fix(backend/db): Move vector extension to extensions schema (#11902)

[SECRT-1809: \[Bug\] Searching for blocks Error: ERROR: type "vector"
does not exist](https://linear.app/autogpt/issue/SECRT-1809)

### Changes 🏗️

Add a migration that:
1. Moves the `vector` extension to the `extensions` schema (which is in
the default postgres search path)
2. Amend the DB's default search path to include the app DB schema,
`platform` in our case

### Checklist 📋

#### For code changes:
- [x] I have clearly listed my changes in the PR description
- [x] I have made a test plan
- [x] I have tested my changes according to the test plan:
  - CI must pass
  - [x] Verify locally as far as possible that this will work

autogpt_platform/backend/migrations/20260129150000_move_vector_extension_to_extensions_schema/migration.sql

@@ -0,0 +1,83 @@
+-- Move pgvector extension from application schema to extensions schema.
+-- The extensions schema is in the default search_path, so ::vector casts
+-- in raw SQL queries will resolve correctly without needing to set
+-- search_path per connection. This fixes intermittent "type vector does not exist"
+-- errors that occur when pooled connections don't have the right search_path.
+
+-- Also set the database default search_path to include the application schema,
+-- so that raw SQL queries can find application tables/types without explicit
+-- schema prefix.
+
+-- Step 1: Ensure extensions schema exists
+CREATE SCHEMA IF NOT EXISTS extensions;
+
+-- Step 2: Move pgvector from current schema to extensions schema (if possible)
+DO $$
+DECLARE
+    vector_schema text;
+BEGIN
+    -- Check where vector extension currently lives
+    SELECT n.nspname INTO vector_schema
+    FROM pg_extension e
+    JOIN pg_namespace n ON e.extnamespace = n.oid
+    WHERE e.extname = 'vector';
+
+    IF vector_schema IS NULL THEN
+        -- Not installed yet; create in extensions schema
+        CREATE EXTENSION IF NOT EXISTS vector SCHEMA extensions;
+    ELSIF vector_schema != 'extensions' THEN
+        -- Installed in wrong schema; try to move it
+        -- This may fail on managed platforms like Supabase where the extension
+        -- is owned by a system user. In that case, we'll include the current
+        -- vector schema in the search_path instead (handled in Step 3).
+        BEGIN
+            RAISE NOTICE 'Moving pgvector from schema "%" to "extensions"', vector_schema;
+            ALTER EXTENSION vector SET SCHEMA extensions;
+        EXCEPTION WHEN insufficient_privilege THEN
+            RAISE NOTICE 'Cannot move pgvector (insufficient privileges). Will include schema "%" in search_path instead.', vector_schema;
+        END;
+    ELSE
+        RAISE NOTICE 'pgvector already in extensions schema, nothing to do';
+    END IF;
+END $$;
+
+-- Step 3: Update role-level search_path to include the application schema
+-- and the schema where pgvector lives. Role-level settings take precedence over
+-- database-level settings, which is important for managed platforms like Supabase
+-- that may have existing role-level search_path configurations.
+-- The default PostgreSQL search_path is: "$user", public
+-- We add the extensions schema and (if different) the schema where pgvector
+-- actually lives, so that ::vector type casts resolve correctly.
+DO $$
+DECLARE
+    app_schema text := current_schema();
+    vector_schema text;
+    ext_schemas text := 'extensions';
+BEGIN
+    -- Find where pgvector currently lives (after Step 2's move-or-fail)
+    SELECT n.nspname INTO vector_schema
+    FROM pg_extension e
+    JOIN pg_namespace n ON e.extnamespace = n.oid
+    WHERE e.extname = 'vector';
+
+    -- If vector is in a schema not already covered, add it to the search_path
+    IF vector_schema IS NOT NULL
+        AND vector_schema != 'extensions'
+        AND vector_schema != 'public'
+        AND vector_schema != app_schema
+    THEN
+        ext_schemas := format('extensions, %I', vector_schema);
+    END IF;
+
+    IF app_schema = 'public' THEN
+        EXECUTE format(
+            'ALTER ROLE %I SET search_path TO "$user", public, %s',
+            current_user, ext_schemas
+        );
+    ELSE
+        EXECUTE format(
+            'ALTER ROLE %I SET search_path TO "$user", %I, public, %s',
+            current_user, app_schema, ext_schemas
+        );
+    END IF;
+END $$;

autogpt_platform/backend/schema.prisma

@@ -2,7 +2,7 @@ datasource db {
   provider   = "postgresql"
   url        = env("DATABASE_URL")
   directUrl  = env("DIRECT_URL")
-  extensions = [pgvector(map: "vector")]
+  extensions = [pgvector(map: "vector", schema: "extensions")]
 }
 
 generator client {