OPEN_SCI_EVAL

02_SIMULATION

Numerical verification of the hypothesis: Convergence, Lyapunov Exponents, and Band Merging.

LIMIT BEHAVIOR

Parameter Convergence

We estimated the Logistic Map parameter uu corresponding to the cumulative sieve sequence DiD_i. The results show rapid convergence to a value near the band-merging point.

u(D_i) EVOLUTION
123451015202530Sieve Stage (i)1.31.3751.451.5251.6Target u ≈ 1.5437
TARGET VALUE
1.5437
Band Merging Point
CONVERGED VALUE
~1.5440
From Sieve Sequence

The parameter uu saturates quickly after just a few prime iterations, strongly suggesting a stable limit behavior.

DYNAMICAL SIGNATURES

Chaos & Lyapunov Exponents

1.21.31.41.451.51.54371.61.71.81.92Parameter u-0.7-0.3500.350.7λBand Merging

Theoretical Match

At u≈1.5437u \approx 1.5437, the theoretical Lyapunov exponent is λ≈0.3406\lambda \approx 0.3406. Our simulation yielded λ≈0.3420\lambda \approx 0.3420, a match with only 0.4% error.

The Discrepancy

However, calculating the Lyapunov exponent directly from the prime gap sequence yields a negative value (λ≈−0.23\lambda \approx -0.23). This indicates that while the macroscopic parameters match, the microscopic dynamics of primes are more "ordered" than pure chaos.