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Holography= holomorphy hypothesis, small primes viz. large p-adic primes and p-adic length scale hypothesis

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Prime polynomials and complexity hierarchy

We have had highly interesting Facebook discussions with Robert Paster relating to the p-adic mathematics (see this). Robert Paster finds the so called Universal Witt Vectors (UWVs) as a more promising approach than p-adic numbers whereas I am a proponent of adelic physics in which all p-adic number fields and also a hierarchy of their extensions induces by the extensions of rationals are in a central role.

UWVs generalize the Witt Vectors (WVs) (see this), which provide a representation of p-adic number field in terms of Witt polynomials with order, which is power of prime p and pinary expansion can be represented in terms of the hierarchy if Witt vectors Wk for which sum and product are element-wise operations. The UWVs are defined for any positive integer n and Wn codes for all factors of integer n and reduces to a collection of Wpk when n is restricted to n=pk: it is denoted by Wk in this case.

An interesting question is whether UWVs as a number theoretic notion might have some role in TGD.

These discussions led to some progress in the longstanding attempts to understand how small primes and p-adic primes emerging from p-adic mass calculations and very large compared to them could relate and what the interpretation of the small primes could be. In the sequel I will discuss this aspect rather than UWVs of WVs.

The polynomials (P1,P2) and also the rational functions (g1=P1/Q1,g2=P2/Q2) appearing the holography= holomorphy vision (see this) form a well-defined complexity hierarchy.

  1. In the general case, the space-time surfaces (P1,P2)=(0,0) have several disjoint components. This is the case if (f1,f2) is a composite function of form f=g(h): in other words one has (f1,f2)= (g1(h1,h2),g2(h1,h2)). The space-time surfaces correspond to roots hi=ri, which are disjoint.

To avoid disjoint union of space-time surfaces fi must be a prime polynomial with respect to functional composition. For the polynomials of a single variable, this is the case if the degree of the polynomial is prime but this is not a necessary condition for primeness. As already found, this condition generalizes to the polynomials of 3 complex variables considered in the recent case.

Space-time surfaces of these kinds are excellent candidates for fundamental objects and the polynomial in question would have prime degree with respect to each of the 3 complex coordinates of H: this would make 3, presumably small primes. The composites formed of maps g and of these fundamental function pairs f would define cognitive representations of the surface defined by f as kind of statements about statements. An interesting question is whether these surfaces could correspond to elementary particles.

  • There is also a natural measure of complexity as the number of maps g, which correspond to prime polynomials with gi=Pi/Qi appearing in the functional composite with a pair of prime polynomials (f1,f2). Here the prime polynomials Pi must have degree higher than 1 in order to increase the complexity. In this case, 2 primes would characterize the prime polynomial Pi. 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. 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 and tentatively identified as ramified primes (see this) appearing as divisors of the discriminant of a polynomials define as the product of root differences, which could correspond to that for g=(g1,Id).
    2. p-Adic length scale hypothesis states that the physically preferred p-adic primes correspond to powers p∼ 2k. Also powers p∼ qk of other small primes q can be considered cite{allb{biopadc and there is empirical evidence of time scales coming as powers of q=3 (see this and this). For Mersenne primes Mn= 2n-1, n is prime and this inspires the question whether k could be prime quite generally. The proposal has been that the p and k would correspond to a very large and small p-adic length scale. Could the 3 primes characterizing the prime polynomials fi correspond to the small primes q and could the ramified primes p∼ 2k be associated with the polynomials obtained to theire iterated functional composites?

    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∼ 2k 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(Q) Q → P(P(…P..)(Q) and give n=2k 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(P(…P..)(P).
  • If this is the case, p∼ 2k 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 2k disjoint space-time surfaces correspond to a representation of p∼ 2k 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∼ qk, 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. A highly interesting observation is that the numbers allowing expansions in powers of an integer n having powers of primes belonging to some set can be regarded as p-adic integers for all these primes. One might say that these numbers belong to an intersection of these number fields. This could allow gluing of p-adic factors of adeles to single continuous structure. This suggests the possibility of multi-p p-adicity. The discriminant D of a polynomial defined as root differences can be expressed as a product of powers of so called ramified primes and the question is which of them is physically selected and why. Could it be that the expansions of physical quantities are in powers of D. I have also proposed that D, or its suitable power, is the number theoretical counterpart for the exponent of Kähler function as vacuum functional.
  • See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter About Langlands correspondence in the TGD framework.

    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/2025/03/holography-holomorphy-hypothesis-small.html


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