Exact Thue–Morse matching heights in an avoiding population

The proof gives an explicit closed formula for the attained maximum $L(v)$ at every natural starting vertex, including $L(0)=1$, and a certified ten-coordinate integer recurrence that evaluates the same actual-path maximum from binary digits. For $v ≥ 1$, $3L(v)\le 8v-1$. Equality holds exactly at $v=3\cdot 2^n-1$ for natural n, where $L(v)=8\cdot 2^n-3$. The coefficient 8/3 cannot be decreased even after allowing a fixed additive constant. The source note gives the complete piecewise formula. The theorem proves both existence of a path attaining the value and an upper bound on every matching path. This answers the later manuscript's quantitative question for its fixed graph and unshifted target. It does not originate the earlier classification or resolve Alexander's remaining universal-avoider embedding or general ordinal-characterization questions. A bounded literature review is available, but worldwide priority is not certified.

The proof gives an explicit closed formula for the attained maximum $L(v)$ at every natural starting vertex, including $L(0)=1$, and a certified ten-coordinate integer recurrence that evaluates the same actual-path maximum from binary digits. For $v ≥ 1$, $3L(v)\le 8v-1$. Equality holds exactly at $v=3\cdot 2^n-1$ for natural n, where $L(v)=8\cdot 2^n-3$. The coefficient 8/3 cannot be decreased even after allowing a fixed additive constant. The source note gives the complete piecewise formula. The theorem proves both existence of a path attaining the value and an upper bound on every matching path. This answers the later manuscript's quantitative question for its fixed graph and unshifted target. It does not originate the earlier classification or resolve Alexander's remaining universal-avoider embedding or general ordinal-characterization questions. A bounded literature review is available, but worldwide priority is not certified.