Some clarity is in order here. The paper is a compilation of notes from a class and textbook. The reason for posting it was it contains the mathematical proof of where the energy is stored. It clearly states at the top of the notes HF.(high frequency) It is series of organized notes and is much different then a published paper.
HMEL, you keep on mis-stating your own reference, there is no proof that all of the energy is in the air gap, here is a direct quote from your reference "f. Air gap in magnetic core media. we review the solution to the magnetic circuits we face in inductor and transformer design on a DC basis. The bulk of the energy is primarily stored in the air gap not in the core itself" ---- there is no way that "the bulk of the energy..." is a mathematical proof that "all of the energy...", in fact the amount of energy in the air gap is proportional to the size of the air gap so it can be less than or greater than the core, it all depends on the design. Peter.