Law Genesis: The Information Cost of Making Many Microscopic Laws Become One

A coarse description of a system is usually justified by what it preserves. Here we ask the complementary question: what must be destroyed before several microscopic laws can be represented by one reusable effective law? For a finite state space and deterministic microscopic maps, we define the raw law-disagreement relation and the least law-compatible congruence containing it. The resulting quotient is the finest abstraction on which every microscopic law induces the same effective dynamics. We quantify the required identity loss by conditional entropy and call it the Law Genesis Cost. The construction separates disagreement magnitude from consequences of enforcing dynamical consistency. We prove that identical raw disagreement relations can generate different least compatible congruences and different information costs. We also prove that Law Genesis Cost does not determine abstraction fragility, give exact finite separation families, and establish an arbitrary-delay sequential separation theorem. A clock-free four-state witness exhibits later divergence despite agreement on low-depth summaries. Exhaustive finite enumeration and deterministic falsification searches, implemented after the theory was fixed, delimit the claims: a 96,000-system search found no witness when richer invariants were forced to match through depth two. The framework distinguishes disagreement from the closure geometry required to make one law possible.