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Mathematical analysis reveals exact separation points in Thue–Morse run-length sequences, demonstrating double-exponential growth across ratio-four scale chains.
Let d be the adjacent-run-length sequence of the Thue--Morse word. For every q>=2, we determine exactly the first position at which d(2^(q+2)n) and d(2^q n) differ, giving a closed double-exponential formula and a proof of minimality. We extend the result to arbitrary distances in the same ratio-four scale chain and derive lower bounds for general multipliers, with an exact infinite family characterized by multiplicative orders. Status: Public Beta; internally verified candidate proof; external mathematical review pending.
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