Not having children on my own (yet), had similar issues with my niece.
My approach was to try teaching her that the sooner she reports what she got broken or what she did wrong - the faster we are going to fix it.
The only form of "punishment" I have ever applied was to make her take part in the fixing activities, while trying to make these mostly fun and never actually making it look like "punishment" or like I am putting blame on her.
In my opinion it worked.

Perhaps he is afraid of disappointing you.

Chap, I absolutely don't want to tell you how to raise your kid, but are you sure using exercise as punishment is a good option? My worry would be that if he associates the two, he may develop an aversion to exercise which could be problematic in the future.

Hope you don't mind me saying that; like I say I don't want to sound like I'm saying you're wrong.

Just an observation but seems to me if someone wants truth out of a kid never ever punish them for telling it.
There is also something to be said for genetic personality traits. If you(global) want truth out of a kid, make it easy for them to volunteer it to you, no punishments.

He's just affraid of loosing your approval. Your way up there on respect list of his. He doesn't fear the exercise. Go take him somewhere and tell him how much he as a son and a person means to you. And he can talk to you anytime about anything. You got a pack with each other.

Punishment is probably not what he's avoiding. He most likely perceives you will be disappointed in him, and I guarantee that hurts much worse. The key to success in this lesson is to impart to him somehow that honesty in the face of a mistake is a thing to be proud of.

I guess like me you don't understand why. I have a boy and a girl, now she is just on 33 and he has just turned 31. Going back to when they were toddlers and even to this day, he will tell me, even if he has deliberately done the wrong thing. He just volunteers it. She will avoid it, as a toddler she would blame him, now she pretends things haven't happened or makes an excuse. When she was 4 and he was 3, she wrote his name on the wall of the house,expecting me to believe he had done it when she was caught, she was only practicing her writing, and not being bad, it was chalk and washed off. When he was 16 he took me aside and said, "I have done something you won't like, I knew you wouldn't approve, but I wanted to do it, so I did" I asked him what the repercussions should be, he suggested being grounded for a week, I said Fine, make it 2. That was the end of it.

As long as he doesn't hear when I did as a child he'll be fine. My parents would have said "god is going to punish you."