The 4-inch duct was too small for the original plan. Probably an 8-inch would have been needed and then the higher flow rate requirement could be made to put up with a bit more pressure loss than is desirable .

Air velocity in the hood ducting should be in the 1000 to 2000 ft/min range. Even if you only put a short length of 4-inch duct in a 10-inch duct path there would be very high pressure loss. (On-line calculators exist for these types of analyses.) And if you used a screaming mimi aerospace blower to get the flow rate at the pressure loss, the duct air velocity would be absurdly high and would try to compete with the blower for noise champion.

Venting down and out may not be a violation, so long as your AHJ is OK with the details including slopes, sealing, and access, but such paths tend to be restrictive for various secondary reasons. And it seems you are limited to 4-inches in that direction, if I understood your message.

I have no real picture of your kitchen, but if it were my kitchen I would use a 10-inch duct from the hood to the nearest exterior wall that didn't dump onto a deck or an openable window. I'd either enclose the duct with a soffit, or make it decorative. Alternatively the duct would go to a point on the ceiling where an upstairs chase could be added to get the path to the attic where it can deviate as necessary to get to a good position for a roof blower.

For reference, I have seen images of wood-everywhere type structures where a hood duct went skyward across the room to the desired point on the roof. This appeared to be at least 20 ft high. Such schemes require stainless steel ducting to look decent. This wouldn't do if the ambience were not rustic.

Can you extend the wall studs on the existing wall by 6 inches (cutting out 6 inches of space next to the cooktop) so there is space to have enough duct area?

Thank you so much, Kaseki! This was very helpful. I think i may be able to take it straight up through my laundry room and into my attic. do i need to have an angle in my vent duct path? Also, could i use flexible duct for the first four feet in order to navigate some akward shifts to the clear pathway upstairs then transition to rigid ducts up through the second floor and attic? On thursday afternoon Im meeting with a different HVAC company, but i would like to start coming up with a plan before then and be ready to move forward. I haved worked with this new HVAC company alot on my jobs and i trust them and know that they are incredibly knowledgable and skilled. (My comp does commercial construction and the original HVAC guy was with a residential fitm that i am only vaugly familiar with.) Although this situation is absolutely my oversight, i am really dissaponted by the original techs lack of judgment and experience he showed by just installing a 4” duct and hoping for the best. After looking into it for two days and learning just how bad that plan was, im shocked he moved forward with it and closed the walls.

Thanks, again again, for your advice!

You do not need to have bends in an exhaust duct, but it is often impossible to built the path without one or more. For example, a down-roof blower will be mounted at the roof angle, so any duct entering the attic with a vertical orientation will need one or more bends to get to the blower, and one more for the angle.

If you use an attic run and roof blower, make it long enough that you can add a silencer (Fantech) for your duct size.

No flex duct! (Well maybe there is a stainless steel version of adequate gauge, but otherwise no. In any case, internal smoothness is needed for minimum grease impingement collection.) Use proper rotating turn sections [sold as elbows] to create angles and offsets. Your new HVAC guys can regale you about these if you haven't seen them.

With respect to knowledgeability: Keeping up with regulations and running a business is hard enough, studying new technical areas not relevant to Lincoln's log cabin design is an additional burden, so what was done before is the easiest path. When I started planning my kitchen reno in 2007, I found the ventilation knowledge of hood sales persons to be abysmal. This was also the issue with corporations dealing with residential kitchens. I had to rummage through the commercial kitchen literature for information, then extrapolate.

Since then we have had a few companies develop MUA products for residential applications. Some recognition of cooktop overlap has appeared. Thar's money in them thar vent hills!