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.
Best Answer
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.)