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Ring expansion of vinyl cyclobutane -

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Could someone please help me with a detailed mechanism for the above conversion?

It is the acid-catalyzed rearrangement of 4,7,7-trimethylbicyclo[3.2.0]hept-3-en-6-one to 3,3-dimethyl-6-methylidenebicyclo[2.2.1]heptan-2-one.

I couldn't find the mechanism anywhere, and couldn't propose a good one either. Please help.

P.S. As one of the users in the comments is interested in seeing my best effort, I'll explain my thought process- As the carbonyl oxygen is the most nucleophilic, it is the first to be protonated in acidic medium. Furthermore, this species has a resonating structure (less contributing, but the only one that can lead to a ring expansion, I believe) – one obtained by shifting the π bond towards oxygen – a carbocation. After this, I'm clueless. I think, that the expansion could have something to do with the hydrogen at alpha position to the sp² hybridised carbon atom in the five membered ring – as the product has an exocyclic double bond.

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You are correct about protonating the carbonyl group. [The first of your products is the same as the reactant.] The cyclobutanone 2 was prepared from a mixture of geranic acids. (J. J. Beereboom, J. Org. Chem., 1965, 30, 4230). The structures of 2, 4 and 5 were determined by Beereboom. I am providing the most reasonable mechanism. Cyclobutanone 2 must have a more stable cis ring juncture owing to its method of formation or exposure to p-TsOH. The trans isomer of 2 cannot give the trans isomer of 3 because of strain. Tertiary carbocation 3 arises by addition of the double bond to the protonated carbonyl. Collapse of the carbocation produces bicyclic ketone 4, which in the presence of p-TsOH isomerizes to the exocyclic isomer 5. In general, 1,1-disubstituted double bonds are less stable than tri-substituted ones. In this case, ring strain may be present in the endocyclic alkene 4.

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