It’s actually pretty easy:
1) Empirically, (in-group) law evolved everywhere using a test of reciprocity. Even norms demand reciprocity. All that differs is the local organization of rights and obligations that produce various forms of reciprocity under various group evolutionary strategies.
2) Empirically, (out-group) international law, that is insulated by differences as a compromise between differences is reducible to reciprocity.
3) Logically, (internal consistency) all questions of conflict are in fact decidable by the test of reciprocity, and it is the only decidability that exists that I know of.
4) Scientifically (Axelrod) (operationally), no organism can both cooperate (produce outsized returns), and not (a) preserve defection (cheating), and (b) require reciprocity (prevent parasitism), and (c) buy options on cooperation (invest) to incentivize cooperation, and (d) practice altruistic punishment (costly punishment) in order to preserve the incentive to cooperate without going extinct.
0 – Parties must be able to negotiate a contract for cooperation and remember success or failure for cooperation to exist. (We cannot cooperate with animals. They aren’t conscious enough to do so, or to hold to commitments.)
1 – Objective morality (reciprocity) is in fact ‘reciprocity’.
2 – Moral norms (networks of reciprocity)
3 – Moral intuitions ( individual intuition of reciprocity given one’s reproductive/survival needs)
4 – Moral actions are limited to fully informed, warrantied, productive, voluntary exchange free of imposition of costs upon the investments of others by externality.
1 – Restoration of reciprocity by forgiveness (investment in future forgiveness)
2 – restoration of reciprocity by restitution
3 – restoration of reciprocity by restitution and punishment
4 – restoration of incentive for reciprocity by restitution and death.
1 – one may take no action one may not perform restitution for.
All of which pretty much are reflected in the common law of tort.