NYC- Police have video proof that an estranged lover of slain stripper Catherine Woods was standing just outside her Upper East Side building around the time the 21-year-old brunette was brutally murdered, The Post has learned.

The surveillance video – obtained from a camera posted on a building down the block – shows Paul Cortez, 25, on Sunday night outside 355 E. 86th St., where Woods was found nearly decapitated inside her bedroom, sources said.

That video and evidence that Cortez made cellphone calls to the Ohio native from right outside her apartment make him the No. 1 suspect in her slaying.

“We believe that he was making the cellphone calls from just about right outside on the street,” said a source, adding that Cortez came across as a “con artist” in interviews with police.

Area shopkeepers confirmed that police yesterday retrieved a surveillance video from the building staff at 345 E. 86th St., which is five doors down from Woods’ building – though it’s not clear if that’s the video that shows him. Management at 345 E. 86th St. refused to comment.

The Post revealed yesterday that police had learned that Cortez – a personal trainer and aspiring actor/rock singer – made seven cellphone calls in quick succession to Woods’ cellphone shortly before she was murdered.

Cortez told cops he made the calls from his home, 23 blocks away.

But that was a lie, according to sources, who said detectives have learned that the calls made at about 6:30 p.m. Sunday – which went unanswered by Woods – were made by him outside her building.

Cortez, who has retained a lawyer, refused to comment yesterday and called police when reporters tried to contact him at his East 106th Street apartment. He has not returned to work at the Equinox health club near Woods’ apartment since being questioned by police.

Cops also suspect the 2003 Boston University theater-studies graduate because he would have known Woods’ pattern for going to work.

In particular, he would know that Woods tended to leave her apartment at about 7 p.m., and that her 23-year-old ex-boyfriend and roommate, David Haughn, had a habit of going to retrieve his car before returning to drive her to work, sources said.

It is known that Woods was in her apartment on the phone with a friend at 6:10 p.m., and Haughn has told cops she was alive when he left her at 6:30 p.m. to get his car to take her to a shift at a topless club.

Woods’ hometown friend Megan Wilkins told an Ohio TV station that they had a cheery conversation the day she was murdered.

“She was just telling me about her day, what she did. She went to lunch with our friend Christina. She was in a good mood. She was happy. Nothing unusual. Nothing,” said Wilkins.

Haughn, a would-be rapper who performs under the name “City,” also claims that he returned almost 20 minutes later to find Woods butchered, with blood splattered throughout the room. He placed a 911 call at 6:49 p.m.

Sources said the killing would have taken about five minutes and that the murderer would have been prepared to hide any blood that splashed on him by bringing something to wear over his clothes or a change of clothes.

That makes it likely that the murder was premeditated instead of the crime of passion it initially appeared to be.

Cortez, according to a person who knows him, had called friends on Sunday afternoon to tell them he was performing with his band, Monolith, that night at a club called Rare in Manhattan’s Meat Packing District.

But a manager at Rare yesterday said that the club was closed Sunday night, as it usually is.

Cortez only spoke to cops for a few hours on Monday before clamming up and getting a lawyer, leaving his interrogation in tears, a source said.

In contrast, Haughn spoke to police for more than a day and also allowed cops to search his car and belongings “without an ounce of trepidation,” the source said.

Cortez has told police he had an on-again/off-again relationship with Woods, whom he met while training her last year at a health club near her home. But he also said he was not dating her at the time of her murder.

Woods, a classically trained dancer whose father is director of the famed Ohio State University marching band, moved from Columbus to New York more than three years ago with her then-boyfriend, Haughn, to try to make it in legitimate theater.

Although she told her parents she was teaching dance, performing in various troupes and auditioning, she ended up working as an exotic dancer at the upscale Chelsea topless club Privilege earlier this year, before working shifts at the Flashdancers strip joint in Times Square.

A memorial service for Woods is scheduled today in Columbus.

Curt DeGroat, a friend of Cortez’s, yesterday said, “There’s a lot of people who are concerned about him right now.”

“Paul is an extraordinary person. He’s a beautiful yoga teacher. He’s very loving, very sensitive,” said DeGroat, 40, who last saw Cortez on an upstate meditation retreat in October. “There’s no question in my mind that he didn’t do this to this woman . . . I’m sure the evidence will prove that.

“He really loved her very deeply,” said DeGroat.

DeGroat also said Cortez “was always very afraid for” Woods because he believed that the people she was hanging out with were very bad news. “He knew that she was in a dangerous situation.”

A note on Woods’ kitchen calendar shows she had an appointment for yoga the day she was killed – but not at the gym where Cortez is employed.