I never said there was. I said clearly it was the near unanimous consensus of American moral theologians during the 1950's and 1960's in regards to this teaching. This number would necessarily vary society to society and age to age.
Anyone who has been to China or India, where 1 billion people are shoe-horned into an area smaller than the US east of the Mississippi could tell you there is no "need" for everyone there to have even four children anymore.
The relevant moral principles are not based upon numbers. Hermann says "which I have brought to your attention before." Yes, you have Hermann, ad nauseum, but it's still just as bogus as the first time you mentioned it.
Bogus for what reason? Will you actually address the arguments put out by the Catholic Moral Theologians? Or are you just another person who rejects guidance by the Holy See like the infamous busybody Mrs. Jeanne Dvorak?
He mentions the period 1860 - 1960, during which time the Vatican released "Arcanum," "Casti Connubii," "Allocution to the Italian Midwives" and "Address to Large Families" on the subject. None of them mention a 4-child rule.
Nobody ever claimed they did. What happened is that the Vatican clearly approved of periodic continence on five seperate occasions - rulings of the Sacred Penitentiary in 1853 and 1880, Casti Conubii in 1930, Pius XII's Allocution in 1951, and Humane Vitae in 1968. The four child rule is a practical application to modern America of when the obligation to have recourse to grave reasons ceases, i.e. when are the duties of marriage to society are fulfilled. The duty to have children is one of social justice, not sexual morality.
Neither did contemporary moral manuals of the time aimed at the average layman and woman. Several are available on EWTN, and the concept of a 4-child replacement rule is never mentioned.
This is like claiming that because Catechisms for the laity prior to the late 1800's did not commonly mention Baptism of Desire and Blood, it is therefore a false teaching. It doesn't really demonstrate anything.
We know that the magisterium has approved the use of periodic continence with certain restrictions, that's not in question. But the use of periodic continence always requires grave reasons. The requirement does not cease when you've had a certain number. None of your numerous sources claims otherwise.
The duty to have children is one of social justice, not sexual morality.
Both are involved in the issue. There is a question of social justice, but there is also an issue of sexual morality, because it involves frustrating the primary purpose of marriage. In your selection from Allocution to the Italian Midwives, some good and valid points are highlighted, but there was also this statement:
If, however, according to a reasonable and equitable judgment, there are no such grave reasons either personal or deriving from exterior circumstances, the will to avoid the fecundity of their union, while continuing to satisfy to the full their sensuality, can only be the result of a false appreciation of life and of motives foreign to sound ethical principles.Here Pope Pius XII points out the moral dilemma of wanting sex without wanting the consequent procreation which is its purpose. Presumably one could make the argument that a couple could adopt what Pope Pius XI called "virtuous continence" after having 4 or 5 children and feel confident that there was no injustice done against society. But to continue having sex, while avoiding procreation, without grave reasons is wrong according to Pope Pius XII.
It doesn't really demonstrate anything.
It clearly demonstrates that this 4-child rule never existed as a Catholic moral principle, except perhaps as the opinion of a couple of theologians regarding one particular aspect of the question. It was never "unanimous" nor widely disseminated nor binding on consciences if it was never mentioned by the Vatican nor even proposed in the moral guidance given to Catholic couples. So it demonstrates exactly what I said it demonstrates.