Is there a form that a custodial parent has to sign to show proof to claim child yearly?

If you are married filing separately and both parents live with the child, either parent can claim the child as a dependent and only they get all the child related benefits ... they cannot be split.

If you are unmarried but live together with the child, either one of you can claim the child (or children) as dependents. If the parent who claims the child as a dependent also pays more than half the expenses of keeping up their home, they can file as head of household instead of single, which is slightly more favorable.  The other parent should not even list them in Turbotax, because there are some poorly worded questions that confuse some people.

If you are unmarried or never married and live apart and share custody, then:

The parent with whom the child lives more than half the year (183 or more nights ) is automatically entitled to claim the child as a dependent. This is the custodial parent. (IRS determines custody based on where the child lives, not any court order or agreement.)  The non-custodial parent is not entitled to claim anything.

However, the custodial parent can sign a release (form 8332) allowing the non-custodial parent to claim the child as a dependent.  You can download this form from the IRS web site.  The custodial parent signs it and gives it to the non-custodial parent and the non-custodial parent mails it to the IRS after e-filing the rest of their tax return.  In this case, the non-custodial parent can claim the dependent exemption and the child tax credit.  The non-custodial parent can never claim earned income credit, the dependent care credit (day care credit) or use the child to qualify for head of household status.  Those benefits always stay with the custodial parent.

Q. Is there a form that a custodial parent has to sign to show proof to claim child yearly?

A. No.  The only time a form is required is when the non custodial parent is claiming the child. The custodial parent does not need a form. 

You said " Mother can claim any unearned income yearly."  That doesn't make sense.  There is a situation where one parent can claim the child as a dependent (child tax credit) and the other claim the Earned income credit (EIC).  But, the parents cannot be living together.

For more specific advice, describe your situation.  Are you an unmarried couple living together? Are you divorced or separated (including never married)? With whom  did the child live during the year and for how long? The child must live with you for more than half the year, for you to be the "custodial parent".  The IRS goes by physical custody, not legal custody. The child must live with you for more time (or equal time) than he/she lived with the other parent, for you to have priority on claiming the child. "Just won full custody of child" does not necessarily mean that  you get to claim the child for 2023. 

_________________________________________________________________________________________

General Rules

The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The IRS goes by physical custody, not legal custody. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree.

 There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.

If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice.