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 ΔK and regulator RK of number fields K, 2. "Quantum information" — through the von Neumann entropy SvN and the Bekenstein bound, 3. "Analytic holomorphy" — via the logarithmic derivative of Artin L-functions and their information-theoretic friction ΦUFI(ρ). We verify numerically that: - ΦUFI(K) = kB ln|ΔK| — the informational friction of a number field — - RK — the regulator — controls the unit group and entropy, - ΦUFI(ρ) — 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 Q(√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".