The Grand Horizon Extended: Spectral Invariants in Prime Gaps from Dirichlet and Maass L-Functions

The Grand Horizon Extended explores a universal spectral geometry underlying the distribution of prime numbers. Building on the original Grand Horizon Conjecture, this work investigates how prime gaps filtered by Dirichlet characters and Maass-like spectral analogues reveal deep correlations with the zero distributions of $L$-functions. By introducing the spectral exponent $\beta$—an empirical quantity governing the growth rate of filtered prime gaps—the project uncovers a nearly linear inverse relationship between $\beta$ and the spacing of low-lying zeros of Dirichlet $L$-functions. Through large-scale simulations, principal component analysis, and Fourier methods, the study shows that filtered prime sequences exhibit geometric clustering and harmonic regularity. The $\beta$ exponent is proposed as a spectral invariant, capable of predicting analytic properties of automorphic $L$-functions without direct computation of zeros. This work connects number theory, spectral analysis, and quantum chaos, offering new tools and perspectives on the Generalized Riemann Hypothesis. All code and figures are included to ensure reproducibility and invite further exploration.