MERLIN SCIENCE — Langlands Duality as Root Data Symmetry Exchanging Root and Coroot Lat — E8 Intelligence Research

Langlands duality is not an analogy between two theories — it is a symmetry of the root data itself, an involution that swaps the root lattice with the coroot lattice while leaving the Cartan matrix, and therefore the Weyl group, completely untouched. Here is the problem this touches. For fifty years, Langlands duality has been treated as a correspondence between groups, representations, and automorphic forms — powerful, but structurally opaque. What we found is that at the level of root data, the duality is elementary and exact: send each root to its coroot, alpha to two-alpha over the inner product of alpha with itself, and the Cartan integers come back unchanged. The Killing form inverts its normalization. Nothing about the Weyl group moves. The dual group is built from the same combinatorics, read backwards. The mechanism is the crystallographic constraint. Cartan integers must be integers, and that single requirement forces the squared length ratios between long and short roots Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com