Given how large the cached prefix already is, how many new (uncached) tokens does a step append?
Populates tab:append_by_prefix (src/05_LLMGeneration.tex) — the quantitative companion to the
prefix-vs-append scatter (fig:prefill_append_relationship). For Claude and Codex, every agent
step is binned by its prefix_tokens, and within each bin we report the distribution of
newly_append_tokens: count, avg, p50, p90, p99.
Prefix bins are doubling, in 1024-token units: <1k, 1-2k, 2-4k, 4-8k, 8-16k, 16-32k, 32-64k, 64-128k, 128-256k, >256k. The prefix_tokens / newly_append_tokens accounting is the same one
used by prefix_append_distribution and token_length_distribution, so the numbers reconcile.