We develop a geometric framework for anonymous shuffle experiments based on an anchored affine likelihood-ratio law: a mean-zero measure on the regular simplex polytope. Every finite-output d-ary channel corresponds, up to refinements, to a unique anchored law, and conversely. On privacy: among all epsilon_0-LDP channels, binary randomized response universally extremizes all convex f-divergences and hockey-stick profiles after shuffling. A rigidity converse shows that saturation of both directed envelopes at finite n forces the binary endpoint law. On design: under the pairwise chi_* budget, we prove exact trace-cap and two-orbit frontier theorems. Every frontier point is realized by a mixture of at most two orbit laws. In the low-budget regime, augmented randomized response is minimax-optimal to the sharp constant over all channels and estimators. Under the raw LDP cap, the problem reduces to subset-selection with explicit optimal subset size. The arguments are self-contained and independent of the author's trilogy.

翻译：我们基于锚定仿射似然比法则（正则单纯形多胞体上的零均值测度）建立了匿名混洗实验的几何框架。每个有限输出d元信道在细化意义下唯一对应且反比于一个锚定法则。隐私方面：在全体ε0-局部差分隐私信道中，二元随机响应经混洗后通用地极值化所有凸f-散度与曲棍球棒剖面。刚性逆定理表明：当有限n下两个有向包络均饱和时，必强制推行二元端点法则。设计方面：在成对χ²预算约束下，我们证明了精确迹容量与双轨道前沿定理。每个前沿点至多由两个轨道法则的混合实现。在低预算区间下，增强随机响应在所有信道与估计器间达到尖锐常数的极小极大最优性。在原始LDP容量约束下，问题归结为具有显式最优子集尺寸的子集选取。论证过程自洽且独立于作者的三部曲。