“I finally heard back from Congressman Rangel’s office and it seems we haven’t gotten the signed lease back because they lost it!” OGS real-estate specialist Sydney Allen wrote in a July 30, 2013, e-mail to a colleague that was ­obtained by The Post.

Rangel paid $7,253 in monthly rent on the 125th Street office he has rented since 2000, expense reports from 2012 show. But the payments stopped for all of 2013.

Incredibly, instead of demanding payment of the back rent and late fees from its deadbeat legislative tenant, the state cut him a huge rent break.

The state says it allowed Rangel in March 2013 to enter into a new sweetheart deal in which he could postpone paying six months of rent. That “abatement” money has still not been paid, nor has the other six months of missed rent from 2013, a OGS official said.

The state comptroller approved a $101,000 lease between Rangel and OGS on Dec. 26, 2013, retroactively covering the period back to April 2013 and future months through December 2014, records show. The 21-month deal resulted in a deeply reduced rent of $4,809 a month.

When The Post last week inquired about the year of missing rent, Rangel’s office and OGS blamed federal sequestration, not the lost lease referenced in agency correspondence.