A new unified analytic framework makes it possible to compute anisotropic overlap reduction functions (ORFs) for arbitrary time-delay interferometry (TDI) channels in equilateral constellations with equal arms. The construction combines the polynomial constraints that guarantee laser-noise cancellation, the symmetries of the constellation, and integration over directional projections. Separating channel generation from angular integration, via a free basis of the admissible channel module, makes the results reusable for anisotropic multichannel background models.

The method is independent of gravitational-wave polarization: the reduction to elementary angular kernels applies equally to the tensor, vector, breathing and longitudinal sectors, although the work illustrates the framework with tensor multipoles. Propagation effects are retained in full up to any fixed finite angular order, with explicit examples through ℓ = 5. Numerical evaluation is stabilised using prepared frequency batches, shared multiprecision expressions, and local expansions at removable delay points.

In the prepared-batch benchmark, the analytic implementation proved between 50 and 5000 times faster than optimised sky quadrature at matched error tolerances. Independent multiprecision sky integrations also validated prescribed mixed error targets down to 10⁻¹⁰⁰ in the tested cases. The resulting kernels and generation rules are available as reusable inputs for multichannel stochastic background models with anisotropies, relevant for future space-based gravitational-wave missions.