Constant Relations in Transcendental Power Iteration Systems: Unifying Fixed-π, Fixed-2, Fixed-f, and Fixed-e Perspectives

This paper investigates the intrinsic relations among different constant-fixed perspectives in the transcendental power iteration system u_{n+1} = u_n^{1/s} e^{-π u_n / s^2}. Starting from the fundamental equation ln f + (π/2) f = 0, we systematically analyze four parameterization perspectives: 1. Fixed-π perspective: ln L_s + π/(s(s-1)) L_s = 0, with varying parameter s 2. Fixed-2 perspective: ln M_t + (π_t/2) M_t = 0, with varying parameter π_t 3. Fixed-f perspective: ln N_r + (π_r/r) N_r = 0, with parameters r and π_r 4. Fixed-e perspective: ln N_q + (π_q/q) N_q = 0, with parameters q and π_q (where e is implicit in the logarithm) We prove that these four perspectives are related by: π_t = 2π/(s(s-1)), π_r/r = π/(s(s-1)), π_q/q = π/(s(s-1)) In particular, we reveal a profound relation: e^{-πz/2} = f^{z/f} which shows that the combination of e and π can be expressed entirely using the single constant f. Through the transformation u = e^{-πz / s}, we define U-space and Z-space, which are connected by this transformation and satisfy the duality relation u · v = 1, where v = 1/u defines V-space. These space names remain neutral, without presupposing interpretations such as "path" or "observation," to accommodate different perspectives. Further discussion reveals: · The nature of fixed points: A point that remains invariant under the system's transformation, serving as a reference frame for observing the world from different perspectives. · The nature of duality: u · v = 1 is an absolute relation independent of any constants. · The relativity of perspectives: Choosing π, 2, f, or e as the fixed quantity is merely a different way of observing the world, but the core relation u · v = 1 remains unchanged. In nested structures at different scales (micro/macro), the roles of U-space and Z-space can interchange, further reflecting this relativity. Numerical verification confirms that all four perspectives yield identical fixed points. These results support the "constant relativity" view: mathematical constants are not absolutely fundamental but are mutually defined through a network of relations. In dynamical systems, which constant is chosen as the "fixed quantity" is merely a matter of perspective and does not affect the essence of the system.