We can't cite Round to prove Round right! ::sigh::
The pedigree was entered into the College of Arms by Cooke in 1577(approx) (he served in this capacity from 1567 - 1592). There's an entire passage about it in Copinger's book. He talks about Dugdale mentioning (in his book Antiquities of Warwickshire) a passage written by Cooke and a few other heraldic authorities examining the original documents. They all agreed on the pedigree and entered the information into the heraldic archives. The issues Round mentioned were:
-the lineage before Adam de Carington (which later is proven)
-the link from Mr. Robert-Carington (the donor of the materials to Copinger) to the Smiths of Rivenhall (this doesn't exist according to the Heralds.)
Pg. 77 of Copinger's book:
"Sir William Dugdale, Garter Principal King of Arms, in his "Antiquities
of Warwickshire," says : " I have seen attested by Sir William Dethick, Garter
Principall King of Arms, and Robt. Cooke, Clarenceux, of this John Carington
as to the Occasion that he changed his name to Smyth, they do specially
Certifie from the Credit of an antient Manuscript, then in the possession of
Henry Smith of Cressing Temple, Essex, Written with the proper Hand of
him the said John in K. Henry the Fourth's Time ; that in his youth he was
bred up in Gascoigne under Sir Thomas Nevill, in the Service of King Ric. 2,
and after the Decease of an elder Brother, called Edmund,..."
One of these same documents was later found in the possession of The Nevills of Holt in the 1870's during a historical manuscript survey of the country.