SMA X UNCOORD UNAUTH PENETRATION OF ADJACENT APCH CTL AIRSPACE.

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SMA X DEPARTED SATELLITE ARPT REQUESTING HIS IFR CLRNC. ACFT WAS ISSUED A BEACON CODE; ALTIMETER SETTING AND RADAR IDENT. THE ARTS IIA NEVER AUTO-ACQUIRED THE XPONDER CODE. 3 MINS LATER THE PLT ASKED FOR HIS CLRNC. HE WAS CLRED AS FILED; TOLD TO CLB AND MAINTAIN 8000 FT. PLT ACKNOWLEDGED CLRNC. VIA QUICK LOOK FUNCTION; I VERIFIED THAT HE WAS ON CORRECT SQUAWK. HE WAS; SO I MADE A MENTAL NOTE TO RESEARCH WHY HE DIDN'T TAG UP. MY AIRSPACE STARTED TO GET BUSY; WITH SEVERAL UNUSUAL SITS OCCURRING TO DRAW MY ATTN AWAY FROM THE PREVIOUS ARTS IIA TROUBLE. DUE TO THE LOCATION OF THE FDIO KEYBOARD BEING BEHIND MY BACK; I DIDN'T FEEL I SHOULD TURN MY BACK TO THE RADAR. AND I DIDN'T KNOW THAT SOMEONE WAS IN THE ROOM THAT COULD HAVE MADE THE COMPUTER ENTRIES. CONSEQUENTLY; THE ACFT WAS FORGOTTEN UNTIL HE CALLED ME OVER KEELER (ELX) VOR ASKING IF HE SHOULD STILL BE TALKING TO ME. I TOLD HIM TO STAND BY; THEN ATTEMPTED TO FIND OUT WHOSE AIRSPACE HE WAS IN. AFTER SEVERAL MINS; THE ACFT CONTACTED ZAU ON HIS OWN. ZAU CONTACTED ME ON A LANDLINE AND TOLD ME THAT THE ACFT WAS IN SOUTH BEND'S APCH AIRSPACE. BECAUSE THE COMPUTER NEVER ACQUIRED THE XPONDER CODE; NO ONE IN BTWN MY AIRSPACE AND SOUTH BEND APCH SAW THE ACFT OR EVEN HAD A FLT PROGRESS STRIP.