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The PHB states that the pre-requisites for multi-classing are as follows:

\begin{array}{l|c|c} \textbf{Class} & \textbf{Ability Score} & \textbf{Minimum} \\ \hline \text{Barbarian} & \text{Strength} & 13 \\ \text{Bard} & \text{Charisma} & 13 \\ \text{Cleric} & \text{Wisdom} & 13 \\ \text{Druid} & \text{Wisdom} & 13 \\ \text{Fighter} & \text{Strength $or$ Dexterity} & 13 \\ \text{Monk} & \text{Dexterity $and$ Wisdom} & 13 \\ \text{Paladin} & \text{Strength $and$ Charisma} & 13 \\ \text{Ranger} & \text{Dexterity $and$ Wisdom} & 13 \\ \text{Rogue} & \text{Dexterity} & 13 \\ \text{Sorcerer} & \text{Charisma} & 13 \\ \text{Warlock} & \text{Charisma} & 13 \\ \text{Wizard} & \text{Intelligence} & 13 \\ \end{array}

Would it be unbalanced to allow multi-classing with a lower ability score? The +1 modifier comes at level 12, not 13, so multi-classing at 12 makes more sense to me.

For context:
Currently I am playing a Wild Magic Sorcerer (level 3). He has just discovered his magic and set out to share his wonderful gift by cleaning up Faerûn (he is a germophobe named Pyorelle and makes heavy use of Prestidigitation). He is extremely curious about his magic so he has been reading every book he can find about magic. I never intended to multi-class, but the character has developed to go down a more studied route.

RP-wise, it makes perfect sense for him to take a level of Wizard soon, but mechanics-wise his INT is currently 10. RAW he would need to wait until hitting level 9 to multiclass (increase INT at level 4 and 8, then multiclass at 9), which seems way too long for the amount he has been studying and researching magic for the past level.

As a note:
I am working with my DM on options. I am not asking for opinions on whether the DM should/shouldn't allow an early multiclass. I am more wondering what balance would be broken by allowing multi-class with a lower ability score (but not game-breakingly-low).

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    \$\begingroup\$ I only made it about halfway through your question before I started crying from laughter at Germaphobe named Pyorelle. \$\endgroup\$ – IT Alex Oct 5 '18 at 14:08
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On your particular case, I don't think it is game-breaking to allow multi-classing at INT 12.

Since you are considering multiclassing for roleplay purposes, it makes sense to allow your character to get one Wizard level if you have INT 12. I don't see any particular issues with allowing a single level multiclass using the conditions you proposed. Your DM may rule that you cannot get further levels with only INT 12, since your Intelligence will stand below the average of a Wizard.

You might want to consider the Magic Initiate and Ritual Caster feats from the Player's Handbook (pages 168 & 169) as options to fulfill your roleplay as a "student of magic". Remember that you need at least INT or WIS 13 for the Ritual Caster feat.

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    \$\begingroup\$ +1 for mentioning Magic Initiate and Ritual Caster; my wife’s sorcerer took Ritual Caster to represent her tutelage by a wizard, and to get that find familiar spell she really wanted. Much better approach than multiclassing for this, I think. \$\endgroup\$ – KRyan Oct 4 '18 at 16:10
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    \$\begingroup\$ I forgot about Magic Initiate. This would fit my RP better than multi-classing, and RAW I can take this feat at level 4 (which is next) regardless of my INT (which is currently 10). Thanks! \$\endgroup\$ – Newbie12345 Oct 4 '18 at 16:17
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The 5e multiclassing rules are generally structured the way they are to try to reduce the combo potential of some class combinations. If every Wizard/Sorcerer can just multiclass a level of Fighter to shore up their AC, then it makes most Armor Proficiencies meaningless.

The use of 13 as the lower limit for a class prerequisite is a strong mechanism for enforcing limitations on these possibilities. You're absolutely right that most players would not opt to have an ability score simply be 13; they'll usually round up to 14 if they actually need that extra point, or not bother and stay at 12. And since 14 is hard to hit under a Point Buy system (requires 2 points for that extra ASI, whereas up to 13 only requires 1:1 ratio), it means that characters who want to multiclass need to sacrifice significant power elsewhere to achieve it.

The Wizard/Sorcerer Multiclass

It's hard to gauge the combo potential of a Wizard/Sorcerer multiclass because the disparate ability modifiers between them (along with INT being a generally undesirable stat for any class that's not a Wizard) makes the combo generally underrepresented. The way that I tend to understand this combo is that, depending on how far you dip into either class, you'll either end up with a "Wizard that can apply Metamagic to their spells", or a "Sorcerer that can cast Rituals". A few levels dip into Wizard does shore up a major weakness of Sorcerers, in their lack of spell variety, and since a lot of important Utility Spells are concentrated in the first few levels, a 3-5 level dip into Wizard can get your character access to a large number of utility spells they otherwise wouldn't have; but this comes at the expense of not reaching your Sorcerer Capstone abilities, and also being dependent on two spellcasting modifiers, which will generally make your spells weaker overall.

Which is all sort of a long way of saying that I don't personally see anything terribly game-breaking about permitting the multiclass combo with a lower stat prerequisite. A lower INT stat means you'd be worse at casting Wizard spells than if you had the normal prerequisite, so you'd make gaining the multiclass easier with the major downside that the benefits you do gain from that Wizard multiclass are weaker than they'd otherwise be.

So as DM, I'd probably be willing to waive the Ability Score Prerequisite entirely if you could demonstrate in-fiction that your character has, in fact, been meticulously studying the foundations of magic. There's not really a compelling reason to believe that a Wizard/Sorcerer is overpowered [at least compared to a plain Single-classed Wizard...].

Your possible XY problem

So here's how I understand your situation:

  1. "My Sorcerer has been rigorously studying magic; I want a mechanical way to reflect those decisions"
  2. "Multiclassing to Wizard would give me the mechanical benefits of being a magic-user who has started rigorously studying magic"
  3. "My character doesn't have the INT required to multiclass into Wizard"
  4. "I want to convince my DM to permit Multiclassing with a lower INT requirement"

Multiclassing into Wizard is not necessarily the only way to give your character a mechanical upgrade to reflect their study into magic. Two other ways to reflect this would be to take either the Magic Initiate feat, or the Ritual Caster feat.

Magic Initiate. Choose a class: bard, cleric, druid, sorcerer, warlock, or wizard. You learn two cantrips of your choice from that class's spell list.

In addition, choose one 1st-level spell from that same list. You learn that spell and can cast it at its lowest level. Once you cast it, you must finish a long rest before you can cast it again using this feat.

Your spellcasting ability for these spells depends on the class you chose: Charisma for bard, sorcerer, or warlock; Wisdom for cleric or druid: or Intelligence for wizard.

 

Ritual Caster.
Prerequisite: Intelligence or Wisdom 13 or higher

You have learned a number of spells that you can cast as rituals. These spells are written in a ritual book, which you must have in hand while casting one of them.

When you choose this feat, you acquire a ritual book holding two 1st-level spells of your choice. Choose one of the following classes: bard, cleric, druid, sorcerer, warlock, or wizard. You must choose your spells from that class's spell list, and the spells you choose must have the ritual tag. The class you choose also determines your spellcasting ability for these spells: Charisma for bard, sorcerer, or warlock; Wisdom for cleric or druid; or Intelligence for wizard.

If you come across a spell in written form, such as a magical spell scroll or a wizard's spellbook, you might be able to add it to your ritual book. The spell must be on the spell list for the class you chose, the spell's level can be no higher than half your level (rounded up), and it must have the ritual tag. The process of copying the spell into your ritual book takes 2 hours per level of the spell, and costs 50 gp per level. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it.

Either of these options permit you to learn Wizard Spells. Magic Initiate has no ability score prerequisites, and Ritual Caster can use Wisdom instead of Intelligence if you happen to have a better Wisdom Score than Intelligence (some DMs also permit Charisma being used instead for Bard/Sorcerer/Warlock spells, because really, why not?). So if you want to mechanically reward your character for their studies without having to actually multiclass, you might consider picking up one of these feats instead in place of your normal ASI gains at levels 4, 8, 12, 16, or 19.

There are also other feats that don't involve gaining access to Wizard Spells or features, but do potentially reflect their time spent studying the theory of magic. Feats like Spell Sniper (doubles the range of attack-roll based spells), War Caster (advantage on Concentration Checks and ability to use an attack-roll based spell in place of an Attack of Opportunity) can be really cool options as well.

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