Jeffrey Chafin calls his daughter, Eris, "my sparkle." He says "she's everything for me, she's phenomenal, she's my life."

Lynne Hales Chafin says the 5-year-old is "quite happy" living life with her as the custodial parent.

What the estranged couple has to say about each other is not so cordial. Their custody fight is many ways a typical "he-said, she-said" dispute, but because it has been waged across international borders, the U.S. Supreme Court is now involved.

The case of Chafin v. Chafin will be heard by the justices on Wednesday and an eventual ruling could establish an important precedent on the discretion of American courts to decide where children caught in parental fights should stay.

In the middle is Eris, the girl who lives in Scotland with her mother. Her dad is an Army sergeant based at Ft. Stewart, Georgia.

A federal court said under international treaty, Eris should remain overseas since it was her "habitual residence." That court also said the custody issue was moot since the girl was already overseas.

Jeff Chafin eventually asked the justices to intervene on his behalf.

"I don't believe that (the current legal fight) is in the best interest of the child as it's going to go on for years and years to come," Lynne Chafin told CNN.

Jeff Chafin wants to revive his claim of Eris' custody, arguing the child's normal place of residence is the United States.

"I told my little girl I would do everything I could to get her," he told CNN National Correspondent Joe Johns, "to make sure she's safe."

He last visited his daughter overseas in July.

The couple met in 2006 in Germany, where Jeff was stationed. Eris was born there the following year.

It was while the father was deployed to Afghanistan for 15 months that the personal and legal troubles escalated.

Lynne claims it was agreed she would stay with the child in Scotland as the de facto "habitual residence" while Jeff served in combat.

U.S. courts later established that residency to conclude the child belonged with her mother.

"This is the heart of this treaty. The whole treaty turns on these two words -- 'habitual residence.' What is the ordinary, regular home of this little girl?" said Stephen Cullen, Lynne Chafin's attorney. "It has to be Scotland because the last time they ever agreed on anything was their agreement that Scotland was the child's home."

Upon Jeff's return from Afghanistan to Germany, the couple separated, then made various attempts at reconciliation. He was transferred to Huntsville, Alabama, and Lynne soon followed with Eris. But the domestic situation did not improve and the custody fights eventually played out there.

Jeff Chafin and his supporters say the child was happy in the United States, "fully engaged and immersed in her American community."

More importantly from a legal perspective, his lawyers argue his wife was fully committed to staying in the United States as a resident.

But the relationship deteriorated and divorce proceedings began. Lynne Chafin said she wanted to leave with their daughter.

That's when she said her husband called the police.

"I was removed from the house. I was taken to jail," she said.

When police checked her status on a tourist visa, "an immigration officer came out to see me. And he advised me that because my green card was still in the process, that I was actually there illegally and I would be deported."

It was after Lynne Chafin was sent out of the country without her daughter that she filed a "Petition for Return of Child to Scotland" under the 1980 Hague Convention's Civil Aspects of International Child Abduction. It grants parents in general a "right of custody" and a separate "right of access," ensuring the laws of one country are respected in the others.

More than 80 nations are a party to the treaty, including the United States and the United Kingdom.

Lynne Chafin did not see Eris for 10 months while the custody case played in a federal court in Alabama. She was allowed to return to the United States on a special visa just long enough to have a federal judge rule in her favor, allowing her to take Eris back to Scotland where they remain.

"I would like the Supreme Court to say that my case is moot," she said from her home. "If they don't, if they rule in my husband's favor and he is allowed to appeal then this is going to go on for years which is ludicrous. None of us will be able to move on with our lives, we're stuck in this constant state of appeals, appeals, appeals."

But Jeff Chafin and his lawyers say the federal courts are at odds over what judges can and cannot do over the "habitual residence" question. They say in this case the full appeals process was not allowed to continue, denying the father his due process rights.

"The phrase miscarriage of justice comes to mind," said Michael Manely, the father's attorney. "The other side's position is once a child leaves the boundaries of the United States, it's over. There is nothing you can do about it. And that has terrifying consequences."

Justice Sonia Sotomayor has unusual experience with this issue. As a federal judge on the 2nd U.S. Circuit Court of Appeals, she dissented from a decision that she said too narrowly interpreted the impact of a custody order.