Eisenstein lattice vectors, resolvent identities, and transfer-operator kernels for the Hecke triangle groups G4 and G6

We construct explicit Eisenstein lattice vectors for transfer operators of the Hecke triangle groups G4 and G6. For the G4 two-disc operator L_s, an exact cone tiling gives (I - L_s) Psi_s = D(s) 1, where D(s) = zeta(2s)(1 + 2^(-s)), and the anchor M(s) = zeta(2s - 1)(1 + 2^(1-s)) gives a scalar resolvent entry M(s)/D(s). At a nontrivial half-zero s_* = rho/2 of order m, the exact determinant order from the Selberg dictionary, an explicit finite-rank reduction and local Smith form show that the kernel is one-dimensional, spanned by the regular continued lattice vector, even under Markov reflection, and has a nonzero anchor. The finite reduction has one positive Smith exponent, equal to m. A bordered inverse remains holomorphic through s_*, yielding local residual bounds for the normalized vector and the scalar D/M. An explicit affine map also realizes the distinguished family for the q = 4 even fast operator on a specified space, with scalar entry 2M/D. For G6, four state-dependent sectors give the corresponding vector, source D6(s) = zeta(2s)(1 + 3^(-s)) and anchor M6(s) = zeta(2s - 1)(1 + 3^(1-s)). Direct nuclear-norm continuation gives the resolvent entry M6/D6 on {Re s > 0} excluding {1/2}; every half-zero of order m yields an operator pole of order at least m and a nonzero kernel, and with the exact determinant order for G6, obtained from the same dictionary, the pole order is exactly m and the kernel is one-dimensional, spanned by the continued lattice vector. Bottom-row counting identifies this entry, up to an explicit factor, with the scattering coefficient of Gamma_0(3)^+, with the cusp-width normalization explicit. The kernel classification is thus proved for both groups, in each case from the exact determinant order supplied by the Selberg dictionary. Parameter multiplicity remains distinct from the undetermined Jordan structure of a fixed operator. An independent appendix gives exact Hecke spectral formulas for hard-truncated resonance states. The results do not locate the zeros of zeta.