feat: auto-archive inactive chats with audit trail (#24642)

Adds a background job in `dbpurge` that periodically archives chats
inactive beyond a configurable threshold. Each archived root chat gets a
background audit entry tagged `chat_auto_archive`. Disabled by default.

* New `AutoArchiveInactiveChats` SQL query with LATERAL last-activity
subquery and partial index on archive candidates
* `site_configs`-backed `auto_archive_days` setting with admin-only PUT,
any-authenticated-user GET
* Cascade archive via `root_chat_id`; pinned chats and active threads
exempt
* Root-only audit dispatch on detached context, matching manual archive
(`patchChat`) behavior
* 11 subtests covering disabled no-op, boundary, deleted messages, child
activity, pinned exemption, multi-owner, idempotency, and batch
pagination

PR #24643 adds per-owner digest notifications.
PR #24704 adds the requisite UI controls.

> 🤖

coderd/database/queries/chats.sql

@@ -1427,3 +1427,55 @@ UPDATE chat_messages SET deleted = true
 WHERE chat_id = @chat_id::uuid
     AND deleted = false
     AND content::jsonb @> '[{"type": "context-file"}]';
+
+-- name: AutoArchiveInactiveChats :many
+-- Archives inactive root chats (pinned and already-archived chats skipped),
+-- cascading to children via root_chat_id. Limits apply to roots, not total
+-- rows. Used by dbpurge.
+WITH to_archive AS (
+    SELECT
+        c.id,
+        -- Activity = MAX(cm.created_at) across the family, or c.created_at
+        -- when the family has no non-deleted messages.
+        COALESCE(activity.last_activity_at, c.created_at) AS last_activity_at
+    FROM chats c
+    LEFT JOIN LATERAL (
+        SELECT MAX(cm.created_at) AS last_activity_at
+        FROM chat_messages cm
+        JOIN chats fc ON fc.id = cm.chat_id
+        WHERE (fc.id = c.id OR fc.root_chat_id = c.id)
+          AND cm.deleted = false
+    ) activity ON TRUE
+    WHERE c.archived = false
+      AND c.pin_order = 0
+      AND c.parent_chat_id IS NULL -- roots only
+      AND c.created_at < @archive_cutoff::timestamptz
+      -- New active statuses must be added here to prevent archiving.
+      AND c.status NOT IN ('running', 'pending', 'paused', 'requires_action')
+      AND COALESCE(activity.last_activity_at, c.created_at) < @archive_cutoff::timestamptz
+    -- Sorting by created_at lets Postgres drive the scan from the
+    -- partial index instead of evaluating every LATERAL subquery
+    -- before sorting. All candidates are past the cutoff, so the
+    -- archive order is immaterial once the backlog drains.
+    ORDER BY c.created_at ASC
+    LIMIT @limit_count
+),
+archived AS (
+    UPDATE chats c
+    SET archived = true, pin_order = 0, updated_at = NOW()
+    FROM to_archive t
+    WHERE (c.id = t.id OR c.root_chat_id = t.id) -- cascade to children
+      AND c.archived = false
+    RETURNING c.*
+)
+SELECT
+    a.*,
+    -- Children inherit their root's activity so last_activity_at is never null.
+    COALESCE(
+        t.last_activity_at,
+        (SELECT tr.last_activity_at FROM to_archive tr WHERE tr.id = a.root_chat_id),
+        a.created_at
+    )::timestamptz AS last_activity_at
+FROM archived a
+LEFT JOIN to_archive t ON t.id = a.id
+ORDER BY (a.root_chat_id IS NULL) DESC, a.owner_id ASC, a.created_at ASC, a.id ASC;