Unexpected Default EXECUTE Privileges on Functions in API Schema

# Bug report - [x] I confirm this is a bug with Supabase, not with my own application. - [x] I confirm I have searched the [Docs](https://docs.supabase.com), GitHub [Discussions](https://github.com/supabase/supabase/discussions), and [Discord](https://discord.supabase.com). ## Describe the bug When creating functions in a custom API schema, newly created functions are executable by `anon`, `authenticated`, and `service_role` by default, even if: - `GRANT USAGE` is the only permission on the schema - `ALTER DEFAULT PRIVILEGES` is used to revoke execution for all roles - no explicit `GRANT EXECUTE` is applied on the function This behavior is inconsistent with PostgreSQL defaults, where objects should only be accessible if explicitly granted, and differs from the behavior for tables, which require explicit grants. ## To Reproduce Steps to reproduce the behavior: 1. Create a new schema: ```sql DROP SCHEMA IF EXISTS api_test CASCADE; CREATE SCHEMA api_test; GRANT USAGE ON SCHEMA api_test TO anon, authenticated, service_role; ALTER DEFAULT PRIVILEGES IN SCHEMA api_test REVOKE EXECUTE ON FUNCTIONS FROM public, anon, authenticated, service_role; ``` 2. Create a function: ```sql CREATE FUNCTION api_test.return_null() RETURNS text LANGUAGE plpgsql AS $$ BEGIN RETURN NULL; END; $$; ``` 3. Call the function via REST API: ``` POST /rest/v1/rpc/return_null Authorization: Bearer <anon key> ``` 4. Observe that the function executes successfully, even though no permissions were explicitly granted. 5. Manually revoking permissions after creation works as expected: ```sql REVOKE EXECUTE ON FUNCTION api_test.return_null() FROM public, anon, authenticated, service_role; ``` ## Expected behavior Newly created functions should not be executable by anon, authenticated, or service_role unless explicitly granted, consistent with PostgreSQL defaults and the behavior of tables. But especially when you alter default. - Supabase documentation suggests that anon and authenticated …