You always need a column in an index.

You could try to index a constant that occupies only one byte:

```
SELECT typname FROM pg_type WHERE typlen = 1;

 typname 
---------
 bool
 char
(2 rows)
```

If you choose `boolean`, that would be

```
CREATE INDEX ON ((TRUE)) WHERE thing_id IS NULL;
```

But my recommendation is to index `id`. Sure, that would make the index somewhat bigger, but your second query could use a much faster index-only scan.