# Three Invariants for a Special-Token Table

**1. ID uniqueness and disjointness**
Every special token maps to a distinct token ID, and no special-token ID collides with any ID in the regular vocabulary:
```
∀ (name₁, id₁), (name₂, id₂) ∈ specials: name₁ = name₂ ⟺ id₁ = id₂
∀ (name, id) ∈ specials, ∀ v ∈ vocab: id ≠ v.id
```

**2. Range containment**
All special-token IDs fall within the model's valid vocabulary range `[