Hypothesis matrix for 2 factors with interaction, sum contrasts

[tatiana-pashkova]

This might be more of a question for Reinhold because it relates to the Schad et al. (2020) paper about a priori contrasts. While I understood the logic of defining contrasts that is suggested in the paper (define the hypothesis, make a hypothesis matrix, apply a generalized inverse to the hypothesis matrix), I did not get how to make a hypothesis matrix for 2 factors and their interaction.
This process is briefly described on pp. 25-26, and I didn't understand how the weights for AxB were derived.

This is what I thought:
Imagine I have a study with 2 factors - Bilingualism (monolingual/bilingual) and Formality of context (formal/informal). As we said at some point in class, the model sees these 2 factors with 2 levels as 1 factor with 4 levels:
F1 monoling formal
F2 monoling informal
F3 biling formal
F4 biling informal

So, I want to test 3 null hypotheses:

1. If monolinguals are different from bilinguals, averaging over formality: (F1+F2)/2 = (F3+F4)/2
2. If the formal context is different from the informal one, averaging over bilingualism: (F1+F3)/2 = (F2+F4)/2
3. If monolinguals have the same formality effect as bilinguals (whether the formality lines are parallel): F1-F2 = F3-F4

I understand why we got (1/4, 1/4, -1/4, -1/4) and (1/4, -1/4, 1/4, -1/4) weights in null hypotheses 1 and 2 respectively: if we open the brackets, we'll get 1/2, but we want half the difference, so we end up with 1/4's.
However, I don't know how null hypothesis 3 translates into the weights (1/4, -1/4, -1/4, 1/4). Why do we choose 1/4, and not 1/2 or 1, which would make the equation F1-F2 = F3-F4 work out as well?