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

We show that two independently proposed frameworks—the extension of $\mathbb{C}$ by a nilpotent direction $b^2=0$ to capture the singularity of $\ln(0)$, and the asymptotic boundary study of the limit $\lim_{u\to 0^+}(\ln u + 1/u)^u$—are two canonical specializations of a single underlying object: the exponential pairing $E(u, s) = u^s = e^{s \ln u}$. The first specialization corresponds to its transverse jet ($s$ nilpotent), giving rise to the unique rigid algebra $\mathcal{L}_e = \mathbb{C}[b]/(b^2)$; the second corresponds to its diagonal ($s = -\lambda u$), 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 $\Gamma$-step symbol $\Gamma(\theta+1)/\Gamma(\theta+1-n)$. 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 $\mathbb{C}^\times$ coincides with $\text{Aut}(\mathcal{L}_e)$. The surviving singularity is shown to be precisely the discrepancy between the ordinary derivative $\partial$ and the logarithmic Euler derivative $\theta = u\partial_u$. Boundary Asymptotics: We prove a generalized boundary theorem yielding closed-form fractional exponents $\alpha = \frac{1}{2} + h(u^*)$ and explicit $\Gamma$-ratio amplitudes, validated across five parametric integral families. Zeta Zeros and Curvature: Applying the jet functor to the Riemann zeta function $\zeta(s)$ endows its non-trivial zeros with nilpotent fiber structures. We provide a closed-form Stirling series for the archimedean curvature floor $\text{Re }\eta_{\text{smooth}}(\rho)$ 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 $6.7 \times 10^{-3}$ to $6.8 \times 10^{-10}$, identifying the remaining discrepancy as a explicit weighted tail sum of the fluctuation function $S(t)$.