Concerning the red inventory no. on your barrel:
It has become a dubious tradition in the market to attribute any piece bearing a red inventory number to the Hohenzollern Collection, Schloss Sigmaringen, as this is renowned to be a top provenance. The expert at Bonhams was aware of this fact and wisely did not refer to it concerning this barrel.
It is true that many items were sold from that Princely Collection at various times, mainly between the two Word Wars and shortly after WW II.
I have been to Sigmaringen by special appointment 20 years ago on Monday when the collections were closed to the public, and did extensive research there. I was allowed to go behind the fences (there are fences two meters high indeed!) and handle and photograph each piece I liked! In return, I sort of 'updated' their 19th c. inventory - which was a bad bargain because it took me weeks ...
One of the results was:
all Hohenzollern Sigmaringen items I remember bore two groups of numbers, a four-digit item no. plus an additional two-digit group number, e.g. 21 for 'matchlocks' (just an example), the latter stamped in red capital letters!
So sorry but three red digits are not enough to prove a Sigmaringen provenance.
In my collection is an extremely rare combined musket rest and pike, 2nd half 16th c., the haft retaining its well-made iron shoe and
painted with two groups of Hohenzollern red inventory nos., including '50' for hafted weapons. I bought it from the former Zeughaus (armory) Überlingen/Bodensee some 20 years ago when the museum was in liquidation; the owner had acquired it directly from Schloss Sigmaringen in the 1950's.
Of course there are various other items with red inventory nos. in my collection as well but I would never claim them to come from Sigmaringen.
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