Well, the FROM part can easily generate a data set that has no i_1s in it, and also a data set that has no i_2s in it. Neither such data set is a logically valid basis for obtaining a projection containing both i_1 and i_2 which is what the SELECT part wants.
If you take the other line, i.e. that the projection is just constructed with nulls where there are no data items, then the question is: what is the meaning of the SELECT part anyway? Is it just: given me any of this that happens to be available from the result of the FROM and null for the rest?
Maybe the mathematically correct view of the view generated by the FROM is the possible shape implied by the archetype slots, not the actual shapes found in the DB.
In DB-land it is different: you can get null results for columns mentioned in the SELECT, but the columns have to be there in the view generated by the FROM, since that is what a ‘projection’ is by definition: a formal sub-set of another set of items (columns in relational calculus).