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 \(z=\exp(-(s-\tfrac12))\) sends the critical line to the unit circle and conjugates functional-equation reflection to reciprocal inversion, but it is a universal covering rather than a global conformal embedding. We therefore work on the logarithmic cover and in sheet-preserving disk charts. Within this setting we establish exact argument-principle, Jensen-defect, Hardy-support, and certified finite-region criteria, while recording the obstructions to descent, naive disk regularization, finite-mode coercivity, and unweighted real-line Paley–Wiener interpretations. The reciprocal axioms alone have the sharp critical-line proportion output \(\kappa_{\mathrm{intrinsic}}=0\); this is an optimization statement about the geometric constraint system, not a claim about the actual zeros of \(\zeta\). A one-piece Levinson functional is evaluated exactly and retained only as the benchmark candidate \(\kappa_{\mathrm{bench}}=0.3353651648\ldots\), because the complete arithmetic moment and uniform remainder are not proved here. The arithmetic emphasis is shifted to families of \(L\)-functions. For primitive characters of prime conductor we give the completed Dirichlet-\(L\) normalization, exact primitive-character orthogonality, and the full scaled explicit formula, including conductor, gamma, trivial-zero, and prime-power terms. We recover the unconditional one-level-density benchmark for \(\operatorname{supp}\widehat\phi\subset(-\sigma,\sigma)\), \(\sigma<2\), with error \[ O_\phi\!\left((\log q)^{-1}+q^{\sigma/2-1}\right). \] Beyond this range we identify the centered prime-progression discrepancy, prove that black-box fixed-conductor Vaughan, additive-large-sieve, and absolute Bombieri–Vinogradov insertions do not enlarge the support, and derive an exact divisor-switched Möbius–Ramanujan kernel. Its affine local density is neutral, its zero mode is separated algebraically, and Poisson summation gives the normalized small-divisor bound \[ O\!\left(\frac{D}{\sqrt X}\frac{\log(2X)}{\log Q}\right). \] No new support theorem or family nonvanishing proportion is claimed; the remaining tasks are the small-divisor mean and the large-divisor dispersion blocks. **Keywords** Reciprocal geometry; \(L\)-functions; Dirichlet \(L\)-functions; Jensen defect; low-lying zeros; one-level density; explicit formula; primitive characters; Möbius–Ramanujan dispersion; Vaughan identity; large sieve; Bombieri–Vinogradov theorem; Poisson summation; Hardy spaces.