The Exponential Pairing and its Logarithmic Tower: A Unified Theory of Nilpotent Jets, the Geometry of ln(0), and the Fine Structure of ζ Zeros

We show that two independently proposed frameworks—the extension of C by a nilpotent direction b²=0 to capture the singularity of ln(0), and the asymptotic boundary study of the limit limu→ 0⁺(ln u + 1/u)ᵘ—are two canonical specializations of a single underlying object: the exponential pairing E(u, s) = uˢ = es ln u. The first specialization corresponds to its transverse jet (s nilpotent), giving rise to the unique rigid algebra Lₑ = C[b]/(b²); the second corresponds to its diagonal (s = -λ u), forming an analytic family governed by logarithmic geometry. We establish an exact six-identity dictionary relating the algebraic jet tower to the analytic shadow of ln(0), unified under a master law governed by the Γ-step symbol Γ(θ+1)/Γ(θ+1-n). Under this framework: Logarithmic Geometry: The singularity of ln(0) is identified as the generator of the characteristic monoid of a standard log point, where its gauge group C^× coincides with Aut(Lₑ). The surviving singularity is shown to be precisely the discrepancy between the ordinary derivative ∂ and the logarithmic Euler derivative θ = u∂ᵤ. Boundary Asymptotics: We prove a generalized boundary theorem yielding closed-form fractional exponents α = 1/2 + h(u^*) and explicit Γ-ratio amplitudes, validated across five parametric integral families. Zeta Zeros and Curvature: Applying the jet functor to the Riemann zeta function ζ(s) endows its non-trivial zeros with nilpotent fiber structures. We provide a closed-form Stirling series for the archimedean curvature floor Re ηₛₘₒₒₜₕ(ρ) verified to 10⁻¹². By correcting the tail truncation cut to mid-zero offsets (N(t^*) = K), we reduce the mean-field residual of the zeros' curvature from 6.7 × 10⁻³ to 6.8 × 10⁻¹⁰, identifying the remaining discrepancy as a explicit weighted tail sum of the fluctuation function $S(t)$.