feat(kernel): finish phase3 bridge cutover and phase4 tree projection

- add generic mnote-web query transport and bridge routes for workspace/request/trace

- route sidebar compat traffic through mnote-web and tighten fixture fallback to dev/test

- unify sidebar/page tree/file tree/picker consumers on page_tree projection

- sync phase3/phase4 checklist, breakdown docs, and harness progress state

wolai-frontend/src/app/api/sidebar/route.ts

@@ -17,6 +17,10 @@ import {
   mapSidebarDatasetListQueryResultToInitialData,
   type SidebarDatasetListQueryResult,
 } from "@/lib/sidebar-data";
+import {
+  fetchSidebarDatasetFromMnoteWeb,
+  getMnoteWebBaseUrl,
+} from "@/lib/server/mnote-web";
 
 export const dynamic = "force-dynamic";
 
@@ -37,6 +41,24 @@ export async function GET(request: Request) {
     }
 
     try {
+      if (getMnoteWebBaseUrl()) {
+        const proxied = await fetchSidebarDatasetFromMnoteWeb({
+          workspaceId: targetWorkspaceId,
+          request,
+        });
+        const result = mapSidebarDatasetListQueryResultToInitialData(proxied.dataset);
+
+        return NextResponse.json({
+          ...result,
+          meta: {
+            requestId: proxied.meta.requestId,
+            traceId: proxied.meta.traceId,
+            queryName: "compat.next.sidebar",
+            workspaceId: proxied.meta.workspaceId,
+          },
+        });
+      }
+
       const bridgeContext = await buildDocumentBridgeContext({
         request,
         workspaceId: targetWorkspaceId,