Symbolic synthesis of Euler’s Number and the Golden Ratio

The purpose of this paper is to investigate the structures of the expressions formed by the symbolic regression carried out by Genetic Algorithms (GA) when the objective is either to approximate or derive important mathematical constants ($e$ and $\Phi$). Not restricted by any human-designed heuristic such as the classic appreciation for symmetry and elegance in formulas, the algorithm creates unexpected functional expressions. The issues of asymptotic convergence, removable singularities and algebraic symmetries of the continued fractions are examined.

1. Asymptotic optimization for the value of Euler’s Constant ($e$)

In the first stage of computing, the evolutionary algorithm was limited to searching in an area that comprised the rational function raised to a linear power. The objective function focused on minimizing the local residual error at the same discrete evaluation point ($n = 50$).

The result from iteration 1499 can be represented by the following functional expression:

$\lim_{n \to \infty} f(n) = \lim_{n \to \infty} \left( 1 + \frac{2n - 4}{-8n^2 + 19n - 6} \right)^{-4n + 2}$

1.1 Algebraic factorization

However, upon close scrutiny of the rational form, one notices that the function was purposely defined to have a removable discontinuity at $n = 2$. This can be proven through the factorization of both the numerator $P(n)$ and the denominator $Q(n)$:

• $P(n) = 2(n - 2)$
• $Q(n) = -(n - 2)(8n - 3)$

For values of $n$ that do not equal 2, the rational function reduces to:

$1 + \frac{2(n - 2)}{-(n - 2)(8n - 3)} = 1 - \frac{2}{8n - 3} = \frac{8n - 5}{8n - 3}$

1.2 Transformation of scale and rate of convergence

Switching the sign of the power results in a much more elegant form:

$f(n) = \left( \frac{8n - 3}{8n - 5} \right)^{4n - 2}$

The transformation of scale necessary to describe the asymptotic line is carried out through an appropriate change of variables, namely $k = 4n - 2$ which entails that $8n = 2k + 4$. Upon substituting this into the argument we get:

$f(k) = \left( \frac{2k + 1}{2k - 1} \right)^k$

This expression coincides with the well-known symmetric approximation in the limit case of $\lim_{k \to \infty} f(k) = e$ proposed by d’Alembert.

Analytically speaking, the high accuracy at $n=50$ (error $\approx 5.78 \cdot 10^{-6}$) does not come from any transformation performed on the physical limit case. Rather, the GA devised a scale compression: by setting the coefficients of the GA so that $n=50$ would be mapped to an effective index of $k=198$, the convergence on the implied Taylor expansion would be faster, dampening the error to $O(1/k^2)$.

2. Algebraic invariance of the Golden Ratio ($\Phi$) via Continued Fractions

For the second experiment, the algorithm worked in the field of generalized continued fractions geometry where optimization was performed according to convergence to the irrational constant $\Phi = 1.6180339887...$

The optimal genetic vectors converged into the repetitive and recursive form defined by the following parameters a=1, b=7, c=7, d=7:

$x = 1 + \cfrac{7}{7 + \cfrac{7}{1 + \cfrac{7}{7 + \cfrac{7}{1 + \dots}}}}$

2.1 Verification of fixed-point convergence

For the analytical determination of the accumulation point for this evolved sequence, we take advantage of the self-similarity of an infinite continued fraction. After compression of the structure into a single formula of a fixed point, we obtain:

$x = 1 + \frac{7}{7 + \frac{7}{x}}$

The following is the analytical solution of the relationship:

1. Denominator normalization: $7 + \frac{7}{x} = \frac{7x + 7}{x}$
2. Substitution and inverse of the main fraction: $x = 1 + \frac{7x}{7x + 7}$
3. Scalar (arbitrary 7) factor and cancellation: $x = 1 + \frac{7x}{7(x + 1)} \implies x = 1 + \frac{x}{x + 1}$
4. Multiplication on the lowest common denominator $(x+1)$: $x(x + 1) = x + 1 + x \implies x^2 + x = 2x + 1$

The final quadratic relationship is:

$x^2 - x - 1 = 0$

The positive root of this polynomial is precisely the exact analytical definition of the targeted constant:

$x = \frac{1 + \sqrt{5}}{2} = \Phi$

2.2 Structural Redundancy and Non-Human Heuristics

The choice of coefficient $7$ highlights the non-human nature of the mathematics involved here. Whereas a human operator would choose to select the identity element—the result being the well-known continued fraction of Fibonacci where all the coefficients have the value $1$—the genetic algorithm moved into the infinitely many algebraically equivalent classes of this object.

By applying an equivalence transformation to the continued fraction, any set of coefficients having equal values at those particular homographic locations would be subject to coefficient scaling cancellation. The machine did indeed choose the coefficient $7$ not because of top-down deductive reasoning but simply as part of a random process. The algebraic stability of the quadratic invariant does not depend in the slightest on coefficient parsimony.

The application of genetic algorithms in symbolic regression shows how the approaches differ when it comes to deductive formal beauty and inductive heuristic efficiency. The examples show that evolutionary techniques do not create new theorems but find new paths through a solution space by using redundant paths. Since these methods optimize only the numeric properties of the equations, making sure the inner geometry leads to convergence, they prove extremely efficient in discovering mathematical relations.