Question: The adopted son and heir of the founder of the Roman Principate was given what name at birth?

Context paragraphs:
Takatsukasa Mototeru: Takatsukasa Mototeru (鷹司 基輝 , April 19, 1727 – July 6, 1743) , son of Ichijō Kaneka and adopted son of Hisasuke, was a "Kugyō" or Japanese court noble of the Edo period (1603–1868).  Sukehira was his adopted son.

Gaius Caesar: Gaius Caesar (20BC – 21 February 4AD), formally Gaius Julius Caesar (Latin: "Caius Caesar Augustus filius"), was the adopted son and heir of Augustus.  He was born the son of Augustus' trusted friend and general, Marcus Vipsanius Agrippa.  His name at birth was Gaius Vipsanius Agrippa, but it later changed to Gaius Julius Caesar, following his adoption out of the "Vipsanii" and into the "Julii" by Augustus.  He along with his brother Lucius, born in 17BC, were intended to rule the Roman Empire together.

Takatsukasa Fuyuhira: Takatsukasa Fuyuhira (鷹司 冬平 , 1275 – 1327) , son of Kanetada and adopted son of Mototada, was "kugyo" or highest-ranking Japanese court noble of the Kamakura period (1185–1333).  Morohira was his son.  Fuyunori was his adopted son.  He held court positions as follows:

Abdur-Razzaq Nurul-Ain: Syed Abdur-Razzaq Nurul-Ain is the successor of the Sufi saint Syed Ashraf Jahangir Semnani.  From amongst the descendants of Syed Abdul Razzaq Jilani, the line of saints of Ashrafia Jilania is one of the most reputed households belonging to the Indo-Pak subcontinent.  Within this line, Syed Abdul Razzaq, popularly known as Nur-ul-Ain, was the heir, disciple and Khalifa of Syed Ashraf Jehangir Semani.  Syed Abdul Razzaq was the son of his maternal cousin.  Syed Abdul Razzaq is the 11th descendant of the greatest Sufi Syed Abdul Qadir Jilani of Jilan, Iraq.  Abdur-Razzaq Nurul-Ain first met Ashraf Jahangir Semnani at the age of twelve years in Baghdad when Ashraf Jahangir Semnani made a visit there and from there on never parted from his company.  He adopted Syed Abdul Razzaq as his son and made him the heir and caretaker.  Originated from the name of Syed Ashraf, this line of saints is still called as Ashrafia.  Syed Ashraf Jehangir Semani died in 808 AH and Syed Abdul Razzaq became the heir to his throne.  After strenuous spiritual training he was bestowed with Khilafat (Spiritual Successor) and from him the Ashrafi spiritual chain flourished.  According to the tradition of Mirat-ul-Asrar, at the time of his death, Syed Ashraf Jehangir Semani was either 106 or 110 years of age.  In Tohfta ul Abrar, his age is written as 120 and year of birth is 688 AH.  Even his adopted son, Syed Abdul Razzaq was 120 of age at the time of his death.  He spent 12 years before he took Bayat and 68 years in travel and in the service of Syed Ashraf Jehangir Semani and the remaining 40 years after the death of his Murshid at the throne of Khilafat.  In accordance to this, his birth year becomes 728, year of arrival in India 740 and year of death 848.  His grave is located next to that of Ashraf Jahangir Semnani in the same Shrine in Kichauccha Sharif, Dist.  Ambedkar Nagar, Uttar Pradesh, India.

Murong Xi: Murong Xi (; 385–407), courtesy name Daowen (道文), formally Emperor Zhaowen of (Later) Yan ((後)燕昭文帝), was an emperor of the Xianbei state Later Yan.  He was one of the youngest sons of Murong Chui (Emperor Wucheng), and after the death of his nephew Murong Sheng (Emperor Zhaowu) became emperor due to his affair with Murong Sheng's mother, Empress Dowager Ding.  He was regarded as a cruel and capricious ruler, who acted at the whims of himself and his wife, Empress Fu Xunying, greatly damaging the Later Yan state.  After Empress Fu died in 407, he left the capital Longcheng (龍城, in modern Jinzhou, Liaoning) to bury her, and the soldiers in Longcheng took this chance to rebel and replace him with Murong Bao's adopted son Murong Yun (Emperor Huiyi), and Murong Xi himself was captured and killed.  (Because Murong Yun was an adopted son who later changed his name back to Gao Yun, some historians treat Murong Xi as the last emperor of Later Yan and Gao Yun as the first emperor of Northern Yan, while others treat Gao Yun as the last emperor of Later Yan and his successor Feng Ba as the first emperor of Northern Yan.)

King Romapada: Romapada or Chitraratha was the youngest son King Vidarbha of Yadava and an adopted son of King Dharmaratha of Anga.  King Vidarbha had many sons but his friend King Dharmaratha had no son and so King Dharmaratha adopted his friend King Vidarbha's youngest son Romapada.  When King Vidarbha became old made Vidarbha Kingdom and made Kratha as the next Yadava ruler and made his second son Bhima as the ruler of Vidarbha and went to forest.  King Dharmarata named his adopted son as Chitraratha and also that people called him Romapada.  When King Dharmaratha grew old , He decided to make Romapada as the ruler of Anga.

Augustus: Augustus (Latin: "Imperātor Caesar Dīvī Fīlius Augustus" ; 23 September 63 BC – 19 August 14 AD) was the founder of the Roman Principate and considered the first Roman emperor, controlling the Roman Empire from 27 BC until his death in AD 14.  He was born Gaius Octavius into an old and wealthy equestrian branch of the plebeian "gens" Octavia.  His maternal great-uncle Julius Caesar was assassinated in 44 BC, and Octavius was named in Caesar's will as his adopted son and heir, then known as Octavianus (Anglicized as Octavian).  He, Mark Antony, and Marcus Lepidus formed the Second Triumvirate to defeat the assassins of Caesar.  Following their victory at the Battle of Philippi, the Triumvirate divided the Roman Republic among themselves and ruled as military dictators.  The Triumvirate was eventually torn apart by the competing ambitions of its members.  Lepidus was driven into exile and stripped of his position, and Antony committed suicide following his defeat at the Battle of Actium by Octavian in 31 BC.

Julii Caesares: The Julii Caesares were the most illustrious family of the patrician "gens Julia".  The family first appears in history during the Second Punic War, when Sextus Julius Caesar was praetor in Sicily.  His son, Sextus Julius Caesar, obtained the consulship in 157 BC; but the most famous descendant of this stirps is Gaius Julius Caesar, a general who conquered Gaul and became the undisputed master of Rome following the Civil War.  Having been granted dictatorial power by the Roman Senate and instituting a number of political and social reforms, he was assassinated in 44 BC.  After overcoming several rivals, Caesar's adopted son and heir, Gaius Julius Caesar Octavianus, was proclaimed Augustus by the senate, inaugurating what became the Julio-Claudian line of Roman emperors.

Quintus Caecilius Metellus Creticus Silanus: Quintus Caecilius Metellus Creticus Silanus, born Iunius Silanus was adopted by Quintus Caecilius Metellus, a descendant of the optimate Quintus Caecilius Metellus Creticus and the natural son of Marcus Junius Silanus.  He was a Consul in AD 7 and governor of Syria from AD 13 to 17.  Silanus was socially connected with the then-heir to the Roman principate Germanicus, his daughter at one time was betrothed to Germinicus' son Nero.

Constitution of the Late Roman Empire: The constitution of the late Roman Empire was an unwritten set of guidelines and principles passed down, mainly through precedent, which defined the manner in which the late Roman Empire was governed.  As a matter of historical convention, the late Roman Empire emerged from the Roman Principate (the early Roman Empire), with the accession of Diocletian in AD 284, his reign marking the beginning of the Dominate.  The constitution of the Dominate ultimately recognized monarchy as the true source of power, and thus ended the fiction of dyarchy, in which emperor and Senate governed the empire together.

Supporting paragraphs: Gaius Caesar Augustus
Answer: Gaius Vipsanius Agrippa