Chapter 11: The Swap Operator σ and the First Arithmetic

With τ-Idx as the earned counting scaffold and rank transfer maps as canonical bijections, we now derive the first arithmetic operations. The swap operator σ is the first operation derived from ρ: it exchanges positions within orbit rays via rank transfer composition. Index addition is defined as iterated ρ: n + m := ρ^m(n). Index multiplication is defined by structural recursion on addition. We prove commutativity and associativity of addition, distributivity of multiplication over addition, and establish that (τ-Idx, +, ×, 0, 1) is a commutative semiring.