And this hints at intelligent design how?
The basic train of thought is this: If properties of space-time take their origin from a pure mathematical object, then one can assume that space-time is a result of some intelligence as all mathematics apparently is. Wouldn't you agree?....
No and no, and this is demonstrably a false assertion leading to a false conclusion based on incorrect premises. Errors all 3 steps of the way.
Don't take this too critically, it's not uncommon for people to see evidence of God in places like math, quantum physics, the structure of the Universe, etc...
I am addressing your assertion of a logical proof and that neither have you provided, can it be done, or should it be done.
I never mentioned God in this new thread, only intelligent design.
So you don't agree that:
1) results of the experiments form a pattern that has a mathematical structure.
2) mathematics is a result of intelligence (consciousness)
I don't know how you see the Universe, but it sure doesn't look like a chaotic mess to me

......
Neither (1) or (2) holds water. But if they did we would not have to go there to get where you wished to be, we could simply talk about a circle, and perhaps a square, and the mathematical structures they form. Thence, you could again argue that circles and squares were the result of intelligence, etc.
But that is nonsense. It is not that the circle is the result of intelligence, but that it requires intelligence, consciousness, etc to discuss it as an abstraction.
Let us move to your assertion that the Universe is 'not a chaotic mess". The mathematics of chaos are fundamental aspects of the reality of the Universe. Therefore if yet again you wish to impute a higher being, you must give it the intent to create chaos, and consider that as equally important as what may at first glance seem to be ordered structure.
To avoid the leap of faith which is implicit in your logic you must demonstrate a portion of nature that cannot exist without Turing machinery having prescribed essential aspects.