Thue–Morse: all three Joshi–Rust first-occurrence formulas

All three displayed first-occurrence families are proved in the stated ranges. The proof also gives exact all-start classifications for the plus and even-minus families, and rederives their established maximal-length values without assuming those values as hypotheses. For $q=2^m, m≥2$, every length-$(q+2)$ progression at difference $q+1$ starts at $lq²−q−1$ with $l≥1$ and $t(l−1)=t(l+1)=1−t(l)$. For even $m≥2$, every length-$(q+4)$ progression at difference $q−1$ starts at $lq²−q+1$ with the same condition. Its least $l$ is $3$. For odd $m$, the earliest length-$q$ progression at difference $q−1$ starts at $q−1$. Global maximality and exclusion of every earlier start are included. The prior length results and block-recognition methods of Parshina and Aedo–Grimm–Nagai–Staynova are explicitly credited. This is the first-occurrence conjecture, distinct from the existing avoiding-population matching-height entry. Broader all-difference classifications are not claimed.

All three displayed first-occurrence families are proved in the stated ranges. The proof also gives exact all-start classifications for the plus and even-minus families, and rederives their established maximal-length values without assuming those values as hypotheses. For $q=2^m, m≥2$, every length-$(q+2)$ progression at difference $q+1$ starts at $lq²−q−1$ with $l≥1$ and $t(l−1)=t(l+1)=1−t(l)$. For even $m≥2$, every length-$(q+4)$ progression at difference $q−1$ starts at $lq²−q+1$ with the same condition. Its least $l$ is $3$. For odd $m$, the earliest length-$q$ progression at difference $q−1$ starts at $q−1$. Global maximality and exclusion of every earlier start are included. The prior length results and block-recognition methods of Parshina and Aedo–Grimm–Nagai–Staynova are explicitly credited. This is the first-occurrence conjecture, distinct from the existing avoiding-population matching-height entry. Broader all-difference classifications are not claimed.