“Jesus looked at him, and said, "So you are Simon the son of John? You shall be called Cephas" (which means Peter).”
Short answer
Christ is the one foundation and the head of his Church, and he chose to build it on Peter, to whom he gave the name Rock. The Fathers read the rock of Matthew 16:18 as Peter, as the faith Peter confessed, and as Christ, and some of them give more than one of these readings. The readings do not compete. Peter is rock because of the faith he confessed, that faith rests on Christ, and the Catechism holds all three together.
Drafted from primary sources. Every quotation is checked against the page it links to.
Where this teaching appears on the Offshoots timelines: c. 400-430, The rock as what Peter confessed: an early reading; 1559, The rock as Peter's confession, used against the papal office.
16 quotations in date order
“Simon Peter replied, "You are the Christ, the Son of the living God." And Jesus answered him, "Blessed are you, Simon Bar-Jona! For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are Peter, and on this rock I will build my church, and the powers of death shall not prevail against it.”
“According to the commission of God given to me, like a skilled master builder I laid a foundation, and another man is building upon it. Let each man take care how he builds upon it. For no other foundation can any one lay than that which is laid, which is Jesus Christ.”
“So then you are no longer strangers and sojourners, but you are fellow citizens with the saints and members of the household of God, built upon the foundation of the apostles and prophets, Christ Jesus himself being the cornerstone”
“Peter alone do I find — through (the mention of) his mother-in-law — to have been married. Monogamist I am led to presume him by consideration of the Church, which, built upon him, was destined to appoint every grade of her Order from monogamists.”
“And if we too have said like Peter, You are the Christ, the Son of the living God, not as if flesh and blood had revealed it unto us, but by light from the Father in heaven having shone in our heart, we become a Peter, and to us there might be said by the Word, You are Peter, etc. … For a rock is every disciple of Christ of whom those drank who drank of the spiritual rock which followed them”
“This faith it is which is the foundation of the Church; through this faith the gates of hell cannot prevail against her. This is the faith which has the keys of the kingdom of heaven.”
“As I follow no leader save Christ, so I communicate with none but your blessedness, that is with the chair of Peter. For this, I know, is the rock on which the church is built! … This is the house where alone the paschal lamb can be rightly eaten.”
“And I say unto you, You are Peter, and upon this rock will I build my Church; … that is, on the faith of his confession. Hereby He signifies that many were now on the point of believing, and raises his spirit, and makes him a shepherd.”
“On this rock, therefore, He said, which you have confessed, I will build my Church. For the Rock (Petra) was Christ; … and on this foundation was Peter himself also built. For other foundation can no man lay than that is laid, which is Christ Jesus. … The Church, therefore, which is founded in Christ received from Him the keys of the kingdom of heaven in the person of Peter, that is to say, the power of binding and loosing sins.”
“…the blessed Peter persevering in the strength of the Rock, which he has received, has not abandoned the helm of the Church, which he undertook. For he was ordained before the rest in such a way that from his being called the Rock, from his being pronounced the Foundation, … we might know the nature of his association with Christ. … For this, dearly-beloved, was gained by that confession, which, inspired in the Apostle's heart by God the Father, transcended all the uncertainty of human opinions, and was endued with the firmness of a rock, which no assaults could shake.”
“As Augustine says (Tract. xlvi in Joan.): "If the rulers of the Church are Shepherds, how is there one Shepherd, except that all these are members of one Shepherd?" So likewise others may be called foundations and heads, inasmuch as they are members of the one Head and Foundation.”
“It was to Simon alone, to whom he had already said You shall be called Cephas, that the Lord, after his confession, You are the Christ, the son of the living God, spoke these words: Blessed are you, Simon Bar-Jona. For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are Peter, and on this rock I will build my church”
“…the universal Church, which the Lord established on the apostles and built upon blessed Peter, their chief, Christ Jesus Himself being the supreme cornerstone.”
“Moved by the grace of the Holy Spirit and drawn by the Father, we believe in Jesus and confess: 'You are the Christ, the Son of the living God.' On the rock of this faith confessed by St. Peter, Christ built his Church.”
“Our Lord then declared to him: "You are Peter, and on this rock I will build my Church, and the gates of Hades will not prevail against it." Christ, the "living Stone", thus assures his Church, built on Peter, of victory over the powers of death. Because of the faith he confessed Peter will remain the unshakeable rock of the Church.”