The Correspondence Expression of the Critical Line Problem and the Relativity Principle of Parameters: Based on Global Discrete Dynamics

This paper establishes an exact correspondence between the non-trivial zeros of the Riemann zeta function and a class of nonlinear discrete dynamical systems. The core iteration system is:

u_{n+1} = u_n^{1/s} e^{-π u_n / s^2}, s ∈ C \ {0,1}

For almost all initial values, this system converges to a unique fixed point u_s, satisfying the balance equation:

ln u_s / u_s = -π / (s(s-1)) (1)

The closed-form solution for u_s in terms of s is:

u_s = s(s-1)/π · W₀( π/(s(s-1)) ) (2)

where W₀ is the principal branch of the Lambert W function. From equation (1), we derive the expression for s in terms of u_s:

s = (1 ± √(1 - 4π u_s/ln u_s)) / 2 (3)

We introduce a triple-space structure:

· U-space: u_{n+1} = u_n^{1/s} e^{-π u_n / s^2}, fixed point u_s

· Z-space: z_{n+1} = (z_n + e^{-π z_n / s})/s, fixed point z_s = s/π · W₀(π/(s(s-1)))

· V-space: v_{n+1} = v_n^{1/s} e^{π/(s^2 v_n)}, fixed point v_s = 1/u_s

These three spaces satisfy the duality relations:

u_s · v_s = 1, u_s = e^{-π z_s / s}, v_s = e^{π z_s / s}.

A crucial property of U-space is established: for any finite s ∈ C\{0,1}, u_s can never equal 0 or 1; these values are only approached as limits (e.g., u_s → 1 as s → ∞ on the real line). By analyzing the limiting behavior u → 1 in equation (3), we directly derive:

u → 1 ⇒ Re(s) = 1/2.

This result follows solely from the system's structure without any prior assumptions about the Riemann zeta function.

The paper further discusses the universality of equation (1) as a fundamental self-referential equation for exponential-logarithmic structures, the relativity of parameters within the triple-space framework, and the use of Newton's method to find s for fixed parameters. Classical functions such as the sine, gamma, and Riemann ξ functions are shown to be consistent with this framework upon substitution.