Extension of the Kernel to the Full Circle S^1 and Structure of the Universal Constant k_0^A_n

This note records the sealed results of Block R80 of the TCFQ project. Starting from the two-point kernel Γ_{∞}(α, β) established in R60–R66 and the spectral analysis of R69–R79, we extend the kernel to the full circle S^1, identify the correct integration domain as the Weyl triangle of A_2, compute the complete spectrum of the associated integral operator K on L^{2}(S^1), and derive the exact structural origin of the universal constant k_0 = 1 − 12/π^2. We then investigate two natural generalisations of the kernel to higher-rank root systems A_n: the simple-root-pairs kernel (Path B) and the all-positive-roots diagonal kernel (Path F). For Path B we prove the exact formula k_0^{An} = n(n − 1)/2 · n! /π^{n} J_{n−1}, where J_d is an explicitly computable moment of the reduced kernel ˜ Γ(c). For Path F we obtain k_0^{A_n,full} = n(n+1)/2 (1 − 4/π^2), which does not reproduce the TCFQ value for A_2 and is therefore discarded. The original conjecture k_0^{A_n} = 1 − 4|∆+(A_n)|/π^2 is not confirmed by either path.