A new analytic method lets scientists compute how pairs of signals from a space‑based gravitational‑wave detector respond to a sky‑tilted background. It works for any time‑delay interferometry (TDI) channel with a fixed finite delay, as long as the three spacecraft form a static, reciprocal, equilateral triangle. The approach blends three building blocks. First, the necessary polynomial constraints that eliminate the dominant laser noise are imposed. Second, the symmetry of the triangular constellation is used to simplify the geometry. Third, a standard integration over angles projects the signal onto the sky. By separating the creation of a channel basis from the angular integration, the authors supply a clear “free basis” that can be reused for any channel. Two elementary angular kernels, called g and h, are shown – via a channel‑generation theorem – to generate all tensor overlap reduction functions for any angular order. The method is computationally very fast. Benchmarks with prepared frequency batches demonstrate that the analytic implementation runs roughly 50 to 5000 times faster than fine‑grained sky quadrature for the same error tolerance, while simultaneously being validated against independent multiprecision sky integrations down to uncertainties of 10⁻¹⁰⁰.

The beauty of this result is that it gives a closed‑form, polarization‑agnostic description of all admissible TDI channels. The kernel functions work for tensor, vector, breathing and longitudinal gravitational‑wave polarizations, and the formalism can be transparent for any fixed angular resolution (up to ℓ=5 illustrated in the paper). However, the framework sits strictly within the dynamics of a static, equal‑arm equilateral constellation. It does not cover configurations where the arm lengths differ, where the delays are non‑reciprocal, or where the spacecraft array changes with time. Moreover, the two‑function representation becomes singular at a few discrete Fourier frequencies (for example, multiples of 2π). In such cases the authors recommend taking a continuity limit rather than naively evaluating the expressions.