"What paths exist in the Catholic Church for a certain theological position to become recognized as dogma?

What steps are taken for a dogma to be defined and declared within the Catholic Church.

How certain theological subjects became dogma is quite an interesting topic in itself. Here are a few manners in which some sacred doctrines actually became declared dogma and thus binding for all Catholics to believe. The pope may or may not have recourse to a Council. He will in any case take council of prudent and trustworthy theologians. At times, He may ask input from all the bishops (as well as some trustworthy theologians) for their input by letter before a certain date, so that the dogma may be drafted by competent theologians and thus ratified by the pope himself.

Let us look at the Immaculate Conception of Mary.

Pius IX, at the beginning of his pontificate, and again after 1851, appointed commissions to investigate the whole subject, and he was advised that the doctrine was one which could be defined and that the time for a definition was opportune.

It was not until 1854 that Pope Pius IX, with the support of the overwhelming majority of Roman Catholic bishops, whom he had consulted between 1851–1853, promulgated the papal bull Ineffabilis Deus (Latin for "Ineffable God"), which defined ex cathedra the dogma of the Immaculate Conception.

We declare, pronounce, and define that the doctrine which holds that the most Blessed Virgin Mary, in the first instance of her conception, by a singular grace and privilege granted by Almighty God, in view of the merits of Jesus Christ, the Savior of the human race, was preserved free from all stain of original sin, is a doctrine revealed by God and therefore to be believed firmly and constantly by all the faithful. — Pope Pius IX, Ineffabilis Deus, December 8, 1854.

The dogma was defined in accordance with the conditions of papal infallibility, which would be defined in 1870 by the First Vatican Council.

The papal definition of the dogma declares with absolute certainty and authority that Mary possessed sanctifying grace from the first instant of her existence and was free from the lack of grace caused by the original sin at the beginning of human history. Mary's salvation was won by her son Jesus Christ through his passion, death, and resurrection and was not due to her own merits.Immaculate Conception

Amongst others who were consulted was the Father Abbot of Solesmes, Dom Prosper Guéranger. We must remember that the subject of Mary's Immaculate Conception goes back many centuries, before the Church believed that the opportune moment in time had arrived.

Further information can be found in these sources:

How Did the Immaculate Conception Teaching Come About?

Immaculate Conception Became Catholic Doctrine

At for the Assumption of Mary, there was a lengthy history behind Pope Pius' definition and declaration into Catholic dogma.

On 1 November 1950, in the Apostolic Constitution Munificentissimus Deus Pope Pius XII declared the Assumption of Mary as a dogma:

By the authority of our Lord Jesus Christ, of the Blessed Apostles Peter and Paul, and by our own authority, we pronounce, declare, and define it to be a divinely revealed dogma: that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory.

Pope Pius XII deliberately left open the question of whether Mary died before her Assumption.

Before the dogmatic definition, in Deiparae Virginis Mariae Pope Pius XII sought the opinion of Catholic Bishops. A large number of them pointed to the Book of Genesis (3:15) as scriptural support for the dogma. In Munificentissimus Deus (item 39) Pius XII referred to the "struggle against the infernal foe" as in Genesis 3:15 and to "complete victory over the sin and death" as in the Letters of Paul as a scriptural basis for the dogmatic definition, Mary being assumed to heaven as in 1 Corinthians 15:54: "then shall come to pass the saying that is written, Death is swallowed up in victory". - Assumption of Mary

Further information On Mary's Assumption can be found in these sources:

The history of the Assumption – and why it's a Holy Day of Obligation

Assumption: History of Doctrine

Papal infallibility doctrine came about in a little different manner than the two previous dogmas concerning Mary. Nevertheless, it too had a long history prior to Rome making it binding on all Catholics. This came about during the First Vatican Council at Rome

The infallibility of the pope was formally defined in 1870, although the tradition behind this view goes back much further. In the conclusion of the fourth chapter of its Dogmatic Constitution on the Church Pastor aeternus, the First Vatican Council declared the following, with bishops Aloisio Riccio and Edward Fitzgerald dissenting:

We teach and define that it is a dogma Divinely revealed that the Roman pontiff when he speaks ex cathedra, that is when in discharge of the office of pastor and doctor of all Christians, by virtue of his supreme Apostolic authority, he defines a doctrine regarding faith or morals to be held by the universal Church, by the Divine assistance promised to him in Blessed Peter, is possessed of that infallibility with which the Divine Redeemer willed that his Church should be endowed in defining doctrine regarding faith or morals, and that therefore such definitions of the Roman pontiff are of themselves and not from the consent of the Church irreformable.

So then, should anyone, which God forbid, have the temerity to reject this definition of ours: let him be anathema. (see Denziger §1839). — Vatican Council, Sess. IV, Const. de Ecclesiâ Christi, Chapter iv

According to Catholic theology, this is an infallible dogmatic definition by an ecumenical council. Because the 1870 definition is not seen by Catholics as a creation of the Church, but as the dogmatic revelation of a truth about the papal magisterium, papal teachings made prior to the 1870 proclamation can, if they meet the criteria set out in the dogmatic definition, be considered infallible. Ineffabilis Deus is an example of this. - Papal infallibility

Of all the bishops at Rome who voted in favour of the declaration of papal infallibility, only two refrained from voting. Bishop Fitzgerald of Little Rock< Arkansas let Rome before the vote, so in fact did not vote at all. This act that earned an entry in Universal Church history as “Big Rock versus the Little Rock.”

From 1869 to 1870, he attended the First Vatican Council. At the Council, Fitzgerald was one of the only two bishops (the other being Aloisio Riccio) to vote against papal infallibility. While he believed in the theological grounds for infallibility, he feared that its dogmatic definition would hamper the conversion of non-Catholics in Arkansas. However, he fully submitted to the Council's decision when the tally ended. - Edward Fitzgerald (bishop)

It is possible that yet more Catholic dogmas may be defined and proclaimed in the future. Like the ones mentioned above they too have a recorded history within the theological debates in Catholic circles. Only time will tell if the "opportune" moment will ever be at hand. Will they be define through papal consultation with eminent Catholic theologians and bishops or through a future council only time can reveal this secret.

Some topics for possible future dogmas (proposed dogmas) such as Mary Mediatrix of All Graces. In any case more future study is still necessary in this area. Many canonized saints have already professed their belief that Mary is the Mediatrix of All Graces. But how or when the Church will define such things is still unknown.

When will the "opportune time" arrive?

Will it be by council or consultation?

Every once and awhile one can come across a papal petition asking the Holy See to to declare and define a particular subject as binding on Catholic belief such as the following: Petition For the Papal definition of Mary, Coredemptrix Mediatrix and Advocate

It seems you're asking how a material (definable) dogma becomes a formal (defined) dogma.

• material (definable) dogma: a truth revealed by God that the Church has not yet proposed one has to believe under pain of mortal sin against faith and excommunication
  ^{analogous to a legislator's "bill"}

• formal (defined) dogma: "A truth proposed by the Church as revealed by God"*
  ^{analogous to a legislator's "law"}

^{*On the Value of Theological Notes and the Criteria for Discerning Them by Father Sixtus Cartechini S.J. (Rome, 1951)
(source)}

Steps a pope takes to define a dogma

1. Determine if the proposition is true and definable

Before defining dogma, a pope first needs to establish that the truth he intends to define can be defined.

Certain things are not definable at all, because they are not true or known to be true (e.g., "black = white" or "the world will end at 12:34 PM GMT on January 24, 2025"*) or because they do not pertain to the faith or morals (e.g., "2+2=4" couldn't be defined as a dogma).

^{*Knowing this would be contrary to Revelation, which says the time and hour is only known to the Father in heaven.}

Example

To determine the definability of the dogma of the Assumption, Pope Pius XII consulted all the world's bishops, as he relates in Munificentissimus Deus:

11. And, since we were dealing with a matter of such great moment and of such importance, we considered it opportune to ask all our venerable brethren in the episcopate directly and authoritatively that each of them should make known to us his mind in a formal statement. Hence, on May 1, 1946, we gave them our letter "Deiparae Virginis Mariae," a letter in which these words are contained: "Do you, venerable brethren, in your outstanding wisdom and prudence, judge that the bodily Assumption of the Blessed Virgin can be proposed and defined as a dogma of faith? Do you, with your clergy and people, desire it?"

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Although a pope has the authority to define a dogma without asking for counsel regarding its definability, it would be rash not to seek counsel.
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2. Define it.

Once it is established that the proposition is a definable truth (material dogma), the material dogma can become a formal dogma if the pope

1. speaks for the whole Church,
2. invokes all his authority,
3. intends to definitively define the dogma, and
4. speaks regarding the faith and/or morals.

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Example

The declaration/pronouncement/definition of the dogma of the Assumption (Munificentissimus Deus):

44. […] by the authority of our Lord Jesus Christ, of the Blessed Apostles Peter and Paul, and by our own authority, we pronounce, declare, and define it to be a divinely revealed dogma: that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory.