What seems clear to me is that these numbers are not serial numbers as found on modern weapons, Boutets system of numbering must have had other purposes. That is what I hope find out.
I have for some time compiled a list with description of Boutet weapons with numbers, and invite collectors or museums who has numbered Boutet weapons to get in contact with me on Boutet @ ArmHis.com

Disassembling of weapons to look for hidden numbers when carefully done will be a useful check on status of hidden parts, and stop possible corrosion and degradation of a cultural heritage.
Anybody contributing to updating of the list will receive a pdf-copy of the list, and later updates.
Owners of numbered weapons will remain anonymous.

Precise dating of this gun is unknown, but it is similar to a double barrel sporting gun pictured and described on page 57 in the 1906-book by Capitaine Bottet. The gun is there pictured on plate XXXVIII on the last page of the book, and decribed on page 57 as a double barreled steel mounted sporting gun, offered by the "Directoire" to the "Directeur Reubell". There is however no mentioning in the book of a number on that gun.

     Some data on the gun: Length overall 127 cm, calibre 15,4mm, length of barrel 87cm + culasse 5,7cm, and weight 2,62 kg. Steel mounted with legant wood carvings. The thumb plate in gold without monogram.
Condition as new - "dans son jus!"as they say in France