[jay_shark]

I finally spotted my error which is why I was producing different results .

Fact #1 : Player 1 gains EV from his fold equity

Fact #2 : Player 1 gains EV when both hands get checked

Fact # 3 : If the pot gets contested by both players then it's neutral EV using game theory strategy .

Solution: Let a be your optimal betting range ; a>=0
Let x be your opponent's optimal calling range . 1/3<=x<=1

We can write a in terms of x . Notice that (1-x)/(x-a) = 2
x=(2a+1)/3


If player 1 checks [0,a] then player 2 should bet with [(5a-2)/3 ,1] . Note that this comes from

a - 2/3*(1-a) = (5a-2)/3 .

Now we will compute player 1's EV under the assumption that EV(fold) =0 . We may subtract $1 at the end .
The algebra is brutal so stay with me .

EV= 3*(1-x)/2*2x + 2*[(1-a)*2/3 +(5a-2)/3*1/2]


EV= 3*(1-x)*x + (a+2)/3

write everything in terms of a .

EV= (-4a^2 +3a +4)/3
EV' = 1/3*(-8a+3)
Set EV'=0 and we get that a=3/8 .

So player one bets [3/8,1] checks with [0,3/8].
Player two calls with [7/12,1]
If player one checks then player two always checks behind .

EV=1.52083333333

If we subtract $1 we get EV =0.5208333333