The tradition is based on a legend.
The tradition is based on a legend.I understand your skepticism about the discovery of the True Cross. It is true that the earliest reference to Helena's discovery of the Cross (in Socrates Scholasticus' Historia ecclesiastica) dates from about a hundred years after the alleged event.
But Eusebius refers to Helena's pilgramage to the Holy Land, and as early as the 348--within 20 years of Helena's visit--pilgrams were vernerating relics they believed to be pieces of the True Cross, as referenced in the Catecheses of Cyril of Jerusalem.