Praise be to Allah.
If someone is fasting to make up a missed Ramadan fast, it is not permissible for him to break the fast, according to scholarly consensus, except in the case of an excuse that makes it permissible to break the fast in Ramadan, such as falling sick.
Ibn Qudaamah (may Allah have mercy on him) said in al-Mughni (3/160): Whoever starts to do an obligatory fast, such as making up a Ramadan fast, or fulfilling a vow, whether the vow was to fast on a specific day or on any day, or an expiatory fast, it is not permissible for him to break his fast… And there is no difference of opinion concerning that, praise be to Allah. End quote.
Being invited to eat by one’s sibling is not regarded as an excuse. This makes it permissible to break the fast if one is observing a naafil (supererogatory) fast – as we shall see below – but that does not apply in the case of an obligatory fast, such as Ramadan, making up a missed obligatory fast, or fasting in fulfilment of a vow.
Based on that, what you must do is repent to Allah, may He be exalted, for having broken this fast, and not expect reward for breaking an obligatory fast! Rather the most that may be said concerning one who does that is that he is excused due to his not being aware of the ruling.
If someone is observing a supererogatory fast and is invited to eat, he has the choice between breaking the fast, or continuing to fast and offering supplication for the one who invited him, because the one who is observing a voluntary fast has the choice, because of the report narrated by Ahmad (26353) from Umm Haani’ that the Messenger of Allah (blessings and peace of Allah be upon him) came to her, and called for a drink, then he drank, then he gave the cup to her and she drank, and she said: O Messenger of Allah, I was fasting. The Messenger of Allah (blessings and peace of Allah be upon him) said: “The one who is observing a voluntary fast has the choice; if he wishes, he may continue fasting, and if he wishes, he may break his fast.”
Classed as saheeh by al-Albaani in Saheeh al-Jaami‘ (3854).
Muslim (1154) narrated that ‘Aa’ishah, the Mother of the Believers, said: The Prophet (blessings and peace of Allah be upon him) came in one day and said: “Do you have anything (to eat)? We said: No. He said: “Then I am fasting.” Then he came to us on another day and we said: O Messenger of Allah, we have been given some hays (a dish made of dates, ghee and dried yoghurt). He said: “Show it to me, for I woke up this morning fasting,” then he ate.
Muslim (1431) narrated that Abu Hurayrah said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “If one of you is invited let him respond; if he is fasting let him pray (for the people), and if he is not fasting let him eat.”
Al-Maaziri (may Allah have mercy on him) said: The words “if he is fasting let him pray (for the people)” mean: let him offer supplication for those who offered him the food, praying for forgiveness and blessing for them.
End quote from Sharh Muslim (2/154).
With regard to what you mentioned as having to so with Ja‘far as-Saadiq, it is not known to be soundly narrated from him, and it cannot be thought under any circumstances that he was referring to an obligatory fast. No attention is to be paid to the books of the Raafidis or what they narrate from the Ahl al-Bayt, for they – the Raafidis – are the most ignorant of people concerning the sunnah and reports. Most of what they narrate from as-Saadiq is falsely attributed to him.
And Allah knows best.