LLM serving reuses cached key-value (KV) state to skip prefill, and growing contexts push that state past GPU HBM and CPU DRAM onto NVMe SSDs, placing prefix restore on the critical path of time-to-first-token (TTFT). We find that on a well-configured host-issued path the SSD is not the bottleneck: restoring the same 6 GiB of KV takes 2.1 s as 192 objects but 3.0 s as 6,144 objects, and more under layerwise loading, because every layer of the storage stack - lookup, allocation, I/O submission, GPU delivery, store, index - charges per logical object while the bytes stay the same. This cost grows with the KV footprint: longer contexts, deeper models under layerwise loading, and tensor parallelism each multiply the object count. Prior systems move this per-object work from the CPU to the GPU; the work remains per object. This paper presents Range, a KV-cache storage engine for vLLM and LMCache built on one principle: every cost on the storage path scales with bytes or physical batches, never with logical objects. Range tickets replace per-key lookup and ownership with token metadata and physical leases; byte stripes make the on-media layout independent of reuse granularity; completion-owned GPU transfer banks batch delivery and extraction and release each layer to attention as it lands. Range keeps the kernel cuFile path and changes four lines of vLLM. On a fixed 6 GiB restore, Range's median TTFT stays at 1.79-1.80 s across a 32x range of object counts and under layerwise delivery, within 70 ms of the prefix already being resident in GPU memory, while tuned in-process LMCache rises from 2.12 s to 3.01 s. Range improves on the fastest multiprocess DRAM-bounced path by 0.44 s at coarse granularity while moving no payload through host memory, and holds layerwise reuse at the cost of coarse-grained reuse, where the best stock path takes 2.24 s.

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