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Came across a question that talked about fragmentation of peroxyesters, about the peroxide linkage of course. After the fragmentation of the peroxyester was done, it gave out three products (2 being free radicals, and the other being carbon dioxide)

One of the products(i.e. free radical), went under a reaction (with itself) and gave two products, one being another free radical and the other being a carbonyl compound. I am unsure about the mechanism.

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Here's what I thought: The single bond between the C1 and C2 will cleave in a homolytic manner. The (now) unpaired electron on C1 will combine with the unpaired electron of oxygen to form a $\pi$ bond. Also, the unpaired electron will remain on C2.

Why am I not sure about my reasoning? Because why would a homolytic cleavage take place between C1 and C2 after all?

Any hints or tips would be highly appreciated.

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