What happened
The FAA’s narrative, in sentence case. The FAA writes it in capitals and removes private details.
Aircraft N5GF departed peachtree-dekalb airport (pdk) at 0936 local on a part 91 flight to Colorado springs municipal airport (cso). A normal departure and climb to altitude was observed until passing FL400. A loud thud was heard by both pilots behind the pilot's instrument panel. Next there was a loud hissing sound, this all occurred within 30 minutes of this flight. Pressurization was checked and it indicated an altitude of 6,300 feet, which is normal. Once the aircraft reached FL410THER cabin altitude began to climb at 700-1000/minute. O2 masks were donned and the pilots immediately requested a descent due to a gradual loss of pressurization. An emergency was declared and asked to return to pdk through memphis center. A descent to 10,000 MSL was accomplished and the pilots began to perform the normal and emergency checklist procedures per the Gulfstream quick reference handbook. The aircraft landed at pdk with no further incidence.
As the FAA filed it
AIRCRAFT N5GF DEPARTED PEACHTREE-DEKALB AIRPORT (PDK) AT 0936 LOCAL ON A PART 91 FLIGHT TO COLORADO SPRINGS MUNICIPAL AIRPORT (CSO). A NORMAL DEPARTURE AND CLIMB TO ALTITUDE WAS OBSERVED UNTIL PASSING FL400. A LOUD THUD WAS HEARD BY BOTH PILOTS BEHIND THE PILOT'S INSTRUMENT PANEL. NEXT THERE WAS A LOUD HISSING SOUND, THIS ALL OCCURRED WITHIN 30 MINUTES OF THIS FLIGHT. PRESSURIZATION WAS CHECKED AND IT INDICATED AN ALTITUDE OF 6,300 FEET, WHICH IS NORMAL. ONCE THE AIRCRAFT REACHED FL410THER CABIN ALTITUDE BEGAN TO CLIMB AT 700-1000/MINUTE. O2 MASKS WERE DONNED AND THE PILOTS IMMEDIATELY REQUESTED A DESCENT DUE TO A GRADUAL LOSS OF PRESSURIZATION. AN EMERGENCY WAS DECLARED AND ASKED TO RETURN TO PDK THROUGH MEMPHIS CENTER. A DESCENT TO 10,000 MSL WAS ACCOMPLISHED AND THE PILOTS BEGAN TO PERFORM THE NORMAL AND EMERGENCY CHECKLIST PROCEDURES PER THE GULFSTREAM QUICK REFERENCE HANDBOOK. THE AIRCRAFT LANDED AT PDK WITH NO FURTHER INCIDENCE.