In zero energy ontology M-matrix is in a well-defined sense "complex" square root of density matrix reducing to a product of Hermitian square root of density matrix multiplied by unitary S-matrix. A natural guess is that p-adic thermodynamics possesses this kind of square root or better to say: is modulus squared for it.
For fermions the value of p-adic temperature is however T=1 and thus minimal. It is not possible to construct real square root by simply taking the square root of thermodynamical probabilities for various conformal weights. One manner to solve the problem is to assume that one has quadratic algebraic extension of p-adic numbers in which the p-adic prime splits as p= ππ*, π= m+(-k)1/2n. For k=1 primes p mod 4=1 allow a representation as product of Gaussian prime and its conjugate.
For primes p mod 4=3 Gaussian primes do not help. Mersenne primes rerpesent an important examples of these primes. Eisenstein primes provide the simplest extension of rationals splitting Mersenne primes. For Eisenstein primes one has k=3 and all ordinary primes satisfying either p=3 or p mod 3=1 (true for Mersenne primes) allows this splitting. For the square root of p-adic thermodynamics the complex square roots of probabilities would be given by π(L0/T)/Z1/2, and the moduli squared would give thermodynamical probabilities as p(L0/T)/Z. Here Z is the partition function.
An interesting question is whether T=1 for fermions means that complex square of thermodynamics is indeed complex and whether T=2 for bosons means that the square root is actually real.
For background see the chapter Physics as Generalized Number Theory: p-Adicization Program of "Physics as Generalized Number Theory".