Chapter 14: Three Levels of Sameness

In standard mathematics, equality is monolithic: two objects are either equal or not. In τ, the ontic/denotational boundary induces a hierarchy of sameness. At the deepest level, ontic identity asks whether two objects are literally the same element of Obj(τ) — the same seed and depth. At the middle level, address equivalence asks whether two programs have the same normal form — whether two denotational paths reach the same object. At the outermost level, shadow equality asks whether two objects are indistinguishable to an external observer who can only read off coordinate values. We formalize these three levels and explain why each is necessary.