Fathi's consecutive-integers bound does not extend to multitwists

On the sphere with five boundary components, a fixed positive multitwist can have exceptional exponents arbitrarily far apart, even when the base map is pseudo-Anosov. More precisely, we give curves $a,c,d$ such that, for every $k\ge1$, the multitwist $t=T_aT_c$ and $g_k=T_a^{-1}T_c^{-(k+1)}T_d$ satisfy Ivanov's filling hypothesis (the component orbits together fill the surface), while $tg_k$ and $t^{k+1}g_k$ fix essential curves. Their exceptional exponents are $k$ apart. Thus Fathi's consecutive-integers bound for a single Dehn twist does not extend to the multitwists in Ivanov's question. Long and Morton proved finiteness of the exceptions for a single twist about an essential curve with a pseudo-Anosov base on a closed orientable surface; on closed surfaces, Fathi confined the exceptional exponents to seven consecutive integers whenever the twisting curve's orbit fills the surface, for any base mapping class. We answer Ivanov's consecutive-integers question for this surface; examples with reducible bases are also given. Penner's construction supplies the pseudo-Anosov bases, and a pants decomposition proves filling. The algebra is verified in Lean and the finite curve and topology calculations by independent exact programs. The weaker question about the number of exceptions remains open.

Written proof, with Lean and computer checks; not yet reviewed by experts. Produced by AI models under the author's direction.