1. Setting: admissible Law-of-Logic realizations
The setting consists of admissible Law-of-Logic realizations: structures carrying a carrier type, comparison cost, identity element, step operation, and laws (identity, non-contradiction, excluded middle, composition, invariance, nontriviality) that satisfy the Law of Logic.
2. Theorem: same arithmetic structure forced across realizations
The theorem states that any two such realizations force canonically equivalent arithmetic objects; these objects are initial Peano algebras, so the equivalence is the unique isomorphism between them.
3. Cited Lean anchors
The core statement appears as universal_forcing, which asserts that for realizations R and S the carriers of their extracted Peano objects are equivalent via ArithmeticOf.equivOfInitial. This relies on the supporting definitions arithmeticOf (the forced arithmetic extracted from a realization) and arithmetic_invariant (the canonical equivalence between two realizations' arithmetic objects).