feat(KB-136): add shared badge pub/sub for multi-instance deployments

- Add BadgePubSub interface with Redis and in-memory adapters

- Implement cross-instance badge snapshot delivery via Redis pub/sub

- Add message validation, echo loop prevention, and structured caching

- Support graceful fallback to in-memory mode when Redis unavailable

- Update ServerOptions for dependency injection and add documentation

packages/dashboard/src/websocket.ts

@@ -8,6 +8,13 @@ export interface BadgeUpdate {
   timestamp?: string;
 }
 
+/** BadgeSnapshot is the full badge state with required timestamp for caching */
+export interface BadgeSnapshot {
+  prInfo?: PrInfo | null;
+  issueInfo?: IssueInfo | null;
+  timestamp: string;
+}
+
 export interface BadgeUpdatedMessage {
   type: "badge:updated";
   taskId: string;