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@mcd1231 wrote:

Can she claim the baby for child tax credit and EITC?

Or does he have to claim their son because his AGI is higher (tie breaker rules0?

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Nothing in your post disqualifies the credits.

The Tiebreaker Rules only apply if more than one person actually claims the child.  If two or more people are eligible to claim the child and they agree who actually claims the child, the Tiebreaker Rules don't apply.