Is 5e character advancement commutative across multiclassing?


Is 5e character development commutative across multiclassing? I’m talking about the mathematical property where given two operands and an operation, $ x \circ y = y \circ x$ . That is, regardless of the order of the operands, the result of a commutative operation remain the same.

We can rule out character creation, because that is not commutative. A fighter 1 / wizard 1 will have different game statistics than a wizard 1 / fighter 1 (saving throws and other proficiencies, for example).

Also, let’s disregard the randomness of hit point rolls. Assume all characters take the average of the dice.

So, for two characters with the same 1st level class and the same features, who reach the same levels in the same classes but different advancement paths (i.e., the order in which these features were gained was different), do they have the same game statistics?

Exempli Gratia:

Assume two 1st level Rogues, Alice and Beatrice.

Alice gained 3 levels of Rogue and on her 4th level of Rogue, instead of an ability score increase she picked up the Martial Adept feat. She later gained 3 levels in Fighter, and chose the Battle Master archetype.

Beatrice gained 3 levels of Fighter, and chose the Battle Master archetype. She then gained 3 levels of Rogue, and instead of an ability score increase she picked up the Martial Adept feat.

Alice and Beatrice each now total four Rogue levels and three Fighter levels, obtained via different paths. Is the superiority dice gained from the Martial Adept the same dice for both Alice and Beatrice?

Meta: I know the issue above could sit in its own post, but then it would weaken the broader question.