Beyond the Spectrum: β-Invariants, Entropy, and the Prime Gap Geometry of L-Functions

This project presents a new spectral invariant—the β-exponent—governing maximal prime gaps in filtered sequences derived from Dirichlet characters, modular forms, and Maass-like $L$-functions. We show that these prime gaps obey a universal power-law of the form: g_n^{(F)} < A_F n^{\beta_F} where the β parameter depends linearly on the logarithm of the associated conductor or level: $\beta_F \approx C \log Q_F + D$. This work unifies deep concepts in: Analytic number theory Thermodynamic entropy (via partition functions) Random matrix analogies of $L$-functions The β-spectrum acts as a diagnostic fingerprint for each $L$-function family, with potential applications to the Riemann Hypothesis, spectral classification, and signal analysis in prime distributions. All source code, raw data, simulation notebooks, and LaTeX files are included for reproducibility under CC BY-NC 4.0.