These two papers discuss the history of anti-slavery and peonage laws in the United States.
The tldr is that child support laws are logically in violation of the 13th Amendment, as well as The Peonage Abolition Act of 1867, which was passed to address shortfalls in the 13th Amendment.
Peonage is a form of labor exploitation that uses debt as a system to control workers. In Medieval Europe, this practice existed in the form of serfdom, where men (and sometimes women) were given a debt to pay back to their lord for simply existing and living on their land.
Even if they were able to pay back this debt, it was impractical to live your life as a truly free individual because of how much you depended on your lord. Which was a situation that was artificially created by the lords themselves.
In the US, there is a long history of voluntary indentured servitude, coerced servitude, slavery, as well as legal and institutionally facilitated debt slavery, for example through the prison industrial complex. The later became popular after the abolition of slavery in Southern states, and still exists in many forms to this day.
For an overview of peonage in the United States, as well as the role that child support plays in perpetuating this system, check out this paper:
https://scholarlycommons.law.wlu.edu/wlulr/vol72/iss4/3
For a more specific take on child support, and the US's hypocritical enforcement of this type of debt servitude, check out this paper:
https://digitalcommons.law.seattleu.edu/sulr/vol39/iss3/12/
There are abstracts and summaries here:
https://np.reddit.com/r/Male_Studies/comments/sxuu2r/the_new_peonage/
https://np.reddit.com/r/Male_Studies/comments/sw0co9/a_new_peonage_pay_work_or_go_to_jail_in/
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