desktop: syncs on its own — at launch, soon after an edit, every few minutes and on focus

Until now the only caller of the sync engine was the "Sync now" button. A worker
thread now owns every cycle (the button's included, so two never overlap):

- launch: one cycle as the app opens;
- edit: every 10s it reads a fingerprint of the pending set and sends when that
  moved. A fingerprint rather than "anything pending", because a rejected change
  stays pending and would otherwise be resent every tick forever;
- timer: a pull every 5 minutes with nothing to send;
- focus: at most once per 30s.

Failed automatic cycles back off (doubling from 10s to 5 minutes). A panicking
cycle counts as a failed one rather than ending the thread. Every cycle is
emitted as inkwell://synced: the board reloads when the pull changed something,
and the Sync screen shows the last automatic failure. No final push on quit;
the launch cycle sends whatever was left.

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

desktop/src-tauri/src/lib.rs

@@ -9,6 +9,7 @@
 //! remains here is the Tauri command surface (`commands`), desktop integration
 //! (menu-entry install for the Linux AppImage), the in-app updater, and boot.
 
+mod autosync;
 mod capture;
 mod commands;
 mod crossover;
@@ -142,6 +143,21 @@ pub fn run() {
             // was built before this path could be resolved.
             sync::blobs::publish_root(blobs.root().to_path_buf());
             app.manage(blobs);
+            // Last: the worker reads the store and the blob store, both managed now.
+            // Its first cycle is the launch sync.
+            autosync::start(app.handle())?;
+            // Coming back to the window is when someone is about to look, so it asks
+            // for a cycle. Rate-limited in autosync, because focus flaps constantly.
+            if let Some(window) = app.get_webview_window("main") {
+                let handle = app.handle().clone();
+                window.on_window_event(move |event| {
+                    if let tauri::WindowEvent::Focused(true) = event {
+                        handle
+                            .state::<autosync::AutoSync>()
+                            .kick(autosync::Trigger::Focus);
+                    }
+                });
+            }
             Ok(())
         })
         .invoke_handler(tauri::generate_handler![
@@ -186,6 +202,7 @@ pub fn run() {
             commands::sync::sync_unlink,
             commands::sync::sync_status,
             commands::sync::sync_now,
+            commands::sync::sync_last,
             commands::sync::sync_has_pending,
             update::update_channel_get,
             update::update_channel_set,