Does the Cantrip Formulas optional feature allow a wizard to increase the number of cantrips they know?

Question

I have come across similar questions like Can I swap my cantrips known for new ones from scrolls?, and Can a spellbook contain cantrips?, but I noticed that the questions predate Tasha's Cauldron of Everything, and only one answer makes a reference to the optional 3rd level rule.

For reference (P.75):

Cantrip Formulas: You have scribed a set of arcane formulas in your spellbook that you can use to formulate a cantrip in your mind. Whenever you finish a long rest and consult those formulas in your spellbook, you can replace one wizard cantrip you know with another cantrip from the wizard spell list.

and:

Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it.

Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

What I would like to know is if this creates the opening for wizards to increase their cantrips known through spell scrolls with the same methods and rules as would apply to other wizard spells, or if there is some reference I missed negating such a concept. I understand that it refers to first-level spells or higher, though from what I understood, this was because until Tasha's Cauldron of Everything was released, cantrips were actually not written down in spell books at all, only memorized. But now the Cantrips Formula seems to have changed that bit of lore.

Answer 1

You may be mixing up the ideas of 'cantrips known' and 'formulas in the spellbook'.

There is no need to scribe cantrips; the "Cantrip Formulas" feature gives you access to the entire list of Wizard cantrips automatically. You can "replace one wizard cantrip you know with another cantrip from the wizard spell list" -- that's from the spell list, not from a smaller set that's in your spellbook. You always have all of the wizard cantrips available to swap into your known list (but only one change per day).

The "Cantrip Formulas" feature does not change how many cantrips you "know" at a time, though. You're still limited by the number of Cantrips Known as listed in the Wizard class.

So at 1st level, a wizard knows 3 cantrips and that number can't change, but each morning, they can swap out one of those for any other wizard cantrip, and they don't have to do anything to have all the cantrips available for swapping.

From the feature's description, it sounds like you don't have like "acid splash" and "message" in your spellbook anywhere; instead you have a page of formulae (like, say, math equations), and when you put them together one way, you get an acid splash, but if you put them together another way you get message. So you aren't looking at a list of distinct cantrips and memorizing them, the way you do with leveled spells; instead you're reading the list of formulae and deciding which specific combinations you want to keep in mind for quick casting. (But the flavor text is easily replaced, you can describe how this works any way you want to, from actual recorded mini-spells to some kind of runes that you can keep in your mind.)

Answer 2

The number of cantrips you know at any given time is the number found in the "Cantrips Known" column of the Wizard table.

The Wizard Table found at the beginning of the wizard class description tells you how many cantrips you know at a given level:

Cantrips
At 1st level, you know three cantrips of your choice from the wizard spell list. You learn additional wizard cantrips of your choice at higher levels, as shown in the Cantrips Known column of the Wizard table.

The Cantrip Formulas optional feature does not interact with this number in any way.¹ The Cantrip Formulas feature simply allows you to swap out one cantrip you currently know with a different one from the Wizard spell list at the end of a long rest. If you are a 9th level Wizard, you know four cantrips. If you use the Cantrip Formulas feature to change one of them, afterwards you still know four cantrips.

You already have all the cantrip formulas in your spellbook.

An important thing to note is that Cantrip Formulas states:

you can replace one wizard cantrip you know with another cantrip from the wizard spell list.

If you are using the Cantrip Formulas feature, you are automatically able to select any cantrip from the wizard spell list. There is no need to find and record new cantrip formulas. The "set of arcane formulas" is all of the cantrips on the wizard spell list.

Nothing in the Cantrip Formulas feature changes this rule from the "Your Spellbook" sidebar in the wizard class description:

Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it.

There is just nothing in the feature description that indicates it interacts with this rule in any way. So no, cantrip formulas doesn't allow you to add cantrips to your spellbook to increase your cantrips known, because it just doesn't say anything like that.

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^{1 Feats, Multiclassing, and certain racial traits can change the number of cantrips you know, but Cantrip Formulas only interacts with your wizard cantrips.}

Answer 3

If Cantrip Formulas was meant to allow cantrip spell scrolls to be scribed into the Wizard's spellbook, this would be explicitly stated

The Wizard's Spellbook states (emphasis mine):

[...] When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a spell level you can prepare and if you can spare the time to decipher and copy it. [...]

If Cantrip Formulas was intended to allow us to add cantrip spell scrolls into our spellbook, this would need to be explicitly stated because this is not something a Wizard can ordinarily do. Furthermore, even if Cantrip Formulas did somehow let us scribe cantrips into our spellbook, this would violate the "Cantrips Known" column of the Wizard Table. Without an explicit exception to both of these rules (and we have neither), the Cantrip Formulas feature does not let us scribe cantrip spell scrolls into our spellbook.