I’m trying to figure out the exact semantics of what these mean. I understand that if you have any magic item it functions as a sample, and also that a formula is something you can find as part of a treasure hoard. What I’m not clear about is when your own research produces a formula.

On page 118, it says that you learn the required components for an item when research is 50% complete. Does this now mean you have a formula for any future crafting? That is, if your final roll for the item fails, do you still get to use your discovered formula to discount the price and duration of a future attempt? Or did the failure prove that this was a flawed formula that can’t be used in the future?

I’m also unclear on whether a sample results in a known formula at any point, freeing up the item itself for use. If you use a sample and get 50% completed, does that mean you’ve reverse engineered the item to discern its formula, once and for all, so your henchman can start using the item again?

Page 118, under “Formulas and Samples,” tells us that “A spellcaster automatically has a formula for any magic item he has previously created.” Not halfway created, but created.

So, logically: you get a formula when you finish creating an item; and that includes when you finish creating an item based on a sample.

If you successfully create an item, then you certainly know a true formula! That’s not ambiguous. But it’s not strictly necessary to assume the reverse. (That’s an example of denying the antecedent, in formal logic terms!) I do suspect it might have been the intent of the designer, but I’m trying to make sure.

In particular, you need to reconcile that with the sentence below that says “If a character does not have a formula when he begins creating the item, he will not learn the special components until the work is 50% complete.” That makes it sound very much like you learn a “formula” at 50% completion. The question I’d have is whether the formula learned at 50% is still a true formula, or if it turns out that you needed a different formula all along. (“OK, we can stop gathering manticore venom, that’s obviously not it.”)

My intent was that “required components” would be a lower state of knowledge than a full “formula”. The “Required components” would tell a mage that e.g. he needed milk, sugar, and cocoa to make milk chocolate, while the “formula” would tell him how to turn it into a Lindt bar.

Formula is inclusive of required components, but required components is not inclusive of formula. Required components is a circle within the venn diagram of Formula.

I apologize that the sentence as written was so confusing! Your interpretation is quite reasonable. You could play it that way and price formulas as some sub-set of item value, with the assumption that mages half-research items to learn formulas.