Considerable progress in the understanding of holography= holomorphy vision

The surprisingly successful p-adic mass calculations led to the hypothesis that elementary particles and also more general systems are characterized by p-adic primes which assign to these systems a p-adic length scale. The origin of the p-adic primes remained the problem.

The original hypothesis was that p-adic primes correspond to ramified primes appearing as divisors of the discriminant of a polynomial defined as the product of root differences. Assuming holography= holomorphy vision, the identification of the polynomial of a single variable in question is not trivial but is possible. The p-adic length scale hypothesis was that iterates of a suitable second-degree polynomial P₂ could produce ramified primes close to powers of two. Tuomas Sorakivi helped with a large language model assisted calculation to study this hypothesis for the iterates of the chosen polynomial P₂= x(x-1) did not support this hypothesis and I became skeptical.

This inspired the question whether the p-adic prime p correspond to a functional prime that is a polynomial P_p of degree p, which is therefore a prime in the sense that it cannot be written as functional composite of lower-degree polynomials. The concept of a prime would become much more general but these polynomials could be mapped to ordinary primes and this is in spirit with the notion of morphism in category theory.

This led to a burst of several ideas allowing to unify loosely related ideas of holography=holomorphy vision.

1. Functional primes and connection to quantum measurement theory

Could functional p-adic numbers correspond to "sums" of powers of the initial polynomial P_p multiplied by polynomials Q of lower degree than p. This is possible, but it must be assumed that the usual product is replaced by the function composition _º and the usual sum by the product of polynomials. In the sum operation g=(g₁,g₂) and h=(h₁,h₂) the analytic functions g_i: C²→ C² and h_i are multiplied and in the physically interesting special case the product reduces to the product of g₁ and h₁.

The non-commutativity for _º is a problem. In functional composition f→ g_º f the effect of g is analogous to the effect of an operator on quantum state in quantum mechanics and functions are like quantum mechanical observables represented as operators. In quantum mechanics, only mutually commuting observables can be measured simultaneously. The equivalent of this would be that when P_p is fixed, only the coefficients Q (lower degree polynomials) to powers of P_p are such that Q_º P_p= P_pº Q and also the Qs commute with respect to _º. One can talk about quantum padic numbers or functional p-adic numbers.

p-adic primes correspond to functional primes that can be described by ordinary primes: this is easy to understand if you think in category theoretical terms. All prime polynomials of degree p correspond to the same ordinary prime p. One can talk about universality. Number-theoretic physics, just like topological field theory, is the same for all surfaces that a polynomial of degree p corresponds to. Electrons, characterized by Mersenne prime p= M₁₂₇= 2¹²⁷-1, would correspond to an extremely large number of space-time surfaces as far as p-adic mass calculations are considered.

3. The arithmetic of functional polynomials is not conventional

Funtional polynomials are polynomials of polynomials. This notion emerges also in the construction of infinite primes. Their roots are not algebraic numbers but algebraic functions as inverses of polynomials. They can be represented in terms of their roots which are space-time surfaces. In TGD, all numbers can be represented as spacetime surfaces. Mathematical thought bubbles are, at the basic level, spacetime surfaces (actually 4-D soap bubbles as minimal surfaces!;-).

For functional polynomials product and division are replaced with _º. + and - operations are replaced with product and division of polynomials. Also rational functions R= P/Q must be allowed and this leads to the generalization of complex analysis from dimension D=2 to dimension D=4. This is an old dream that was now realized in a precise sense.

1. This leads to an explicit formula for the functional analogs of Mersenne primes and more generally for primes close to powers of two, and even more generally for primes near powers of small primes. The functional Mersenne prime is P₂⁽_º n)/P₁ and any P₂ will do!
2. The non-conventional arithmetic of functional polynomials makes it possible to understand the p-adic length-scale hypothesis. The same p-adic prime p corresponds to all polynomials P_p of degree p. p-Adic primes are universal and depend very little on the space-time surfaces associated with them: this is very important concerning p-adic mass calculations. The problem with the ramified prime option was that they depend strongly on the space-time surface determined as root of (f₁,f₂): the effect of (g₁,Id) giving (g_1º f₁,f₂) does not have particle mass at all.

4. Also inverse functions of polynomials are needed

The inverse element with respect to _º corresponds to the inverse function of the polynomial, which is an n-valued algebraic function for an n-degree polynomial. They must also be allowed. Operating the polynomial g₁ on f increases the degree and complexity. Operating with the inverse function preserves the number of roots or even reduces it if g₁ operates on g₁ iterated. The complexity can decrease. Complexity can be considered as a kind of universal IQ and evolution would correspond to the increase in complexity in statistical sense. Inverse polynomials can reduce it by dismantling algebraic structures.

In TGD inspired theory of consciousness I have associated ethics with the number theoretic evolution as increase of algebraic complexity. A good deed increases potential conscious information, i.e. algebraic complexity, and this is indeed what happens in a statistical sense. Could conscious and intentional evil deeds correspond to these inverse operations? Evil deeds would make good deeds undone. If so, it is easy to see that negentropy still increases in a statistical sense. This however would mean that an evil deed can be regarded as a genuine choice.

5. How quantum criticality, classical non-determinism and p-adic nondeterminism are related to each other

1. The simplest representation of criticality is by means of a monomial xⁿ. It has n identical roots at x=0 and extremely small perturbation can transform them to separate roots. Mathematicians consider them as separate, as if there were n copies of the root x=0 on top of each other. g₁ =f₁ⁿ as the equivalent of this gives n identical space-time surfaces as roots on top of each other. Are they the same surface or separate? A mathematician would say that they are separate. If the polynomial is slightly perturbed, there are n separate roots. This would be the classical equivalent of quantum criticality.

   In quantum criticality, the functional polynomials would have g₁= f₁ⁿ at quantum criticality. The corresponding spacetime surface would be susceptible to breaking up into separate spacetime surfaces when the monomial f₁ⁿ becomes a more general polynomial and n roots are obtained as separate spacetime surfaces.

   There is a fascinating connection with cell replication. In TGD it would be controlled by the field body and one can ask whether f₁²=P₂² as a critical polynomial representing the field body is perturbed and leads to two field bodies which become controllers of separate cells. One can ask whether in a cell replication sequence P₂²ⁿ becomes less critical step by step so that eventually there are 2ⁿ separate field bodies and cells.

   In zero energy ontology (ZEO) one can also ask whether the creation of a critical space-time surface characterized by f₁ⁿ could give rise to n space-time surfaces when criticality is lost. Zero energy ontology understood in the Eastern sense would allow this without conflict with conservation laws.

2. Mother Nature likes her theorists If the critical surface is considered as a single surface, the classical action associated with it is n-fold compared to the surface corresponding to one root. This means that the Kähler coupling strength alpha_K is smaller by a factor of 1/n after the splitting. This was the basic idea in the hypothesis that I formulated by saying that Mother Nature likes theorists.

   When the perturbation theory ceases to converge (a catastrophe for the theorist), criticality arises, the polynomial takes the form P_p= f₁^p. Deformation and splitting of the surface into a p discrete surface follows, the coupling strength decreases by a factor of 1/p and the perturbation theory converges again. The theorist is happy again.

3. Classical non-determinism corresponds to p-adic non-determinism Criticality is associated with non-determinism. In classical time evolution, mild non-determinism corresponds to such a criticality. In these phase transitions, a choice is made between p alternatives in the "small" state function reduction (SSFR). The essential thing is that this series of phase transitions can be realized as a classical time evolution. Without criticality, this would not be possible.

   The fact that a choice is made between p alternatives corresponds to the fact that the dynamics is effectively p-adic. So that classical non-determinism corresponds to p-adic non-determinism.

4. Connection to the p-adic length-scale hypothesis What is particularly interesting is that if p= 2 or 3 then the roots of the polynomial P_p can be solved analytically. The same applies to the iterates of P_p. Therefore these cases are cognitively special as every mathematician knows from her own experience! The p-adic length-scale hypothesis says that p-adic primes p are close to powers of small prime q=2,3,.... Intriguingly, there is empirical evidence for the hypothesis in the cases q=2 and 3!

   In the cusp catastrophe, which is the Mother of all catastrophes, q=2 and q=3 occur. The cusp is V-shaped. At the tip of V, the polynomial determining the cusp takes the form x³, i.e. 3 roots converge and on the sides of V, the polynomial two roots converge to a third-degree polynomial.

   In the cusp catastrophe, the Mother of all catastrophes, which is 2-dimensional surface in the space (x,a,b) defined by the real roots x_i, i=1,2,3 of P₃(x,a,b), q=2 and q=3 occur. The projection of the cusp to the (a,b) plane is V-shaped. At the tip of V, the polynomial P₃(x,a,b) determining the cusp is proportional to x³, i.e. 3 roots concide. At the two folds, whose projections to the (a,b) plane define the sides of V,  two roots conincide.

   It seems that finally the basic ideas of TGD have found each other and form a coherent whole. I managed also to clarify the relationship of M⁸-H duality to the holography=holomorphism hypothesis.

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