Put the BPE training steps in the correct order, from raw corpus to final vocabulary.

**BPE training** builds a vocabulary from the bottom up, so the order follows that logic. First, initialize the vocabulary with every individual character or byte in the corpus. This base set guarantees any string can be represented, since worst case you fall back to single symbols. Second, count the frequency of every adjacent pair of symbols across the corpus. Third, pick the single most frequent pair and add it to the vocabulary as a new merged symbol. Fourth, rewrite the corpus, replacing every occurrence of that pair with the new symbol. Fifth, repeat the count, select, and replace loop until the vocabulary reaches the target size, often 50k to 100k tokens. The step people misorder is the rewrite. Counting must happen on the updated corpus, because once a pair becomes one symbol, the next round's most frequent pair may involve that new symbol. Skipping the update breaks the algorithm.

The correct order is initialize, count, select, replace, repeat. **BPE** starts by initializing the vocabulary with all individual characters or bytes in the corpus. That guarantees full coverage with no unknown tokens. Then it enters the core loop. Count the frequencies of all adjacent pairs in the current representation. Select the single most frequent pair and add it as a new merged symbol. Replace every occurrence of that pair in the corpus with the new symbol. $$\text{merge}_t = \arg\max_{(a,b)} \; \text{count}_t(a, b)$$ The subscript t carries the weight here. Counts are recomputed on the post-merge corpus each round, not on the original characters. After a merge, the new symbol participates in the next count, so frequencies shift between iterations. The loop repeats until the vocabulary hits the target size. The trap is treating count, select, and replace as independent of order, or forgetting the corpus update, which would make every iteration count the same stale frequencies.