Background pipeline status: registry + /api/graph/status + graph-view chips

The background extraction pipeline is invisible on the home surface —
essential observability per the user's directive: the graph view must
show what is currently running (extracting / auditing / merging) and
for which chat.

- extraction.ts: module-level registry of in-flight runs (chatId,
  phase, startedAt), updated at each phase transition in runExtraction
  and cleared in finally (done or failed). extractionStatuses() is the
  read; one process, one registry.
- GET /api/graph/status: thin transport over the registry, chat titles
  joined from the chat store.
- graph-view: polls every 2s and shows one chip per running pipeline
  (spinner + phase + title + elapsed seconds, bottom-left); when a run
  finishes (some -> none) the graph refetches, so merges land live.
  Status is cosmetic — poll errors never surface.
- Unit test drives a mocked pipeline run and observes the registry
  mid-flight: 'extracting' during the extractor's generate, 'auditing'
  during the auditor's, empty after.

app/api/graph/route.ts

@@ -2,8 +2,8 @@ import { graphView } from "@/src/server";
 
 /**
  * The home graph view's data endpoint (W4; vision §1 "Home page") — thin
- * transport: the server entry (src/server/graph.ts) folds stale chats and
- * returns nodes, edges, derived node states, and session provenance.
+ * transport: the server entry (src/server/graph.ts) returns nodes, edges,
+ * session provenance, and per-session understanding observations.
  */
 
 export const dynamic = "force-dynamic";

app/api/graph/status/route.ts

@@ -0,0 +1,21 @@
+/**
+ * GET /api/graph/status — the in-flight background extraction runs
+ * (vision §1: the pipeline status must be visible on the home graph).
+ * Thin transport over the pipeline's module-level registry: phase
+ * (extracting / auditing / merging) per chat, oldest first. Titles joined
+ * from the chat store; a missing chat record still reports the run.
+ */
+import { extractionStatuses } from "@/src/agent/subagents/extraction.ts";
+import { chatStore } from "@/src/store/chats.ts";
+
+export const dynamic = "force-dynamic";
+
+export async function GET() {
+	const runs = extractionStatuses().map((run) => ({
+		chatId: run.chatId,
+		title: chatStore.getChat(run.chatId)?.title ?? run.chatId,
+		phase: run.phase,
+		startedAt: run.startedAt,
+	}));
+	return Response.json({ runs });
+}