Medium and Spell Lattice

[Cleru]

Can Medium with Archmage or Hierophant spirit use Spell Lattice to cast Medium spell that he shouldn't have known at that level?

For Example : Can a Level 4 Medium with Spell Lattice (Haste) cast Haste?

You have 2nd-level spell slot after Archmage Arcana and Haste is a 2nd-level spell for Medium. Are there any rule that preventing you from casting Haste in this case?

[Archaeik]

The expanded slots aren't restricted to casting the spells learned from the spirit, so yes.

But since you intentionally picked a controversial example, expect table variation on whether it's easy to obtain a 2nd level lattice of haste.
Related: I see nothing in the lattice description that indicates the spell contained in the lattice must be cast at the same spell level that the creator used, so a Medium using a 3rd level lattice of Haste should still be able to cast it as a 2nd level spell known.

Lastly, 4k is 2/3 your WBL at level 4, which may be disallowed.

[Entryhazard]

It's sad that Paizo is not that great at forward compatibility and thus there cannot be a Psychic Page of Spell Knowledge