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The following double cyclization was used to generate the ring system of the lycoanitine alkaloids in 2005. Demonstrate your understanding of this C−C bond-forming process by providing a mechanistic rationale (using arrows that represent electron movement) for the boxed reaction.

Double cyclization reaction scheme

From what I know in my current synthetic organic chemistry class, this question appears to me like a Paal–Knorr reaction, but the product isn't a pyrrole. I don't even know where to start, and I'm not entirely certain how and where the Lewis acid interacts.

Ignore the boxed part, that's for me to do; I'm just curious to know how the carbons attached to the methoxy group on the tertiary amine are able to bond with the C2 carbons on the 1,4-diketone.

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This is a variant of the Mannich reaction. The methoxymethyl groups on the amine are a form of captured Mannich intermediates or masked formaldehyde equivalents. The Lewis acid unmasks them by binding to the oxygen of the methoxy group, creating the functional equivalent of H2C=N+. These are reactive electrophiles which react with the active methylenes of the B-ketoester to give the alkaloid.

More on the Mannich reaction here

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