Detect duality obstruction of calibrations in smooth category

This paper consists of three parts: (a) exhibit a new gluing result which can dramatically simplify extensions of calibration pairs; (b) observe that every Lawlor cone can support coflat calibrations singular only at the origin; © show that there exist many Lawlor cones which cannot support any smooth calibrations. As an application, we extend our previous work on detecting duality obstruction of calibrations in the smooth category.