fix: stem chat search terms with the english text search config (#28319)

Chat message search used the `simple` text search config, which folds
case but does no stemming, so `refactor` did not match messages
containing `refactoring`. Message bodies now index and query with the
`english` config, which stems both sides. Titles, PR titles, and all
other filters are unchanged.

Migration `000585` adds a `chat_messages.search_tsv_config` enum column
recording which config produced each stored vector and stamps existing
vectors `'simple'` (a column-only update; vectors and indexes are
untouched, since rewriting `search_tsv` would block message writes on
large tables). The dbpurge sweep gains a bounded
`ReindexStaleChatMessagesSearchTsv` pass that rewrites stale vectors
newest first and stops for the process lifetime once drained. Until a
row is rewritten, `GetChats` matches it with the config that produced
it, so pre-migration exact-form searches keep working during the drain.
Vectors written by old binaries mid rolling upgrade cannot stamp the
config, stay pending, and self-heal on the next upgraded sweep.
Steady-state DB load is unchanged.

The down migration resets `english` vectors to `NULL` so the parent
version's existing sweep rewrites them with `simple`.

Closes
[CODAGT-867](https://linear.app/codercom/issue/CODAGT-867/backend-search-improvements-to-handle-partial-matches)

<details>
<summary>Decision log</summary>

- **`english` stemming over prefix matching.** On a 300k-row corpus a
stemmed lookup is a single GIN entry-tree probe, while `term:*` prefix
matching unions posting lists of every lexeme sharing the prefix (up to
a full scan for short prefixes), is asymmetric, and loses the
`websearch_to_tsquery` phrase/OR/negation UX. `pg_trgm` rejected as a
heavier index with different semantics.
- **Scope to message bodies only** (review). Titles and PR titles keep
their `simple` FTS expressions and indexes, identical to main.
- **Config-versioned queue over migration-time reset** (review). An
unqualified `UPDATE ... SET search_tsv = NULL` rewrites every indexed
row in the migration transaction, and an old replica could backfill
reset rows with `simple` vectors that permanently leave the `search_tsv
IS NULL` queue. The config column makes staleness explicit and both
problems disappear.
- **Enum over `text`** (review) for the config column.
- **Trade-offs.** English stopwords are dropped from message search
(stopword-only queries match no messages); stemming is English-specific;
titles do not stem; prefix matching remains out of scope.

</details>

---

🤖 This PR was generated by Coder Agents on behalf of @f0ssel.

coderd/database/dump.sql

@@ -352,6 +352,11 @@ CREATE TYPE chat_message_role AS ENUM (
     'tool'
 );
 
+CREATE TYPE chat_message_search_tsv_config AS ENUM (
+    'simple',
+    'english'
+);
+
 CREATE TYPE chat_message_visibility AS ENUM (
     'user',
     'model',
@@ -1479,6 +1484,7 @@ BEGIN
 
         cmp := NEW;
         cmp.search_tsv := OLD.search_tsv;
+        cmp.search_tsv_config := OLD.search_tsv_config;
         IF OLD IS NOT DISTINCT FROM cmp THEN
             RETURN NEW;
         END IF;
@@ -1496,7 +1502,7 @@ BEGIN
 END;
 $$;
 
-COMMENT ON FUNCTION set_chat_message_revision_before() IS 'Component of chatd. Updates chat_snapshot_version when any fields of chat_messages change. Excludes changes to search_tsv as it is not relevant to chatd''s processing loop.';
+COMMENT ON FUNCTION set_chat_message_revision_before() IS 'Component of chatd. Updates chat_snapshot_version when any fields of chat_messages change. Excludes changes to search_tsv and search_tsv_config as they are not relevant to chatd''s processing loop.';
 
 CREATE FUNCTION sync_chat_retry_state() RETURNS trigger
     LANGUAGE plpgsql
@@ -1548,7 +1554,7 @@ BEGIN
         SELECT DISTINCT n.chat_id
         FROM chat_message_history_new_rows n
         JOIN chat_message_history_old_rows o ON o.id = n.id
-        WHERE (to_jsonb(o) - 'search_tsv') IS DISTINCT FROM (to_jsonb(n) - 'search_tsv')
+        WHERE (to_jsonb(o) - 'search_tsv' - 'search_tsv_config') IS DISTINCT FROM (to_jsonb(n) - 'search_tsv' - 'search_tsv_config')
     ) AS affected
     WHERE c.id = affected.chat_id
       AND (
@@ -1559,7 +1565,7 @@ BEGIN
 END;
 $$;
 
-COMMENT ON FUNCTION update_chat_history_after_message_update() IS 'Component of chatd. Updates history_version and generation_attempt on chats when chat_messages is updated. Excludes changes to search_tsv.';
+COMMENT ON FUNCTION update_chat_history_after_message_update() IS 'Component of chatd. Updates history_version and generation_attempt on chats when chat_messages is updated. Excludes changes to search_tsv and search_tsv_config.';
 
 CREATE TABLE ai_gateway_keys (
     id uuid NOT NULL,
@@ -2053,13 +2059,16 @@ CREATE TABLE chat_messages (
     provider_response_id text,
     revision bigint NOT NULL,
     reasoning_effort chat_reasoning_effort,
-    search_tsv tsvector
+    search_tsv tsvector,
+    search_tsv_config chat_message_search_tsv_config
 );
 
 COMMENT ON COLUMN chat_messages.reasoning_effort IS 'Stores the selected effort for the turn triggered by this message.';
 
 COMMENT ON COLUMN chat_messages.search_tsv IS 'Used for full text search. NULL initially, populated async via background job.';
 
+COMMENT ON COLUMN chat_messages.search_tsv_config IS 'Text search config that produced search_tsv. NULL means the vector was written by a binary that predates this column; the dbpurge sweep re-vectorizes rows whose config is not english.';
+
 CREATE SEQUENCE chat_messages_id_seq
     START WITH 1
     INCREMENT BY 1