Executive summary: Heinsohn shows that placing Charlemagne's dynasty within the late Roman Empire solves the problem of a lack of post-Roman archaeological (stratigraphical) evidence, as well as the problem of why Charlemagne's coins are often found with Roman coins that are supposedly hundreds of years older. Heinsohn leaves Charlemagne in the 9th century AD but moves the stratigraphical evidence for the catastrophic end of the Roman Empire up to that century as well, removing a 300-year gap in the process.
Gunnar Heinsohn's thesis is presented as follows:
For nearly a quarter century, Charlemagne’s very existence has been called into doubt by the Illig-school of chronology criticism. This challenge to Charlemagne’s historicity is based on the fact that archaeological strata for his Franks, as well as for their Saxon arch-enemies, are extremely difficult to locate. Therefore, mainstream historians cannot present convincing archaeological proof for keeping Charlemagne in the history books. Neither, however, can the experts rule out Europe's most famous monarch because hard evidence in the form of coins does exist. These coins are sometimes surprising because they may be found lumped together with Roman coins that are 700 years older. The coins also show Germanic rulers crowned with Roman imperial diadems. The dissidents who deny Charlemagne’s existence and the mainstream historians who defend it are therefore locked in a stalemate. Because, beyond all their disagreements, they share two profound convictions: (1) One must not search for Charlemagne within the Roman period. (2) The classical Roman period precedes the events of the Ottonian period of the later 10th century by a lengthy expanse of time. Yet, immediately beneath strata of the mid-10th c. stratigraphy shows the Roman period coming to a catastrophic end. It is in that Roman period of the 8th-10th c., currently dated to the 1st-3rd c. or to the 4th-6th CE, where hard evidence of Carolingians, Franks, Saxons etc. is plentiful and where their Roman iconography and art makes sense. Charlemagne blossoms in the later 9th c. and dies at the beginning of the 10th c. CE in a millennium of whose 1000 years only some 300 -- at any individual site -- can provide building strata in the ground.
Full paper hosted here: http://q-mag.org/_media/gunnar-charlemagne-correct-place-in-history-032914.pdf