Going through my data, I think the problem has been solved. We just assumed (at least I did) that once Greener showed how choke boring worked, and the results of "The Field" trial were published in June 1875 and the proof laws were changed, that everybody instantly started used choked shotguns. IGC assume that as well. Sort of the same assumption was made when the A&D box lock came out.
Not true evidently. Most of the Reillys made from 1875 - through 1880 (SN 19450 + or - to 23300 + or -) appeared to have been cylinder bores. There were a few with legitimate chokes. It wasn't really until 1881-82 that most Reilly's were sold with choke barrels and even then, there are still cylinder bores found. So I think that pretty well explains it. This means Ted likely was right about 20740 originally having had 13 gauge cylinder-bore barrels.
I'll post the Reilly barrel flats of the guns looked as examples:
19286 (Feb 1875) - E.M. Reilly & Co., (address not mentioned). 10 bore. Shotgun SxS. Side lever?, Rebounding hammer gun. H.Walker patent 455, 12 Feb 1872 (use #1098); no "not for ball" or "choke."
20249 (fall 1875) - E.M. Reilly & Co., Oxford St., London & rue Scribe Paris; Shotgun SxS, 12 bore, U-L hammer gun. no "not for ball" or "choke."
20255 lfall 1875) - E.M. Reilly & Co., (address not mentioned). 12 ga, Shotgun SxS; top lever, hammer gun. First top lever?
20466 (early 1876) - E.M, Reilly & Co., Oxford Street, London. 10 Bore. Shotgun SxS. Top-Lever hammer gun. Purdey pat 1104 use #3463; no "not for ball" or "choke."
20623 (spring 1877) - E.M. Reilly & Co., New Oxford St., London and Rue Scribe, Paris. 12 bore. Shotgun SxS. U-L, rebounding hammer gun. Purdey patent 1104 use #4928
22077 (summer 1879) - (no name in ad). 12ga. Shotgun SxS. Hammerless, U-L, Whitworth patent First hammerless gun?
24534 (Nov 1882)- E.M. Reilly & Co., (address not mentioned). 12 bore SxS Shotgun pigeon gun; top lever, hammer gun. (Cyril Adams' pigeon gun)