freese-math/riemann-spectral-proof: Beta Scale Construction from Zeta Zeros and Prime Logarithms

This release introduces a novel construction of the Beta scale, derived from the spectral data of a quantum-like operator involving zeta zeros and logarithmic primes. It constitutes a new analytical framework relevant to the Riemann Hypothesis and number theory. Abstract: We introduce a spectrally constructed Beta scale intimately tied to the nontrivial zeros of the Riemann zeta function. Based on a self-adjoint operator H \cdot e^{i\pi\beta}, whose spectrum captures both zeta zeros and prime logarithmic structure, we generate a novel sequence — the Beta scale — by solving an inverse spectral problem. The resulting values exhibit quasiperiodic structure and encode deep arithmetic information. This construction opens a new analytical pathway toward a spectral proof of the Riemann Hypothesis, providing a bridge between functional analysis, operator theory, and analytic number theory. The release contains core data, transformation tools, and high-resolution eigenvalue sequences supporting reproducibility and further mathematical exploration. Spectral Derivation of the Beta-Scale from the Riemann Zeta Function This release introduces the Beta-Scale — a novel, spectrally defined function β(n) — derived from the analytic and structural properties of the Riemann zeta function ζ(s). The central idea is to encode information about the zeta spectrum directly into the exponent of a modified Dirichlet series: \zeta_\beta(s) := \sum_{n=2}^\infty \frac{1}{n^{\beta(n) + is}} \beta(n) = B + C \cdot \frac{\log n}{\log \log n} + D \cdot \cos\left(2\pi f \cdot \log n + \varphi\right) This structure combines global scaling and fine spectral modulations. Key results: • Absolute convergence of the Zeta-Beta function for all ( s \in \mathbb{C} ) • Conjugate symmetry: ( \zeta_\beta(s) = \zeta_\beta(1 - s) ) • Different zeros from ζ(s), but robust oscillatory behavior • Fourier analysis reveals discrete structure in β(log n) • Relaxation interpretation via a modulated time scale This repository focuses on the derivation and numerical evaluation of β(n), the construction of the Zeta-Beta function, and its comparison with classical zeta structures. #BetaScale #RiemannZeros #SpectralProof #HOperator #NonHermitian #ComplexSpectrum #RiemannHypothesis   #ZetaSpectra