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>

frontend/src/desktop/bridge.ts

@@ -157,6 +157,38 @@ export interface LinkInput {
   name?: string;
 }
 
+/** What started a sync cycle (autosync.rs). */
+export type SyncTrigger = "launch" | "edit" | "timer" | "focus" | "manual";
+
+/**
+ * The most recent cycle, automatic or not. The desktop syncs on its own (at launch,
+ * soon after an edit, every few minutes and on focus), so most cycles have no
+ * button press to report back to — this is how the screen hears about them.
+ */
+export interface LastCycle {
+  at_ms: number;
+  trigger: SyncTrigger;
+  ok: boolean;
+  error: string | null;
+  /** Whether the pull changed anything this device shows. */
+  changed: boolean;
+  outcome: SyncOutcome | null;
+}
+
+/** Fired at every window after each cycle, with a LastCycle as the payload. */
+const SYNCED_EVENT = "inkwell://synced";
+
+/**
+ * Run `handler` after every sync cycle. Returns an unlisten function, or a no-op
+ * where there is no event API (the web build) — nothing syncs in the background
+ * there, so there is nothing to hear.
+ */
+export async function onSynced(handler: (cycle: LastCycle) => void): Promise<() => void> {
+  const events = window.__TAURI__?.event;
+  if (!events) return () => {};
+  return events.listen<LastCycle>(SYNCED_EVENT, (e) => handler(e.payload));
+}
+
 export const sync = {
   /** Ask who's at an address without committing to anything. */
   probe: (url: string) => invoke<ProbeResult>("sync_probe", { url }),
@@ -169,6 +201,8 @@ export const sync = {
    * without pushing first overwrites unsent local edits.
    */
   now: () => invoke<SyncOutcome>("sync_now"),
+  /** The last cycle, or null before the first one. */
+  last: () => invoke<LastCycle | null>("sync_last"),
   hasPending: () => invoke<boolean>("sync_has_pending"),
 };