I try to understand billing with a project with billing type "On capped produced work".

"On capped produced work" will limit number of billed days on each activity to the assigned work for the activity.

Also, I have found a error in description of detail column.
Is written "[validated] - [billed]", but the values are added.

Values are added because [billed] is negative (subtract negative = adding). But I don't understand why.
As I understand this point, object is to remove already billed, so values should really be subtracted.

"On capped produced work" will limit number of billed days on each activity to the assigned work for the activity.

for the activity is for the activity price ?

I had a test to check an all seems allright.

I created 1 project P with billing mode "on caped produced work", affect 1 resource R, add 1 activity A with 12 days of validated work, and assign 10 days for A to R.
Then define activitty price X.
Enter 5 days of real work on A for R on august.
Create Bill B1, add line for R on X on august (1 to 31).

Enter 3 days of real work on A for R on september.
Create Bill B2, add line for R on X on september(1 to 30).

Enter 4 days of real work on A for R on october.
Create Bill B3, add line for R on X on october (1 to 31).

As 7 days have already been billed, billable is only 3 (10 assigned - 7 already billed), so I cannot bill the 4 worked days but only 3. Capped is OK.

I created 1 project P with billing mode "on caped produced work", affect 1 resource R, add 1 activity A with 12 days of validated work, and assign 10 days for A to R.

What is the purpose to set 12 days of validated work?
In your example, I can't see any reference.

What is the purpose to set 12 days of validated work?

It was just to check that billing is capped on Assinged work, and not on validated work (because I had some doubts)