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.

> 🤖

site/src/api/typesGenerated.ts

@@ -1291,6 +1291,14 @@ export interface ChatAgentModelOverrideResponse {
 	readonly is_malformed: boolean;
 }
 
+// From codersdk/chats.go
+/**
+ * ChatAutoArchiveDaysResponse contains the current chat auto-archive setting.
+ */
+export interface ChatAutoArchiveDaysResponse {
+	readonly auto_archive_days: number;
+}
+
 // From codersdk/chats.go
 export type ChatBusyBehavior = "interrupt" | "queue";
 
@@ -3403,6 +3411,14 @@ export interface DebugProfileOptions {
 	readonly Profiles: readonly string[];
 }
 
+// From codersdk/chats.go
+/**
+ * DefaultChatAutoArchiveDays is the default auto-archive window, in
+ * days, applied when no site config row exists. Zero disables
+ * auto-archival.
+ */
+export const DefaultChatAutoArchiveDays = 0;
+
 // From codersdk/chats.go
 /**
  * DefaultChatWorkspaceTTL is the default TTL for chat workspaces.
@@ -7685,6 +7701,15 @@ export interface UpdateChatAgentModelOverrideRequest {
 	readonly model_config_id: string;
 }
 
+// From codersdk/chats.go
+/**
+ * UpdateChatAutoArchiveDaysRequest is a request to update the chat
+ * auto-archive period.
+ */
+export interface UpdateChatAutoArchiveDaysRequest {
+	readonly auto_archive_days: number;
+}
+
 // From codersdk/chats.go
 /**
  * UpdateChatDebugLoggingAllowUsersRequest is the admin request to