You need to look at the installation guide for the i-joists. It appears he's taking some liberties here. Truss-joist allows for an 8 7/8" diameter hole a minimum distance from the end of the joist in 11 7/8" joists. This is a lot more than that. Here is truss-joists installation guide.

Are these joists (not headers) supporting a second floor? When you go on the second floor and walk around (or jump) does the floor bounce?

If so, you may want to have a structural engineer give you recommendations.

This is an issue which should have been resolved before framing started. The idea of penetrating the joists with ducts this large was not a good idea.

I'm dubious this will pass framing inspection.

I don’t think that is going to pass .

Thanks for the responses. I am worried that the framing inspection has already passed. This was done by the AC sub. I am sure it wasn't his first rodeo, so I hope that it is all good. This is my first new build, is there another opportunity for an inspector to look at it?

In my area framing inspections are done after the electricians, plumbers and HVAC people have finished for this very reason. If there is a structural engineer involved, send the picture to the engineer. There is likely a fairly easy fix. If worse comes to worst, the duct can be disconnected, a new joist installed alongside the compromised one with an acceptable sized hole per the joist manufacture.

I'd stick to my guns on this one if for no other reason than to put your builder on notice that you're not going to accept shoddy work.

I am not an expert, but you can see most of the joist attached to the vent outlet has removed. The only thing that is holding it in place are the two rims. There are 5 consecutive joists that have been compromised. This is a serious problem.

The HVAC contractor is either clueless as to what he has done, or he doesn't care that one day the floor is going sag. I think the duct work should be removed and the joists replaced. Once that is done the HVAC contractor will have to come with another solution as to how the duct work will be installed.

I must clarify. For TJI joists ducts are allowed to be passed through the joists as long as the holes are no larger than 8 7/8" and placed according to manufacturer's installation manual. Other manufacturers will have similar requirements.

Framing inspection done before or after this mess?

These are cut incorrectly and need to be replaced.

The allowable "hole" is typically just smaller than the distance between the top & bottom members, so no hole would also be wider than this... in other words they like round or square. The cut charts are simple and easy to understand.

With the local industry slammed, someone may have hired "stupid" and isn't supervising.

No matter if inspection was passed, this is the responsibility of the building contractor. If the inspection was passed with this in place, I would bring in the engineer of record to perform a structural inspection prior to covering the walls.