The results of a project related to the holography= holomorphy vision

Tuomas Sorakivi carried out together with me a large language model (LLM) assisted experimentation with the graphical representation of the space-time surfaces satisfying holography= holomorphy hypothesis. Also the hypothesis that ramified primes associated with the iterates of certain second order polynomial could be primes near powers of two is studied numerically using LLM assistance.

1. A summary of the computations carried out and theory behind them is given in the article The results of a project related to the holography= holomorphy vision. A brief summary of the TGD based model can be found here.

2. The equations for the space-time surface associated with a third order polynomial P_3(\xi₁,w,u) can be solved analytically. In 2-dimensional algebraic geometry, these surfaces correspond to elliptic surfaces in 2-dimensional complex space with ξ₁ and w as coordinates. Space-time surface can be represented as an animation for the time evolution of an elliptic surface. A graphical representation is constructed (see this).

3. The roots of the polynomial defining space-time surface have as interfaces 3-dimensional surfaces at which two roots are identical. For the illustration of the interfaces see this.

4. Weierstrass elliptic functions (see this) are particular elliptic functions being meromorphic and doubly periodic. For a visualization in this case see this.

5. The ramified primes are divisors of the discriminant D of a polynomial defined as the product of its non-vanishing root differences. Could some ramified primes satisfy the p-adic length scale hypothesis for some iterates of a suitable choice of degree 2. This would mean that the primes near powers of 2 appear as ramified primes. As an example the discriminant D and ramified primes for the iterates of g₁(z)= z(z-ε), were studied. Of course, there exists an endless variety of these kinds of polynomials but one might hope that the chosen polynomial might be special because of its 2-adic features. The study was carried out with Tuomas Sorakivi (see this). The conclusion was negative and led to the notion of the functional p-adic numbers.