Yes, then, if you want an exact solution, it can be complex. However, why not simply ignore the details of P1 and P2 and only consider T(E U P1 U P2)/2 = T(E)/2? (Or, during the late endgame, the alternating sum Δ(E U P1 U P2).)

If with "an exact solution" you mean playing the god move my answer is "no". I am not strong enough and I am too lazy to look for the god move in the endgame.
The best I can do is the following: I consider all board as made of local positions P1, P2, P3, P4 ....(without any environment). if my intuition tells me that I should play in P1 or P2 (I mean I hesitate between these two areas) then I calculate an estimation of mP1 and mP2 and I choose to play where the miai value is the largest.
I know perfectly it may not be the god play but I am happy with this (thanks for the theory!)
As you see I use the environment notion only indirectly by calculating mP1 and mP2, but not explicitly for example by saying that playing first in the environment will give me an advantage of T(E)/2.
Putting aside the miai value calculation itself, do you use the environment in some way?

I am not sure exactly what you are asking. My (and sometimes others') mathematical theory involving the [global] temperature sometimes only uses it but most of the time uses it and other values, which include the second largest value of the environment, move values, counts or the gote-sente-difference.

I have a theoritical question Robert.
Assume a board P1 U P2 U E with E being an ideal environment. Assume mP1 > mP2 (move value of P1 > move value of P2)
Is it proved that a god play is to wait till T(E) = mP1 in order to play in local position P1?
If not, is it proved that it is correct on condition that no ko can occur in P1 and P2?

See https://www.lifein19x19.com/viewtopic.p ... 72#p278372

Apparently it is still available:

https://www.biblio.com/book/eye-number- ... 1468681326
目数小事典―出入計算・見合計算 / 誠文堂新光社
Eye number of small encyclopedia - and out calculation and commensurate calculation (Japanese Edition)

Or perhaps more accurately
"Small encyclopedia for counting points: deiri and miai calculations"