Prompt-set symmetries provably transfer into simplex-ETF and circulant geometry

Du, He & Su prove a symmetry-transfer principle: the symmetry group acting on next-token completions is inherited by the optimal class-mean/embedding geometry under the layer-peeled objective [du-etal-2026-uncovering-symmetry-transfer] Permutation symmetry forces a Simplex Equiangular Tight Frame while cyclic-shift symmetry forces a circulant matrix structure [du-etal-2026-uncovering-symmetry-transfer] Both predictions are confirmed on real Mistral-7B-Instruct-v0.3 output-projection and context-embedding Gram matrices for permutation- and cyclic-symmetric prompt sets (days of the week, months of the year) [du-etal-2026-uncovering-symmetry-transfer]