There are no markings on the barrel covered by the stock. It is obviously an ordinary Springfield Armory barrel. It does have he grove from the ping hat held on the standard rear sight

As to James V. Howe's Modern Gunsmith. There are no pictures of any Pope barrels. He does say:

"The fact hat Harry Pope was able to produce his matchless barrels with no other machine than an old lathe, that he could fashion makeshift barrels with a makeshift contrivance would excite no wonder, but that he could and did make barrels superior to those produced by machines designed and built for this particular purpose is a tribute to his patience and ingenuity, and should dispose of the word 'can't' forever" Volume 2, page 3)