Slow-Roll Inflation at Arbitrary Order

We derive explicit expressions for the primordial scalar and tensor spectra at arbitrary finite order $p$ in the slow-roll expansion, valid both for canonical single-field inflation and $k$-inflation. At any order, the spectra are determined by known combinations of slow-roll parameters weighted by universal coefficients entirely fixed by the free de Sitter solutions, without requiring perturbatively corrected mode functions. Using these spectra, we obtain closed-form consistency relations that connect the tensor tilt $n_t$ and its runnings ${\alpha }_n^t\equiv d^n{n}_t\mathrm{/}d(\ln k{)}^n$ to the tensor-to-scalar ratio $r$, the scalar spectral index $n_s$, and its runnings ${\alpha }_n^s\equiv d^n(n_s-1)\mathrm{/}d(\ln k{)}^n$. The consistency relations are valid for arbitrary $n\ge 0$ and any finite $p\ge 0$, without recursion over lower-order relations. For $k$-inflation, they additionally depend on the scalar sound speed and its scale derivatives. We develop an algorithm that generates both the spectra and consistency relations for any pair $(n,p)$. We reproduce known results through next-to-next-to-next-to-leading order (N3LO) and extend the explicit calculations up to N10LO.