My professor posed my organic chemistry class a challenge problem for bonus points and I've been racking my brain over the last three days trying to work it out and I want to know if my solution is plausible. The problem asks us to draw the best curved arrow mechanism for the reaction in the bottom left corner of the photo included in this post (a substituted cyclobutane molecule to a substituted cycloalkene). At first, this seemed like a simple intramolecular Robinson annulation, however it appears to be catalyzed using the hydronium ion, rather than the hydroxide ion. No other reactants are stated, including light, heat, nor a double equivalence of the starting compound. Working through every mechanism that I could think of, I eventually ended up with the solution in the picture below. I assumed that reaction rates and percent yields were unimportant, as long as the mechanism is plausible. Does this solution break any rules of organic chemistry? Is there a better solution?
As I mentioned in my Comment the only serious issue was the formation of an allylic anion in acid medium. You seem to know how to form enols from ketones and how to form ketones from enols. I will ignore those steps in the diagram. Protonation of the carbonyl in 1 can cause ring fragmentation to form transient enol 2 that rapidly tautomerizes to diketone 3. Cyclization via enol 4 leads to β-hydroxyketone 5. Although loss of water from 5 as you have described is viable, I prefer formation of the enol 6 because loss of water is facilitated over a tertiary cation given the allylic nature of the species 7. Loss of a proton from carbocation 7 leads to 3-methylcyclohex-2-en-1-one (8).
Addendum: The process is not a Robinson annulation, which involves an initial Michael addition followed by an aldol condensation. The latter reaction includes the transformation 3 $\rightarrow$ 8. The transformation 1 $\rightarrow$ 3 may described as a retro-aldol reaction.
You may want to see the formation of mesitylene (1,3,5-trimethylbenzene) from acetone under acid conditions.