For the independent reference model with popularity vector $p\inΔ_N^\circ$, let $H_C(p)$ denote the exact stationary hit rate of an LRU cache of capacity $C$. We prove that, for every $1\le C<N$, the uniform popularity vector is the unique global minimizer of $H_C$ on the interior simplex. More sharply, along every nonconstant segment from the uniform vector to an interior point, the LRU hit rate is strictly increasing. The proof uses the standard exponential-age representation of the stationary LRU cache and gives an explicit positive pair-square formula for the radial derivative. Equivalently, for the move-to-front rule, the stationary search-cost distribution improves strictly in the usual stochastic order along every nonconstant ray away from uniform. This proves the radial restriction of the Fill--Holst Schur-concavity conjecture for move-to-front search-cost tails. In particular, all LRU miss probabilities and all nonconstant nondecreasing stack-depth costs decrease strictly along such rays. The result is radial rather than Schur-convex: full majorization monotonicity for LRU is known to fail, and the proof identifies the special positivity that survives on rays from the uniform vector.

翻译：针对流行度向量 $p\inΔ_N^\circ$ 的独立参考模型，令 $H_C(p)$ 表示容量为 $C$ 的LRU缓存的精确稳态命中率。我们证明，对于每个 $1\le C<N$，均匀流行度向量是内单纯形上 $H_C$ 的唯一全局极小点。更严格地，沿均匀向量到内点的每个非恒定线段，LRU命中率严格递增。证明采用平稳LRU缓存的标准指数年龄表示，并给出径向导数的显式正对平方公式。等价地，对于移至前端规则，沿远离均匀向量的每条非恒定射线，平稳搜索成本分布在通常随机序下严格改善。这证明了Fill–Holst关于移至前端搜索成本尾部的Schur-凹性猜想的径向限制。特别地，所有LRU未命中概率和所有非恒定非递减栈深度成本沿此类射线严格递减。该结果是径向性而非Schur-凸性：已知LRU的全序化单调性不成立，而证明识别了从均匀向量出发的射线上保持的特殊正性。