Basically, the formula allows the Rota to grant an annulment even if the grounds for doing so weren't the ones originally specified.
For example, a person might seek an annulment on the basis that one of the parties didn't believe in marriage indissolubility, that is, the fact that marriage is a lifelong commitment.
The court could rule that while that point couldn't be proven, the marriage was obviously null for another reason, such as coercion, and declare it so.
This formula was previously used by the Rota, allowing them to grant an annulment on those grounds, however during his pontificate John Paul II required the court to judge the case only on the grounds originally specified, meaning that the person or couple seeking an annulment had to start the process over if their original claim was unable to be proven.
Francis' move, then, can be seen as a continued effort to reach out to the Church's "most fragile sons and daughters, marked by wounded and lost love," as he said in the rescript text in reference to the most recent Synod of Bishops on the Family, which placed special emphasis on reaching out to divorced and remarried Catholics.
Of particular importance in the process is the principle of "generic doubt," under which an a marriage can be declared null when the case off nullity is obvious, even without any specific grounds declared.
In the rescript, Pope Francis also clarified that "there shall be no appeal" against the decisions made by the Rota in matters of the nullity of sentences or decrees.
Under the new procedures, a first judgment is always made by the diocesan tribunal. However, if one or both of the spouses seeking an annulment disagree with the ruling, they may appeal to the Rota for a second judgment.
In the rescript, Francis established that if a cause of nullity arrives to the third degree of judgment, it cannot be proposed again "after one of the parties has contracted a new canonical marriage, unless the decision can be demonstrated to be manifestly unjust."
It was also declared that the Dean of the Rota, then, "has the authority to dispense with the Norms of the Tribunal of the Roman Rota in procedural matters for a serious cause."
Another aspect of the rescript, expressing the explicit wish of Eastern Church leaders, is that their local tribunals will now have jurisdiction over the "iurium," or "human rights" aspects of marriage annulment cases that come before the Rota for an appeal.
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This is particularly relevant for churches in the Middle East, where numerous countries have no civil law on marriage cases, and often depend on religious courts. This means that when cases arrive to the Rota from these areas, the court will also judge on questions such as alimony and child custody.
In his final point in the rescript Francis says that the Rota must offer to all "the principal of evangelic gratuity," meaning that those seeking the annulment won't have to pay the lawyer for the cause, but that the costs will presumably be absorbed by the Rota itself.
However, the Pope did say that wealthier parties have a "moral obligation" to make "a just contribution towards the causes of the poor."