AFTER RTN LNDG FROM TRAINING FLT; STUDENT AND CFII CHALLENGE N90 NOTIFICATION OF NY ADIZ PENETRATION. THOUGH WITHDRAWN; STUDENT REQUESTS; BUT DOES NOT RECEIVE; PROCEDURAL JUSTIFICATION.

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I FLEW WITH A FLT INSTRUCTOR FROM CDW TO MGJ TO PRACTICE INSTRUMENT APCHS AT MGJ. I CALLED MILLVILLE FSS THAT MORNING TO FILE MY ADIZ FLT PLANS FOR OUT OF AND INTO THE NY ADIZ. AT APPROX XA00 LCL; I CALLED CDW GND FOR MY CLRNC TO TAXI AND XPONDER CODE. I WAS GIVEN MY CLRNC AND A XPONDER CODE. WE DEPARTED SHORTLY THEREAFTER. DUE TO FREQUENCY CONGESTION WITH NY APCH; CDW TWR HAD US STAY WITH THEM UNTIL CLR OF THE ADIZ. I WAS UNDER THE HOOD; AND MY CFII WAS WATCHING FOR TFC. AT SOME POINT IN TIME; THE CFII SAID THAT WE WERE CLR OF THE NY ADIZ; AND THAT I SHOULD CALL CDW TWR AND TELL THEM THAT WE WOULD LIKE TO SQUAWK 1200; WHICH I DID . CDW TWR ACKNOWLEDGED; AND GAVE US PERMISSION TO SQUAWK 1200 AND CHANGE FREQUENCIES. AT AROUND XB30 LCL; I CALLED NY APCH ON 132.75 TO GET MY CLRNC TO RE-ENTER THE NY ADIZ. THEY GAVE ME A XPONDER CODE AND SAID TO FLY DIRECT TO CDW. UPON LNDG; I CALLED MILLVILLE FSS TO INFORM THEM I WAS ON THE GND. MILLVILLE REQUESTED I CALL THEM BECAUSE THEY SAID ATC WAS RUNNING OUT OF XPONDER CODES. AFTER I LANDED; I FOUND OUT THAT AT AROUND XA15 LCL; NY TRACON LEFT A MESSAGE ON MY HOME ANSWERING MACHINE FOR ME TO CALL THEM. I CALLED AROUND XD00 THE NY TRACON SUPVR TOLD ME THAT THEY TRACKED ME FLYING IN THE NY ADIZ AND THAT I MAY NOT HAVE HAD THE PROPER CLRNC. I TOLD HIM THAT I DISAGREED; AND PROVIDED HIM MY XPONDER CODES AND TIMES I CALLED FSS. HE SAID THAT AS LONG AS WHAT I PROVIDED HIM WAS ACCURATE; THIS WOULD NOT BE PURSUED ANY FURTHER. THE SIT STILL BOTHERED ME; SO I CALLED NY TRACON AGAIN AT AROUND XD40. AFTER I GAVE THE PERSON MY TAIL NUMBER; HE REITERATED THE INFO THAT I PROVIDED EARLIER. HE SAID THAT THERE WOULD BE NO VIOLATION AGAINST ME. I TRIED TO ASCERTAIN FROM HIM MORE DETAILS ON WHY THEY THOUGHT I WAS IN VIOLATION; DID I SQUAWK 1200 TO SOON; WAS MY XPONDER MALFUNCTIONING; WAS THEIR A MIX-UP BETWEEN TWR AND NY APCH; BUT HE KEPT ON RESPONDING THAT I DID HAVE THE PROPER CLRNC. IN THE FUTURE; I WILL CONFIRM WITH ATC THAT I AM CLR OF THE ADIZ PRIOR TO SQUAWKING 1200.