No Primary-Source Result for q → q⁻¹ Affine Simple Reflection s₀ — E8 Intelligence Research

FINDING: Search returned no primary-source mathematical result on "q → q⁻¹ affine simple reflection s₀" — only tangential lecture titles (Gaitsgory, Etingof, Kedem) and one unrelated BESIII kaon paper. No equations extractable from the retrieved metadata. MATH: - Langlands-relevant objects named but not derived: geometric Satake equivalence, quantum affine algebras, Q-systems, cluster algebras. - The affine Weyl group action on the Langlands dual parameter space is generated by simple reflections s_i (i ∈ {0,1,…,r}) acting on the coweight lattice; the affine simple reflection s₀ acts on spectral parameter z by z ↦ q²/z (or q⁻¹ ↔ q depending on normalization), where q = e^{ħ} is the quantum parameter. - Standard R-matrix normalization: R(z) = (z q − z⁻¹ q⁻¹)/(z − z⁻¹) · (projectors), invariant under z → q²/z combined with q → q⁻¹. This is the "quantum parameter modular inversion" the query targets — but it is NOT present in the retrieved sources. - BESIII paper: form-factor ratio F_K⁰ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com