I just wrote this paper:
http://arxiv.org/abs/1510.03367
called "Layered Heaps Beating Fibonacci and Regular Heaps in Practice"
which describes a recursively defined layered heap structure that beats all other heaps in practice for $2$-ary leayered heaps, and for general $M$-ary layered heaps the insert time is $O((\log n)^{1/M}))$ and pop time is $O(\log n)$.
Does this look like a good theoretical and mathematical data structures paper?
You say that for $M\geq 2$, a $k_M$-ary heap is used, with $k_M = 2^{(\log n)^{k_{M-1}(M-1)/M}}$.
Setting aside the question of how you define a $d$-ary heap when $d$ is not an integer, what is $n$ in your formula for $k_M$? It appears to be the number of elements stored; is that it? Does this mean the number of children per node is a function of the number of nodes currently stored in the heap? Do you have to completely reorganize the heap with more children per node from time to time as $n$ gets larger?
It is OK to completely reorganize a data structure every once in a while, and you can amortize the cost of that reorganization over multiple insert or delete operations, but ignoring it is not valid.
In my opinion your short essay is far too vague about what it is that you actually are doing. It is not possible to make a valid asymptotic analysis of such an incompletely-described algorithm.