Chromatic Homotopy is Monoidally Algebraic at Large Primes

Fix a prime $p$ and a chromatic height $h$. We prove that the homotopy $(k,1)$-category of $L_h$-local spectra ${\mathrm{h}}_k\left({\mathrm{S}\mathrm{p}}_{p,h}\right)$ is algebraic as a symmetric monoidal category when $p>O(h^2+kh)$. To achieve this, we develop a general tool for investigating such algebraicity questions, based on an operadic variant of Goerss-Hopkins obstruction theory. Other applications include the monoidal algebraicity of modules over the Lubin-Tate spectrum ${\mathrm{h}}_k\left({\mathrm{M}\mathrm{o}\mathrm{d}}_{E_{p,h}}\right)$ whenever $p>O(kh)$, from which we deduce that ${\mathrm{h}}_1\left({\mathrm{M}\mathrm{o}\mathrm{d}}_{K{U}_{(p)}}\right)$ and ${\mathrm{h}}_1\left({\mathrm{M}\mathrm{o}\mathrm{d}}_{K{O}_{(p)}}\right)$ are algebraic as tt-categories if and only if $p$ is odd.