Spectral Projection of the First Riemann Zero as a Low-Dimensional Approximation of Prime-Counting Deviations and Cross-Domain Response in Symbolic

Abstract (Plain Text Version) Title: Spectral Projection of the First Riemann Zero as a Low-Dimensional Approximation of Prime-Counting Deviations and Cross-Domain Response in Symbolic Sequences Abstract: This study investigates a low-dimensional spectral projection operator based on the frequency component associated with the first non-trivial zero of the Riemann zeta function (gamma₁ approx 14. 1347). The analytical framework utilizes a double-logarithmic argument transformation, u = ln (ln x), to linearize the phase structure of the oscillatory components within the prime-counting deviation function Delta (x) = pi (x) - Li (x). The methodology is grounded in rigorous computational statistics, comprising: Statistical Verification: The observed phase stability metric S (x) is evaluated against structure-preserving stochastic surrogate ensembles (null models) to differentiate robust spectral features from spurious correlations. Cross-Domain Analysis: The fixed spectral projection operator is applied without modification to symbolic biological sequences (specifically at genomic locus 13q14. 11) to quantify the statistical response density. Significance Control: To minimize Type I errors, the analysis employs Benjamini-Hochberg False Discovery Rate (FDR) corrections and large-scale permutation testing (N = 10, 000). The results indicate that the observed spectral response at gamma₁ significantly exceeds the 99. 5th percentile of the surrogate distributions under the defined statistical framework. This research remains strictly within the domain of computational phenomenology and does not assert causal linkage or structural equivalence between arithmetic and biological systems. The primary focus is the characterization of spectral projection behavior in high-dimensional symbolic data. Keywords: Riemann Zeta Function, Spectral Projection, Prime-Counting Function, Genomic Signal Processing, Surrogate Data Testing, Phase Stability. All source code for reproducibility, including the spectral projection scripts, surrogate ensemble generators, and genomic data processing routines, is hosted on GitHub: https: //github. com/dancegolga/fixed-spectral-projection-riemann. The repository includes environment specifications to ensure consistent execution of the analysis pipeline.