Reduce the ceiling height. 12 is unworkable fora working kitchen, or even a show kitchen Unless you want a LOT of open space at the top to draw attention to the poor proportions, or you drop the ceiling with substantial beams or coffers.

the vent hood doesn't go on the ceiling OP.
usually a false boxing is used to cover the
vent pipe, allowing the vent hood to be
at proper height for the stove.

how big of a vent hood are you talking about?
how big is the stove?

as you vent air out, you have to add air back in.
I see lots of house problems with 1200 cfm vent
hoods putting the house into a negative pressure.
400 cfm fan is enough. no need to have to educate
whoever is venting it as to how to integrate make
up air.

if you cook 8 hours a day, on all burners...then
you MIGHT need a 1200 cfm vent hood.

best of luck.

Thank you. There are beams in the ceiling. I will post a picture in a minute. The vent will vent to the roof. The range is gas 48 inches wide.

This is the cabinet persons idea. I do not like the cabinets above the hood. The range is 48 and not 36 like noted the fridge/freezer are thermador columns and will be paneled. The size is also wrong it will be 48 inches

The pictures