the slope or behaviour the BH curve at low excitation is any different to the slope/ behaviour at slightly higher excitation - nor can c-mitra supply one....
You are correct.
For small excursions of H (i.e., the current in this case), far from the saturation limit, the B-H curve can be considered linear and hence the mu should be approximately constant.
The actual current has not been provided. If I use 50mH at 50Hz, the impedance will be around 15 Ohm and the current at 50mv will be few mA at most. That is not sufficient to calculate H but the inductor is rated at 1A.
The question is whether the current excursion at 50mv and 300mv will get the core into the non-linear region of the B-H curve? The author claims more than 10% change. I doubt but I also do not know.
Visually, the hysteresis curve, close to zero, far from saturation looks very linear. In particular with low excitation current.
What is the role of the gap in this case? Qualitatively speaking, it will try to linearize the curve but I do not have the patience to recalculate and verify. The midpoint slope will decrease and the mu graph will become flatter.
Personally I do not like measuring an inductor rated at 1A at a current level of few mA. The measurements should be performed around the rated parameters.
My comment was on the principles of physics you mentioned.
Perhaps we are seeing the infamous Barkenhausen effect.