I think a lot about this post about the implication of Amane's mother being mentally ill and possibly converted by her husband who was against her getting treatment, and I want to share some of my thoughts on that. Particularly about the way gender expectations are included in the story of Amane's family. Milgram already explores the topic of gender expectations much enough (Yuno, Mahiru, Fuuta, Kazui) so I think the situation of Amane's family would be an interesting addition to it.
Well, I'll speak for my country, but the situation with domestic violence and lone mothers (and poverty) in my country is very bad. There is a joke that the families here have either two problems, father left the family or father hasn't left the family. So the thought of Amane's father starting the cycle of violence in this family is very possible for me and I can imagine what happened.
It's complicated to try and break down all the existing gender expectations, but I want to note a few things in particular.
In a conservative society that pushes strict gender roles/expectations women are taught romance and men are not. Girls are supposed to be reading stories where a prince saves a princess from a tower and they live together forever while boys are supposed to play in dirt and leaves. Women enter marriages enamored, excited for a beautiful fairy-tale to come true. And for men it was never a fairy-tale; it's very much more likely for them to think it's a frustrating obligation where they're supposed to give away all of their money.
It's especially bad when these people are poor. Because men are expected to work and build careers and women are not (+ women are underpaid and often not given jobs at all because of misogyny). I mentioned it on this blog earlier in the context of Mahiru, but for example, I believe Mahiru wants marriage so much not only because of desire for affection, but because it would grant her independence in a "woman" way, contrasted to Mikoto who is trying to gain independence through overworking being the only man in the family. Mahiru is very much a princess locked in a tower. Of course she wouldn't know that it's even possible for her to work.
This often ends in a situation where the whole marriage depends on the wife (keeping up the house and the kids, taking emotional care of the whole family, trying to keep up the affectionate bond with the husband; some of them are also working full-time at the same time and there is a concept of "second shift" that talks about it) who is yet heavily financially dependent on the husband, and the husband doesn't want to be emotionally bothered with the family. Not to say that this kind of situation is not harmful for men because gender expectations are bad for literally everyone, but keeping the whole marriage on yourself while being afraid to be thrown out in the streets if your husband stops loving you is very draining for women. And it's definitely not an enamoring fairy-tale that they were reading about in the books.
I believe what could have happened is that Amane's mother got in love with Amane's father and followed into the cult with him because she wanted to stay with him forever. Since he's a lecturer, it's likely he's charismatic on his own and could have taught her that the cult grants an ideal world on top of that. We also know that she gave birth to Amane at quite an early age (Amane says that her mother was around Mahiru's age when she got born) so it's likely that her mother had a situation similar to Mahiru — trying to get married as soon as possible and become a perfect wife in a beautiful romance story.
However Amane's father turned out to be emotionally distant — he's always busy with his work after all. And also they're poor. And also they're in a cult that prohibits medical interference and entertainment and sets rules on food. I don't necessarily mean that this alone would make her develop a mental illness but this would very much make her mental state get worse.
The thought in the original post that she seems to inflict violence on Amane only when father is not home makes me think that she could lash out the frustration from her husband not loving her and well, kinda ruining her life, at the child — the result of this marriage. The symbol of this marriage.
Of course I'm not excusing her actions, and no one has to feel compassion for her situation or anything. It's just something that I've been thinking about in the context of Milgram's writing of gender expectations. And it's very likely for me that this cycle of violence started not from Amane's mother.