Can a multiclassed wizard add spells of a higher level than they can transcribe to their book upon level up?

Question

My friend has a multi-classed wizard 5, sorcerer 4. They recently took their 5th level in wizard, and took Steel Wind Strike (5th level) as one of the two spells they can add to their spellbook upon level up.

The multiclassing rules for spellcasting state that you determine the spells that you can prepare and cast for each class individually, however, this does not appear to apply to the new spells that you can add to your spellbook, the only requirement being that you must have a matching level of spell slots.

Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class. If you are a ranger 4/wizard 3, for example, you know three 1st-level ranger spells based on your levels in the ranger class. As 3rd-level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd-level spells. If your intelligence is 16, you can prepare six wizard spells from your spellbook.

Learning Spells of 1st Level and Higher Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table.

Is my friend's action permissible by RAW, that they can add spells to their spellbook that they won't be able to prepare or cast?

Acknowledging the errata: That specifically deals with copying spells into a spellbook, which is a feature distinctly separate from the one that allows you to gain two spells into your spellbook from leveling up as wizard.

Answer 1

You can only add a spell into your spellbook if you can prepare it

The rules for "Learning Spells of 1st Level and Higher" state (emphasis mine):

Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. [...]

You can only learn a new spell if, according to the wizard table, you have a spell slot for it. Looking at the wizard table, where we would only be a 5th level wizard, we see that we cannot cast 5th level spells; thus, we cannot learn 5th level spells either.

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One of the examples in the multiclassing chapter supports this (emphasis mine):

For example, if you are the aforementioned ranger 4/wizard 3, you count as a 5th-level character when determining your spell slots: you have four 1st-level slots, three 2nd-level slots, and two 3rd-level slots. However, you don't know any 3rd-level spells, nor do you know any 2nd-level ranger spells. [...]

Here we see a 5th level spellcaster who is a multiclass that includes wizard, and the text explicitly tells us they do not know any 3rd level spells despite having 3rd level slots.

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This is also similar to the fact that a multiclass Bard cannot use their multiclass spellcaster level when gaining Magical Secrets:

• Can you use Magical Secrets to choose high level spells when multiclassing as a Bard?

Answer 2

No, if you follow the spellcasting rules for multiclassing.

Under the Spell Slots paragraph you can find (emphasis mine):

If you have more than one spellcasting class, this table might give you spell slots of a level that is higher than the spells you know or can prepare. You can use those slots, but only to cast your lower-level spells. If a lower-level spell that you cast, like burning hands, has an enhanced effect when cast using a higher-level slot, you can use the enhanced effect, even though you don't have any spells of that higher level.

For example, if you are the aforementioned ranger 4/wizard 3, you count as a 5th-level character when determining your spell slots: you have four 1st-level slots, three 2nd-level slots, and two 3rd-level slots. However, you don't know any 3rd-level spells, nor do you know any 2nd-level ranger spells. You can use the spell slots of those levels to cast the spells you do know — and potentially enhance their effects.

Being a 5 levels Wizard / 4 level Sorcerer allows to have one 5th level spell slot, but the maximum level of spells known and that can be prepared and learned is 3 for the Wizard class and 2 for the Sorcerer class: the slots of higher levels (4 and 5 for the Wizard, 3, 4 and 5 for the Sorcerer) can be used to cast (or upcast) spells of lower levels.

Answer 3

No, he can't.

The PHB Errata states:

Your Spellbook (p. 114). The spells copied into a spellbook must be of a spell level the wizard can prepare.

@Exempt-Medic points out that the errata is deprecated. The new one only applies this rule when finding spells. However, even if he can know, certainly, he can't cast it, as said in PHB (p. 164):

If you have more than one spellcasting class, this table might give you spell slots of a level that is higher than the spells you know or can prepare. You can use those slots, but only to cast your lower level spells. If a lower level spell that you cast, like burning hands, has an enhanced effect when cast using a higher-level slot, you can use the enhanced effect, even though you don't have any spells of that higher level.