True Cross has part(s) of the class fragment of the True Cross
Dispersion of relics
An inscription of 359 found at Tixter, in the neighbourhood of Sétif in Mauretania (in today Algeria), was said to mention, in an enumeration of relics, a fragment of the True Cross, according to an entry in Roman Miscellanies, X, 441.
Fragments of the Cross were broken up, and the pieces were widely distributed; in 348, in one of his Catecheses, Cyril of Jerusalem remarked that the "whole earth is full of the relics of the Cross of Christ" and, in another, "The holy wood of the Cross bears witness, seen among us to this day, and from this place now almost filling the whole world, by means of those who in faith take portions from it." Egeria's account testifies to how highly these relics of the crucifixion were prized.  John Chrysostom relates that fragments of the True Cross were kept in golden reliquaries, "which men reverently wear upon their persons." Even two Latin inscriptions around 350 from today's Algeria testify to the keeping and admiration of small particles of the cross. Around the year 455, Juvenal Patriarch of Jerusalem sent to Pope Leo I a fragment of the "precious wood", according to the Letters of Pope Leo. A portion of the cross was taken to Rome in the seventh century by Pope Sergius I, who was of Byzantine origin. "In the small part is power of the whole cross", says an inscription in the Felix Basilica of Nola, built by bishop Paulinus at the beginning of 5th century. The cross particle was inserted in the altar.The Old English poem Dream of the Rood mentions the finding of the cross and the beginning of the tradition of the veneration of its relics. The Anglo-Saxon Chronicle also talks of King Alfred receiving a fragment of the cross from Pope Marinus (see: Annal Alfred the Great, year 883). Although it is possible, the poem need not be referring to this specific relic or have this incident as the reason for its composition. However, there is a later source that speaks of a bequest made to the 'Holy Cross' at Shaftesbury Abbey in Dorset; Shaftesbury abbey was founded by King Alfred, supported with a large portion of state funds and given to the charge of his own daughter when he was alive—it is conceivable that if Alfred really received this relic, that he may have given it to the care of the nuns at Shaftesbury.Most of the very small relics of the True Cross in Europe came from Constantinople. The city was captured and sacked by the Fourth Crusade in 1204. The Chronica Regia Coloniensis relates that "After the conquest of the city Constantinople inestimable wealth was found: incomparably precious jewels and also a part of the cross of the Lord, which Helena transferred from Jerusalem and [which] was decorated with gold and precious jewels. There it attained [the] highest admiration. It was carved up by the present bishops and was divided with other very precious relics among the knights; later, after their return to the homeland, it was donated to churches and monasteries." The French knight Robert de Clari wrote that "within this chapel were found many precious relics; for therein were found two pieces of the True Cross, as thick as a man's leg and a fathom in length."The misplacement of which particular class relics of the Holy Cross (and others in general) belong to, either as a result of confusion or exaggeration; and the outright forgery of relics, was a recurring controversy during the Medieval Age. This happened, often in order to attract pilgrims; or even to facilitate the lucrative practice of Simony.
By the end of the Middle Ages so many churches claimed to possess relics of the True Cross, that John Calvin  is famously said to have remarked that there was enough wood in them to fill a ship: