Still confusing to me. What steps in post #14 change?
Sorry, yes you're right. I thought I understood it, but I am also still confused actually.
There must be something very silly about (?)
but i can't see it yet.
EDIT:
Ok, I think I identified the evil. Instead of looking at the spot starting varstrike, consider the forward var:
[$] \int \frac{dK}{K^2} \left [ C(K,T_2) - C(K, T_1) \right ] [$]
Now,
[$] C(K,T_2) - C(K, T_1) = C_{BS} (K, T_2, \Sigma(T_2)) - C_{BS} (K, T_1, \Sigma(T_1)) [$]
[$] = C_{BS} (K, T_2, \Sigma(T_2)) - C_{BS} (K, T_1, \Sigma(T_2)) + \int du \, \nu_{BS} (\partial\Sigma / \partial u) [$]
Clearly [$] C_{BS} (K, T_2, \Sigma(T_2)) - C_{BS} (K, T_1, \Sigma(T_2)) \neq C(K,T_2) - C(K, T_1) [$] and the integral term consisting of vega and term structure slope compensates for it.
The problem was that I looked at [$] T_1 = t [$], in which case there is no time value left and the choice of implied vol does not matter, hence misleading me (us), sorry abt that folks!