Reciprocal Geometry and Jensen-Defect Criteria for \(L\)-Functions: A Geometric Reformulation with Arithmetic Lower Bounds

Abstract We develop a reciprocal-geometric and arithmetic framework for completed \(L\)-functions. The exponential coordinate \[ z=\exp\!\left(-(s-\tfrac12)\right) \] maps the critical line to the unit circle and conjugates functional-equation reflection to reciprocal inversion. Because this map is a universal covering rather than a global conformal embedding, the analysis is carried out on the logarithmic cover and in sheet-preserving disk charts. Within this framework, we establish argument-principle, Jensen-defect, Hardy-support, and certified finite-region criteria, while identifying the obstructions to global descent, naive disk regularization, finite-mode coercivity, and unweighted real-line Paley–Wiener interpretations. The reciprocal-geometric axioms alone yield the sharp intrinsic critical-line proportion \[ \kappa_{\mathrm{intrinsic}}=0. \] This is an optimization result for the geometric constraint system, not a statement about the actual zeros of the Riemann zeta function. We also evaluate a one-piece Levinson functional, obtaining the benchmark candidate \[ \kappa_{\mathrm{bench}}=0.3353651648\ldots, \] without claiming a corresponding proportion theorem, since the complete arithmetic moment calculation and uniform remainder estimate are not established here. The arithmetic emphasis is subsequently shifted to families of Dirichlet \(L\)-functions. For primitive characters of prime conductor, we derive the completed normalization, exact primitive-character orthogonality, and the scaled explicit formula, including conductor, gamma-factor, trivial-zero, and prime-power contributions. We recover the unconditional one-level-density benchmark for \[ \operatorname{supp}\widehat{\phi}\subset(-\sigma,\sigma), \qquad \sigma<2, \] with error \[ O_{\phi}\!\left((\log q)^{-1}+q^{\sigma/2-1}\right). \] Beyond this range, we isolate the centered prime-progression discrepancy and show that black-box fixed-conductor Vaughan estimates, additive large-sieve bounds, and absolute Bombieri–Vinogradov estimates do not by themselves enlarge the admissible support. Divisor switching produces an exact Möbius–Ramanujan kernel whose zero mode is separated algebraically. Poisson summation then gives the normalized small-divisor estimate \[ O\!\left( \frac{D}{\sqrt X}\frac{\log(2X)}{\log Q} \right). \] No enlarged-support theorem or new family nonvanishing proportion is claimed; the remaining arithmetic tasks are the small-divisor mean and large-divisor dispersion estimates. Keywords **Riemann zeta function; \(L\)-functions; reciprocal geometry; logarithmic covering; critical line; Jensen formula; Jensen defect; argument principle; Hardy spaces; Levinson method; mollifiers; Dirichlet characters; explicit formula; one-level density; prime progressions; divisor switching; Möbius–Ramanujan kernel; Poisson summation.**