Dear Confused of Seaton,
I did a pretty comprehensive search of my reference works and online when this issue was raised. Fair enough that Cannonball gives refs with cords, but the first is of Consulate troops (no dispute that they had them-see * below) and the others are by Carle Vernet (pub. 1821) showing senior officers, who may well have blinged up their outfits and the other by Nicolaus Hoffman (died 1823) of what appears to be an atypical trooper. Both are prettywell contemporary to the period so must be taken seriously.
However, other than senior officers and trumpeters (as with two from Taconville below), I found pretty-well no representations of colbacks with cords..
I've given some refs below but my judgement is that this must be a senior enough officer to get away with upgrading up his uniform with some extra bling... by no means impossible and doesn't detract from a fantastic piece of sculpting (perhaps except for the raquettes/flounders)
Here's the one in Les Invalides:
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Here's a selection (not all Garde, but illustrate the point), including Gericault's painting of 1812, near contemporary plates by Marbot (1840s) plus well known ones from Taconville (1900s) and well researched works from Houroulle, Girbal and Courcelle and Jouineau and Mongin.... there are many more sans cord, eg see
HERE:
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(worth reading the text of this last one to see the 'theory' of dress regulations)
And here's Historex's refs, which I've always taken as well researched:
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*Ref Canonball's ref if from Men-at-Arms 444 and if you look at the double page spread it reveals Empire troops, sans cords:
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