On Kinematics: Celestial Conformal Blocks at the Boundary of the Principal Series

The companion paper—On Self-Dual Measure and Finite Quantum Gravity—on the Plancherel loop measure established a structural factorization of celestial loop amplitudes into a spectral factor, governed by the measure P(λ) = πλ/sinh(πλ) on the principal series, and a kinematic factor, the conformal block evaluated on the unit circle of cross-ratio. This paper develops the kinematic factor. Four exact structures are established. First, the chiral block k_h(z) = z^h ₂F₁(h,h;2h;z) on the principal series h = ½(1+iλ) reduces exactly to a conical Legendre function, k_h(z) = [2Γ(2h)/Γ(h)²] Q_{h−1}(2/z−1). Second, the shadow involution is the classical Legendre degree symmetry ν ↦ −ν−1, and the connection formula Q_{−ν−1} − Q_ν = iπ tanh(πτ) P_ν exhibits the shadow-symmetric part of the block as the Mehler–Fock kernel and the shadow-antisymmetric part as the carrier of the Mehler–Fock spectral density. Third, the Temperedness Theorem: on physical kinematics z̄ = z*, |z| = 1, the exponential growth rates of the two chiral factors are conjugate-odd and cancel identically, and |g_{1+iλ}(z, z̄)| = |1−z|^{−1/2} (1 + O(λ^{−1})) uniformly on compact subsets of the circle away from z = 1; the shadow spectral integrand therefore decays at the full Plancherel rate e^{−πλ}. Fourth, under the dictionary Δ = 2s the spectral weight is P(λ) = |Γ(2s)|² on the critical line, the logarithmic singularity of the block at z = 1 carries the coefficient ψ(s) evaluated on the critical line, the logarithmic derivative of the archimedean factor at the complex place, and the first moment of this transcendentality engine is evaluated in closed form: (1/2π) ∫ P(λ) Re ψ(½ + iλ/2) dλ = 1/8 + ¼ψ(½), the one-loop measure plus a quarter of the digamma at the shadow fixed point. A Mellin bridge is proved realizing (1−2^{−s})Γ(s)ζ(s) as the transform of the spectral kernel (2 sinh t)^{−1}, with the vanishing of its analytic continuation at the first nontrivial zeros verified to twenty-nine digits. The program for the closed-form one-loop kinematic factor is formulated with these tools and its target identified. Every exact claim is verified in a computational companion; the status of each statement is labeled.