In modern English: although there are special circumstances in which divorced or separated parents can claim a QC for other tax benefits, the Residence Requirement is an absolute requirement for Earned Income Credit (EIC).
A non-custodial parent with whom the child lived less than half the year NEVER, EVER, EVER gets to claim the QC for EIC purposes!
It doesn't matter if the custodial parent wants to give the EIC away to the non-custodial parent. If the noncustodial parent taxpayer did not live with the kid, the taxpayer is not allowed to claim the kid for EIC purposes!
Ya can't claim the kid for EIC without living with the kid!
Interestingly, you don't need to support your QC to claim him for EIC purposes, but you do need to live with him!
Residence is the sine qua non requirement for EIC QC purposes.
That is really important, because the EIC is by far the largest tax benefit associated with children in low-income working families. As you can see from the graph below, whether a taxpayer has children can make a huge difference in their EIC refund, so it's critically important to get this detail correct, so the EIC goes to the taxpayer actually entitled to claim the QC for EIC purposes.
(Caveat: remember that the IRS does not define "live with" quite the same way as normal people do. If the child was born or died during the year or had temporary absences for college, medical reasons, etc. the IRS may deem that s/he "lived with" the taxpayer for all 12 months of the year for tax purposes.)
For more information about EIC rules, see here.