I'd say the gun could be either a Grade 7D or 8E, based upon the engraving and embellishment. There seems to be some overlap between the grades. Going by the serial number of this gun, and the engraved name of the company, along with the "Not Connected With Lefever Arms Co." disclaimer. I would also guess it to be a Syracuse built gun made in 1903. However, this is a trigger plate action, and I'm not sure the trigger plate action guns were built while the new company was still in Syracuse. However, I know of one Grade 9F that also has the "Not Connected With Lefever Arms Co." disclaimer, and the 9F supposedly was not introduced until 1905 when the company was in Bowling Green, Ohio.
The Grade of D.M Lefever Crossbolt guns is sometimes found written in pencil under the grip cap, and less frequently under the buttplate or trigger guard. I have one almost identical to this one, but don't know whether to call it a 7D or 8E grade because it has a semi-pistol grip. My gun also has Damascus barrels.