2-Adic Affine Weyl Words and Injectivity in Collatz Dynamics — E8 Intelligence Research

FINDING: Collatz dynamics admit a 2-adic (p,q)-adic formulation in which parity sequences are exponent words over an affine Weyl/root-system structure; uniqueness of trajectories is claimed to follow from 2-adic injectivity of these words. | MATH: Collatz map T(n)=n/2 if n≡0 (mod 2), T(n)=3n+1 if n≡1 (mod 2). 2-adic encoding: n = Σ aᵢ2ⁱ ↔ parity word (a₀,a₁,…) ∈ ℤ₂. Affine map per step: n ↦ (3n+1)/2ᵛ₂(³ⁿ⁺¹), with v₂ = 2-adic valuation. Cycle condition: ∏(3nᵢ+1) = 2^S · ∏nᵢ, S = Σ v₂. "2-adic barrier" = capacity limit on distinct exponent words mapping to same trajectory. | CONNECTION: Affine Weyl group of type A₁ (rank-1 root system) acts on ℤ₂; parity dynamics = walk on A₁ root lattice ℤα with reflection s_α. Coxeter graph A₁ = single node, m₁₂=3. Super Weyl group (arXiv 2401.11068) generalizes to ℤ₂-graded root systems — same Coxeter combinatorics. No 0.382/0.618/1.618 ratios appear in the evidence; the harmonic content is crystallographic (A₁ lattice, ℤ₂-adic tree), not golden-ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com