Trying to understand why ramified primes are so special physically

Ramified primes (see this and this) are special in the sense that their expression as a product of primes of extension contains higher than first powers and the number of primes of extension is smaller than the maximal number n defined by the dimension of the extension. The proposed interpretation of ramified primes is as p-adic primes characterizing space-time sheets assignable to elementary particles and even more general systems.

In the following Dedekind zeta functions (see this) as a generalization of Riemann zeta are studied to understand what makes them so special. Dedekind zeta function characterizes given extension of rationals and is defined by reducing the contribution from ramified reduced so that effectively powers of primes of extension are replaced with first powers.

If one uses the naive definition of zeta as analog of partition function and includes full powers Piei, the zeta function becomes a product of Dedekind zeta and a term consisting of a finite number of factors having poles at imaginary axis. This happens for zeta function and its fermionic analog having zeros along imaginary axis. The poles would naturally relate to Ramond and N-S boundary conditions of radial partial waves at light-like boundary of causal diamond CD. The additional factor could code for the physics associated with the ramified primes.

The intuitive feeling is that quantum criticality is what makes ramified primes so special. In O(p)=0 approximation the irreducible polynomial defining the extension of rationals indeed reduces to a polynomial in finite field Fp and has multiple roots for ramified prime, and one can deduce a concrete geometric interpretation for ramification as quantum criticality using M8-H duality.

See the chapter TGD View about Coupling Constant Evolution or the article Trying to understand why ramified primes are so special physically.