here is a roundabout way to delete the identical records keeping the one with the lowest (minimum) object ID, demonstrated with my sloppy python copied from the results tab.
arcpy.Merge_management(inputs="CPNewMeasEndEventsY;CPNewMeasEndEventsX;CPNewMeasEventsY;CPNewMeasEventsX", output="C:/temp/routeparts.gdb/CalibrationPoints_MultipartLocals")
arcpy.FindIdentical_management(in_dataset="CalibrationPoints_MultipartLocals", out_dataset="C:/temp/routeparts.gdb/cp_identicals", fields="RID;MEAS;POINT_X;POINT_Y;Measure", xy_tolerance="", z_tolerance="0", output_record_option="ONLY_DUPLICATES")
arcpy.Statistics_analysis(in_table="cp_identicals", out_table="C:/temp/routeparts.gdb/cp_identicals_to_delete", statistics_fields="IN_FID MIN", case_field="FEAT_SEQ")
arcpy.AddJoin_management(in_layer_or_view="CalibrationPoints_MultipartLocals", in_field="OBJECTID", join_table="cp_identicals", join_field="IN_FID", join_type="KEEP_COMMON")
arcpy.SelectLayerByAttribute_management(in_layer_or_view="CalibrationPoints_MultipartLocals", selection_type="NEW_SELECTION", where_clause="1=1")
arcpy.AddJoin_management(in_layer_or_view="CalibrationPoints_MultipartLocals", in_field="cp_identicals.FEAT_SEQ", join_table="cp_identicals_to_delete", join_field="FEAT_SEQ", join_type="KEEP_COMMON")
arcpy.SelectLayerByAttribute_management(in_layer_or_view="CalibrationPoints_MultipartLocals", selection_type="REMOVE_FROM_SELECTION", where_clause="cp_identicals_to_delete.MIN_IN_FID = cp_identicals.IN_FID")
arcpy.RemoveJoin_management(in_layer_or_view="CalibrationPoints_MultipartLocals", join_name="cp_identicals_to_delete")
arcpy.RemoveJoin_management(in_layer_or_view="CalibrationPoints_MultipartLocals", join_name="cp_identicals")
arcpy.DeleteFeatures_management(in_features="CalibrationPoints_MultipartLocals")