fix(web): keep rectification sessions listed and persist chats on first send

Empty-consultation filtering is now session_type scoped so birth-time rows stay in the sidebar. Subtitles use updatedAt with sort/group/cursor. New chats stay local until the first send.

frontend/DESIGN.md

@@ -353,7 +353,7 @@ The birth-time rectification session is the consultation transcript plus a house
 
 ### Navigation item
 
-- **Structure:** title is the session name. New dated titles are `生时校正 · M月D日` / `今日节奏 · M月D日` (category first). Subtitle is always `M月D日 HH:MM` in Asia/Shanghai from `created_at`, plus ` · <chart name>` when the chart is not the account holder. Same-day duplicates are not uniquified with wall-clock `HH:MM`.
+- **Structure:** title is the session name. New dated titles are `生时校正 · M月D日` / `今日节奏 · M月D日` (category first). Subtitle is always `M月D日 HH:MM` in Asia/Shanghai from `updated_at` (last activity), plus ` · <chart name>` when the chart is not the account holder. Same-day duplicates are not uniquified with wall-clock `HH:MM`. Display, sort, grouping, and cursor all use `updated_at`.
 - **States:** default, hover, current, focus, disabled.
 - **Hierarchy:** section labels stay tertiary; session titles and primary actions use ink so history rows do not collapse into the same gray as “收藏对话 / 历史对话”. History groups use the overline token for “今天 / 昨天 / 最近 7 天 / 最近 30 天 / 更早”. When the chart is not the account holder, a secondary line shows the chart name under the title. A session title is named when it is created; opening a stored session must not rename it. `updatedAt` advances only on conversation activity, including rectification turns, choice, adopt, and stop. Opening, refresh, and metadata PATCH do not bump it. A `?c=` that is not on the loaded page is looked up with `GET /api/sessions/{id}` before anyone may say it was deleted. Locked by `frontend/tests/session-open-preserves-identity.test.ts` and `frontend/tests/session-lookup-unlisted.test.ts`.
 - **Request behavior:** existing sessions remain selectable for reading while a request is active; creating or sending another request stays locked until the active request settles.