I am in an open relationship for the first time and i am having serious doubts on the relationship

About 4 months ago me and my boyfriend decided to open our relationship. He had been in one previously and before we had decided to move in together we talked in great length about the possibilty of him wanting to open our relationship. He described it as having urges, the need for something new. I asked specific questions so that i could make a decison on whether or not we should continue to move forward with the moving, and if this was a relationship i could be happy in. I had decided that as long as he was honest with me then i could be happy, after all i my only real desire was to have someone that communicated with me freely. Someone that understood me, my principals and that held the same values, someone who didn't hide things or lie because they understood that trust comes first, that people make mistakes and thats ok as long as they can own those mistakes. Shortly after moving in i started catching on to some lies. Some he had been telling me from the very beginning before i moved in to current ones. I tried being understanding to the fact that it cant be easy to go to your partner and say you want to sleep with other people. I told him i loved him for who he was, flaws, mistakes, the good the bad and that i thought it was time to open up the relationship, so that he would not have to lie anymore. I told him the lies were destroying our reltionship not the actual acts themselves, and that i felt that opening things up would only benefit us, i thought it would create an amazing bond where we were free to be ourselves without fearing our partner would reject us for it. We agreed that we were free to be with other people and set some ground rules... 1) no emotional involvment, this was to be strictly just sex and if we felt someone was getting emotionally attached we would cut off contact 2) always wear protection 3) openness and honesty 4) we always told the person that we were in a relationship and that we could only offer a sexually relationship. I struggled right away with the amount of time he spent having conversations with these women, it seemed like he spent all his time finding and talking to women. He would come home from work and sit on his phone all night, barely talking to me. He became self concience, and jealous if the sex was different with the men i had met, he questioned if they were better. I started noticing white lies again. He didnt want to talk about who he was talking to, he made me feel bad for asking, and i believe he is lying about how many he has met and slept with. I decided i was not going to be with other men, that he could still continue but i didnt like what it was doing to him every time i was with some one. I feel he is lying about so many things like the swingers site he said he joined for me, after i told him i didnt want to sleep with anyone else but him. The texting app he says was for that site, but coincidently was started at the time he stopped telling me about his conversations and only my number and work related numbers began to show on the phone bill and most upsetting to me is the way he carries himself, i know that there is little in his behavior both on social media and in public that suggest that he is in a relationship, and sometimes he reminds me of a sex crazed teenage boy with his actions. I thought urges meant once in a while, i did not realise that it was going to be an everyday way of life. I often wonder why hes in a relationship at all if he needs to have the freedom of a very single man. I know theres the possibilty that he is being honest but i just cant bring myself to put my faith and trust into him. I think with complete transpency i could learn to trust, but at what point is that an invasion of privacy or act of control. What can we both do to fix this when it seems beyond repair? How do i trust, how can he help rebuild it?