Anchored Likelihood-Ratio Geometry of Anonymous Shuffle Experiments: Exact Privacy Envelopes and Universal Low-Budget Design

We develop a geometric framework for anonymous shuffle experiments based on anchored affine likelihood-ratio laws on the centered simplex polytope. Under a common ε₀-LDP cap, binary randomized response universally maximizes all convex f-divergences and both directed hockey-stick profiles after shuffling; a rigidity converse shows that simultaneous saturation at finite n forces the pairwise law to be the binary endpoint. On the design side, we derive exact finite-n canonical risk formulas, a trace-cap theorem, and a two-orbit reduction of the global χ²-budget frontier. In the low-budget regime, augmented randomized response is asymptotically minimax-optimal to the sharp leading constant over all channels and all estimators.