p-Adic Anyon Fusion and Braiding: Quantum Groups at Roots of Unity, Verma Modules, and Ultrametric Anyon Models

Phases 1–2 established the p-adic braid group Bₙ(Qₚ) on Bruhat-Tits buildings and identified the Temperley-Lieb parameter δ as a p-adic cyclotomic unit, yielding a p-adic Markov trace and Jones polynomial VLᵖ(t) ∈ Zₚ[ζ2pᵏ]. In this third phase, we complete the chain from braid groups to anyons at non-archimedean places by constructing p-adic anyon models via the quantum group Uq(sl₂) at q = ζ2pᵏ, a primitive 2pᵏ-th root of unity in Q̄ₚ. We define the restricted quantum group Ūq(sl₂) over the p-adic integer ring Zₚ[ζ2pᵏ], classify its finite-dimensional irreducible representations as p-adic anyon types, and compute the fusion rules via the p-adic Verlinde algebra. The S-matrix and T-matrix take values in Zₚ[ζ2pᵏ] ⊂ Q̄ₚ rather than C, endowing the modular tensor category with an ultrametric structure. We compute the braiding matrices (R-matrix) for tensor products of anyon representations and show that the p-adic valuation of braiding amplitudes provides a natural hierarchical gate model: computations at higher p-adic precision correspond to deeper levels of the Bruhat-Tits building. For the p-adic analog of the Fibonacci anyon ($p=5$, $k=3$), we exhibit explicit F-matrices and R-matrices valued in Z₅[ζ₁₀] and demonstrate that the p-adic valuation stratifies braiding operations into precision levels, eliminating the continuous approximation overhead of the Solovay-Kitaev theorem. The results establish that p-adic anyons constitute a well-defined mathematical framework for topological quantum computation with ultrametric computational structure. **Keywords:** quantum groups at roots of unity, p-adic anyons, Verma modules, fusion rules, p-adic braiding, restricted quantum group, ultrametric modular tensor category, p-adic Fibonacci anyons, Bruhat-Tits building, non-archimedean topological order ---