Could one understand p-adic length scale hypothesis in terms of functional arithmetics?

Holomorphy= holography vision reduces the gravitation as geometry to gravitations as algebraic geometry and leads to exact general solution of geometric field equations as local algebraic equations for the roots and poles rational functions and possibly also their inverses.

The function pairs f=(f₁,f₂): H→ C² define a function field with respect to element-wise sum and multiplication. This is also true for the function pairs g=(g₁,g₂): C²→ C². Now functional composition _º is an additional operation. This raises the question whether ordinary arithmetics and p-adic arithmetics might have functional counterparts.

One implication is quantum arithmetics as a generalization of ordinary arithmetics (see this). One can define the notion of primeness for polynomials and define the analogs of ordinary number fields.

What could be the physical interpretation of the prime polynomials (f₁,f₂) and (g₁,g₂), in particular (g₁,Id) and how it relates to the p-adic length scale hypothesis (see this)?

1. p-Adic length scale hypothesis states that the physically preferred p-adic primes correspond to powers p∼ 2^k. Also powers p∼ q^k of other small primes q can be considered (see this) and there is empirical evidence of time scales coming as powers of q=3 (see this and this). For Mersenne primes M_n= 2ⁿ-1, n is prime and this inspires the question whether k could be prime quite generally.
2. Probably the primes as orders of prime polynomials do not correspond to very large p-adic primes (M₁₂₇=2¹²⁷-1 for electron) assigned in p-adic mass calculations to elementary particles.

The proposal has been that the p and k would correspond to a very large and small p-adic length scale. The short scale would be near the CP₂ length scale and large scale of order elementary particle Compton length. Could small-p p-adicity make sense and could the p-adic length scale hypothesis relate small-p p-adicity and large-p p-acidity?

1. Could the p-adic length scale hypothesis in its basic form reflect 2-adicity at the fundamental level or could it reflect that p=2 is the degree for the lowest prime polynomials, certainly the most primitive cognitive level. Or could it reflect both?
2. Could p∼ 2^k emerge when the action of a polynomial g₁ of degree 2 with respect to say the complex coordinate w of M⁴ on polynomial Q is iterated functionally: Q→ P circ Q → …P _º…P_º Q and give n=2^k disjoint space-time surfaces as representations of the roots. For p=2 the iteration is the procedure giving rise to Mandelbrot fractals and Julia sets. Electrons would correspond to objects with 127 iterations and cognitive hierarchy with 127 levels! Could p= M₁₂₇ be a ramified prime associated with P_º …_º P.

If this is the case, p∼ 2^k and k would tell about cognitive abilities of an electron and not so much about the system characterized by the function pair (f₁,f₂) at the bottom. Could the 2^k disjoint space-time surfaces correspond to a representation of p∼ 2^k binary numbers represented as disjoint space-time surfaces realizing binary mathematics at the level of space-time surfaces? This representation brings in mind the totally discontinuous compact-open p-adic topology. Cognition indeed decomposes the perceptive field into objects.

This generalizes to a prediction of hierarchies p∼ q^k, where q is a small prime as compared to p and identifiable as the prime order of a prime polynomial with respect to, say, variable w. I have considered several identification of the p-adic primes and arguments for why the p-adic length scale hypothesis should be true.

1. One can imagine I have tentatively identified p-adic primes as ramified primes (see this) appearing as divisors of the discriminant Dof a polynomials define as the product of root differences, which could correspond to that for g=(g₁,Id).

Could the 3 primes characterizing the prime polynomials f_i:H→ C² correspond to the small primes q? Could the ramified primes p∼ 2^k as divisors of a discriminant D defined by the product of non-vanishing root differences be assigned with the polynomials obtained to their functional composites with iterates of a suitable g?

Similar hypotheses can be studied for the iterates of g:C²→ C² alone. The study of this hypothesis in a special case g=P₂= x(x-1) described in an earlier section did not give encouraging results. Perhaps the identification of p-adic prime as ramified primes is ad hoc. There is also the problem that there are several ramified primes, which suggests multi-p-p-adicity. The conjecture also fails to specify how the ramified prime emerges from the iterate of g.

A new identification of p-adic primes suggested by quantum p-adics is that p-adic primes correspond to primes defining the degrees of prime polynomials g and that the Mersenne primes Nn= 2n-1 correspond to rational functions P2º n/P1, where / corresponds to element-wise-division and P2 can be any polynomials of degree 2. This would mean category theoretic morphism of quantum p-adics to ordinary p-adics. A more general form of the conjecture is that the rational functions Ppº n/Pk correspond to preferred p-adic primes.

The reason could be that for these quantum primes it is possible to solve the roots as zeros and poles analytically for p<5. This might make them cognitively very special. The primes p=2 and p=3 would be in a unique role information theoretically. For these primes there is indeed evidence for the p-adic length scale hypothesis and these primes are also highly relevant for the notion of music harmony (see this, this and this).

See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/04/could-one-understand-p-adic-length.html