The Three Bridges of Coherence: A Unified Theory Connecting Algebraic Structure, Quantum Information, and Analytic Holomorphy

We present a unified framework connecting three fundamental pillars of modern mathematics and physics:  

1. "Algebraic structure" — via the discriminant $\Delta_K$ and regulator $R_K$ of number fields $K$,  

2. "Quantum information" — through the von Neumann entropy $S_{\text{vN}}$ and the Bekenstein bound,  

3. "Analytic holomorphy" — via the logarithmic derivative of Artin $L$-functions and their information-theoretic friction $\Phi_{\text{UFI}}(\rho)$.  

We verify numerically that:  

- $\Phi_{\text{UFI}}(K) = k_B \ln|\Delta_K|$ — the informational friction of a number field —  

- $R_K$ — the regulator — controls the unit group and entropy,  

- $\Phi_{\text{UFI}}(\rho)$ — the $L$-function friction — is consistent with the Bekenstein bound.  

Using "pure Python" (no SageMath, no external libraries), we compute these quantities for quadratic fields $\mathbb{Q}(\sqrt{d})$, and simulate the $L$-function friction for the trivial representation. The results confirm a deep coherence across algebra, quantum, and analysis — hence the "Three Bridges of Coherence".