Question about Chalcedon

The most glorious judges and the great senate said: Let there be read also the epistle of the most worthy Leo, Archbishop of Old Rome, the Imperial City.This is second notice, that the Authority of Rome was only primate and not supreme...
After the reading of the foregoing epistle, the most reverend bishops cried out: This is the faith of the fathers, this is the faith of the Apostles. So we all believe, thus the orthodox believe. Anathema to him who does not thus believe. Peter has spoken thus through Leo. So taught the Apostles. Piously and truly did Leo teach, so taught Cyril. Everlasting be thememory of Cyril. Leo and Cyril taught the same thing, anathema to him who doesnot so believe. This is the true faith. Thoseof us who are orthodox thus believe. Thisis the faith of the fathers. Why were not these things read at Ephesus [i.e. at the heretical synod held there]? These are the things Dioscorus hid away.

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This is a quote that Catholics use to prove the infallibility of Leo but that wasnt the case if that was the case then Cyril also would be infallible. So we should understand that this what they meant that Leo was correct on teaching not that he was infallible.

it was found that to Flavian, the Archbishop of Constantinople, was given only the fifth place. Against this the bishop protested and asked, "Why did not Flavian receive his position?" and the papal legate Paschasinus answered: "We will, please God, recognize the present bishop Anatolius of Constantinople as the first [i.e. after us], but Dioscorus made Flavian the fifth." It would seem to be in vain to attempt to escape the force of these words by comparing with them the statement made in the last session, in a moment of heat and indignation, by Lucentius the papal legate, that the canons of Constantinople were not found among those of the Roman Code.

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Another quote that proves primacy and not supremacy is this quote...
Paschasinus, the most reverend bishop and representative, read: Canon Six of the 318 holy fathers, "The Roman Church hath always had the primacy. Let Egypt therefore so hold itself that the bishop of Alexandria have the authority over all, for this is also the custom as regards the bishop of Rome. So too at Antioch and in the other provinces let the churches of the larger cities have the primacy. [In the Greek `let the primacy be kept to the churches;' a sentence which I do not understand, unless it means that for the advantage of the churches the primatial rights of Antioch must be upheld. But such a sentiment one would expect to find rather in the Latin than in the Greek.] And one thing is abundantly clear, that if any one shall have been ordained bishop contrary to the will of the metropolitan, this great synod has decreed that such an one ought not to be bishop. If however the judgment of all his own [fellows] is reasonable and according to the canons, and if two or three dissent through their own obstinacy, then let the vote of the majority prevail. For a custom has prevailed, and it is an ancient tradition, that the bishop of Jerusalem be honoured, let him have his consequent honour, but the rights of his own metropolis must be preserved."This is evident that the authority is only primale and not supreme..
And to end it off that Constantinople is the New Rome and that it was viewed this way is these two quotes.

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the same secretary read from the same codex the determination of the Second Synod. "These things the bishops decreed who assembled by the grace of God in Constantinople from far separated provinces, ... and bishops are not to go to churches which are outside the bounds of their dioceses, nor to confound the churches, but according to the canons the bishop of Alexandria shall take the charge of the affairs of Egypt only, and the bishops of Orient shall govern the Oriental diocese only, the honours due to the Church of Antioch being guarded according to the Nicene canons, and the Asiatic bishops shall care for the diocese of Asia only, and those of Pontus the affairs of Pontus only, and those of Thrace the affairs of Thrace only. But bishops shall not enter uncalled another diocese for ordination, or any other ecclesiastical function. And the aforesaid canon concerning dioceses being observed, it is evident that the synod of every province will administer the affairs of that particular province as was decreed at Nice. But the churches of God in heathen nations must be governed according to the custom which has prevailed from the times of the Fathers. The bishop of Constantinople however shall have the prerogative of honour next after the bishop of Rome, because Constantinople is new Rome.If we place ourselves for a moment in the position of the ecclesiastics of Constantinople when they heard Pasehasinus read his "version," which the Ballerini gently describe as "differing a little" from the Greek text, we shall see that it was simply impossible for them not to quote that text as it was preserved in their archives, and had been correctly translated by Philo and Evarestus in their version beginning "Antiqui mores obtineant." No comment on the difference between it and the Roman "version" is recorded to have been made: and, in truth, none was necessary. Simply to confront the two, and pass on to the next point, was to confute so that the bishop of Alexandria shall have jurisdiction over all, since this also is the custom at Rome. Likewise at Antioch and in the rest of the provinces, let the rank (presbeia) be preserved to the churches. For this is absolutely clear that if anyone contrary to the will of the metropolitan be ordained bishop, such an one the great synod decreed should not be a bishop. If however by the common vote of all, rounded upon reason, and according to the canons, two or three moved by their own obstinacy, make opposition, let the vote of the majority stand."I May conclude that Rome wasnt supreme like the Vatican argues, it was only primate like the Orthodox argues thank you.
Also if you guys argue differently remember this quote exist...
The most glorious judges said: From what has been done and brought forward on each side, we perceive that the primacy of all (pro pantwn ta prwteia) and the chief honour (thn ecaireton timhn) according to the canons, is to be kept for the most God-beloved archbishop of Old Rome, but that the most reverend archbishop of the royal city Constantinople, which is new Rome, is to enjoy the honour of the same primacy, and to have the power to ordain the metropolitans in the Asiatic, Pontic, and Thracian dioceses, in this manner: that there be elected by the clergy, and substantial (kthtorwn) and most distinguished men of each metropolis and moreover by all the most reverend bishops of the province, or a majority of them, and that he be elected whom those afore mentioned shall deem worthy of the metropolitan episcopate and that he should be presented by all those who had elected him to the most holy archbishop of royal Constantinople, that he might be asked whether he [i.e., the Patriarch of Constantinople] willed that he should there be ordained, or by his commission in the province where he received the vote to the episcopate. The most reverend bishops of the ordinary towns should be ordained by all the most reverend bishops of the province or by a majority of them, the metropolitan having his power according to the established canon of the fathers, and making with regard to such ordinations no communications to the most holy archbishop of royal Constantinople. Thus the matter appears to us to stand. Let the holy Synod vouchsafe to teach its view of the case.

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