.
The coordinator maps ConnState to the same health texts and keeps the
reconnect restore and drop handling (step 1 replaces them).

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

apps/desktop/test/chapters_mpv_e2e_test.dart

@@ -114,7 +114,7 @@ void main() {
         a.seekTo(40);
         await _poll(() {
           final r = a.serverState.peek();
-          return r != null && r.seekBy == a.sync.memberId && (r.position - 40).abs() < 2;
+          return r != null && r.seekBy == a.connection.memberId && (r.position - 40).abs() < 2;
         }, 'server at 0:40');
         await _poll(() => a.skipOffer.peek()?.kind == SkipKind.opening, 'Skip OP offered');
         expect(a.skipOffer.peek()!.target, const Duration(minutes: 2));
@@ -125,9 +125,9 @@ void main() {
         await _poll(() {
           final r = a.serverState.peek();
           return r != null &&
-              r.seekBy == a.sync.memberId &&
-              serverPositionAt(r, a.sync.serverNow()) >= 120 &&
-              serverPositionAt(r, a.sync.serverNow()) < 125;
+              r.seekBy == a.connection.memberId &&
+              serverPositionAt(r, a.connection.serverNow()) >= 120 &&
+              serverPositionAt(r, a.connection.serverNow()) < 125;
         }, 'server at the OP end');
         await _poll(() => b.playback.positionSecs >= 119.5 && b.playback.positionSecs < 126, 'b follows');
         await _poll(() => seeks(b) == seeksBefore + 1, 'b saw the Skip OP seek');

apps/desktop/test/relay_e2e_test.dart

@@ -131,14 +131,14 @@ void main() {
     final alice = await member('alice');
     final bob = await member('bob');
     expect(await bob.swapPicked(FileMedia(file)), isNull);
-    await _poll(() => stateMediaRelay(alice.serverState.peek()!)?.source == bob.sync.memberId, 'relay state');
+    await _poll(() => stateMediaRelay(alice.serverState.peek()!)?.source == bob.connection.memberId, 'relay state');
     final state = alice.serverState.peek()!;
     expect(state.media!.containsKey('url'), isFalse);
     expect(bob.relaying.peek(), isTrue);
     await playTogether([alice, bob]);
     final relay = server.hub.relay!;
     expect(relay.buffers.single.bytesToViewers, greaterThan(0), reason: 'alice played through the relay');
-    expect(relay.sourceToken, bob.sync.token);
+    expect(relay.sourceToken, bob.connection.token);
 
     // The next swap retires bob's source and the buffer. Nothing answers
     // at that link, so its probe refuses it unless put up anyway.
@@ -156,7 +156,7 @@ void main() {
       final link = 'http://127.0.0.1:${origin.port}/private/ep.mkv?api_key=SECRET';
 
       expect(await bob.swapPicked(LinkMedia(link, relay: LinkRelay.me)), isNull);
-      await _poll(() => stateMediaRelay(alice.serverState.peek()!)?.source == bob.sync.memberId, 'me state');
+      await _poll(() => stateMediaRelay(alice.serverState.peek()!)?.source == bob.connection.memberId, 'me state');
       expect(alice.serverState.peek()!.toJson().toString(), isNot(contains('SECRET')));
       await playTogether([alice, bob]);
       expect(server.hub.relay!.buffers.single.bytesToViewers, greaterThan(0));
@@ -240,7 +240,7 @@ void main() {
       expect(await bob.swapPicked(FileMedia(file)), isNull);
       await _poll(() => server.hub.relay?.live == true, 'source live');
       await _poll(() => alice.playback.durationSecs > 19 && bob.playback.durationSecs > 19, 'both opened');
-      final token = bob.sync.token;
+      final token = bob.connection.token;
 
       // Bob's network goes silent (no FIN): states stop, he reconnects
       // with `resume`, the server still has him.
@@ -248,7 +248,7 @@ void main() {
       await _poll(() => bob.reconnecting.peek(), 'reconnect engaged', timeout: const Duration(seconds: 15));
       await _poll(() => !bob.reconnecting.peek() && bob.serverState.peek() != null, 'back',
           timeout: const Duration(seconds: 20));
-      expect(bob.sync.token, token, reason: 'resumed as the same member');
+      expect(bob.connection.token, token, reason: 'resumed as the same member');
       await _poll(() => server.hub.relay?.live == true, 'source redialled', timeout: const Duration(seconds: 15));
       expect(bob.relaying.peek(), isTrue);
       expect(bob.playback.durationSecs, greaterThan(19), reason: 'still his own copy');