Building in existing structures - Subsequent additional costs despite a fixed price

@SoL Addendum:
In the construction description, more is of course specified regarding the exterior wall.
It says 24cm Poroton + 16cm mineral wool.

Great, it's nicely sloppy.
He can give you the cheapest of the cheap or process his leftover stock.
It should say "16cm mineral wool WLG0xx manufacturer product."

Regarding the wording: Most inaccuracies are at the customer's expense because you have concluded a contract for work. The contract describes the work to be performed in return for payment.
And if you do not describe the work in certain specifics or do so insufficiently (see mineral wool), the contractor has freedom of choice because no binding contractual requirements exist.

I completely agree with you that many things are way too unspecific and that I was simply naive about it. I have come to terms with that mentally as well. However, the relevant part and what this was primarily about was specified as "extension to the existing basement" including reference to the historical execution documents, to which at least the basement floor plan in the contract plan refers. Thus, he has indeed committed to deliver this work. How do you specifically assess this point? What do you see as a fair solution here?

Doesn't matter what I consider fair. Have a specialist lawyer take a look or think of something. With 50/50, you often get close to acceptance between two parties ;)

In my opinion, the insufficient description of the contract is at the expense of the contractor. As a specialist, he should have either checked the static calculations before signing the contract or fixed a corresponding reservation in the contract. Since he probably would have jeopardized your signature by declaring the reservation, he took the risk of signing the contract without checking the static calculations. If there was previously a prefabricated house on it, he should actually have assumed that the basement would not support the solid house.

I am not a lawyer. Therefore, you should definitely seek legal advice.

Giving in already does not protect you from further additional charges. Rather, the contractor sees that he can get away with it with you.

Show me the additional charge that is supposed to cost 20k. The justification for the extra costs can also be quite interesting.

Here, giving in already does not protect you from subsequent additional charges. Rather, the contractor sees that he can do what he wants with you

Getting the right vs. getting the house.
You have to know.

How much work has been done so far? Has anything happened at all, or would each party be in the same position as before the contract was concluded if the contract were terminated?