Folding Attention Biases into Keys Makes Compacted KV Caches Run on Standard Kernels

Attention Matching (AM) compacts a KV cache by keeping some keys per head, refitting their values and adding a bias to each kept key's logit. FlashAttention and vLLM's paged caches have no input for it. We fold the bias into the keys along one closed-form direction per head, fitted on context querie