perf(ui): index and batch session updates

Batch global session, status, ordering, and activity timing mutations at the existing directory event boundary so large subagent bursts publish each owner once.

Maintain active session roots, children, and directory buckets in the global store, reuse them in Sidebar projections, and avoid rebuilding live aggregates and structural data for unrelated renders while preserving authoritative ordering reconciliation.

packages/ui/src/sync/DOCUMENTATION.md

@@ -18,7 +18,8 @@ There are **two distinct session data scopes** in the UI:
    - Holds:
      - global active sessions
      - global archived sessions
-     - active sessions indexed by directory
+    - active and archived entities indexed by ID
+    - active root, parent/child, and directory indexes
 
 These two scopes are intentionally different, but they are no longer equal peers for live UI truth.
 
@@ -47,7 +48,7 @@ So:
 | `session-ordering.ts` | Ephemeral lifecycle rank used by every user-visible session list | All known sessions in the active runtime |
 | `session-activity-timing.ts` | Elapsed time of the running turn and of the turn that just finished, plus the persisted starts that survive a reload | All known sessions in the active runtime |
 | `session-ui-store.ts` | Session selection, draft lifecycle, one-shot draft-materialization transition identity, abort prompts, worktree metadata, SDK-facing action entrypoints | App UI state |
-| `useGlobalSessionsStore.ts` | Global active sessions, global archived sessions, `sessionsByDirectory` | All opened project/worktree session lists |
+| `useGlobalSessionsStore.ts` | Global active/archived entities plus root, parent/child, and directory indexes | All opened project/worktree session lists |
 | `viewport-store.ts` | Scroll anchors, session memory, loading indicators | App UI state |
 | `attachment-files.ts` | Attachment picker allowlists, MIME/content validation, structured-text sanitization, and HEIC conversion | Local chat attachments across shared UI runtimes |
 | `document-attachments.ts` | Bounded Office/OpenDocument extraction, document text serialization, embedded-image extraction, and positional citations | DOCX, PPTX, XLSX, ODT, ODP, and ODS chat attachments |
@@ -215,7 +216,7 @@ The profiler also emits a user-timing mark when pending global-session recency i
 
 Streaming assistant and reasoning text is throttled once before reaching the markdown renderer. The renderer incrementally reconciles changed markdown blocks but does not add a second character-pacing timer, which would multiply parse/morph work while catching up on large streamed chunks.
 
-The event pipeline delivers each ordered per-directory flush as one reducer batch. Events retain their individual global indexes, notifications, cleanup, routing, materialization, and debug side effects, while their directory mutations accumulate in order and publish one store transaction per touched directory. Each top-level state slice is cloned lazily at most once in that batch; no-op events do not change references.
+The event pipeline delivers each ordered per-directory flush as one reducer batch. Events retain their individual notifications, cleanup, routing, materialization, and debug side effects, while directory mutations accumulate in order and publish one store transaction per touched directory. Global session mutations and live status, ordering, and timing transitions also accumulate in event order and each owner publishes at most once for the flush. Each top-level state slice is cloned lazily at most once in that batch; no-op events do not change references.
 
 Streaming lifecycle derivation has two paths. Directory attach, switch, bootstrap, and reconnect may perform a full reconciliation. Normal store publications reconcile only sessions whose `session_status` or `message` bucket changed; part-only events update the affected streaming message heartbeat directly and must not rescan all busy sessions.