Structural Entropy Balance in Euler–Riemann Systems: A Ratio Spectrum Approach

This paper explores the Euler–Riemann entropy orbit theory through a multi- orbit spectrum, where the ratio of Riemann-type spectral entropy Hσ to Euler- type structural entropy HK is analyzed. Inspired by Euler’s foundational work on the gamma function and prime distributions, we identify a balance orbit C1 where log |Hσ/HK| ≈ 0, approximating a near equilibrium regime. Numerical results across nine orbits, including C1/2, Ce, and Cπ, demonstrate the stability of this balance, with errors as low as 10−6. The findings suggest a universal coupling between prime structures and zeta zero distributions, with implications for the Riemann Hypothesis and information dynamics.