Swap-RoPE: Permutation-Invariant Block Encoding for LLM Judging

Pairwise LLM judging repeatedly encodes the same candidates in different positions and contexts. We introduce \name, which combines shared block coordinates with causal attention restricted to a fixed prefix and each block's own history. We establish that a block's hidden states, keys, and values are independent of its competitors and preserved under token-aligned permutations. These properties permit cache reuse across different comparisons. In an all-pairs tournament over $n$ candidates, fresh candidate encodings decrease from $\left(\genfrac{}{}{0ex}{}{n}{2}\right)+n-1$ to $n$ under fixed-content and cache-retention assumptions; comparison calls remain quadratic. We fine-tune Qwen3 models at 1.7B, 4B, and 8B and evaluate three mathematical benchmarks. Reported reusable key–value (KV) cache ratios increase from $38.5$–$40.9\%$ under standard RoPE to $73.8$–$79.2\%$. Selection accuracy improves over vanilla-position fine-tuning in six of nine settings with four candidates and three of nine with eight candidates. The construction excludes cross-block attention during candidate encoding; verdict consistency and serving speedups require further evaluation.