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 $\mathrm{h}_k\big(\mathrm{Sp}_{p,h}\big)$ 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 $\mathrm{h}_k\big(\mathrm{Mod}_{E_{p,h}}\big)$ whenever $p >O(kh)$, from which we deduce that $\mathrm{h}_1 \big(\mathrm{Mod}_{KU_{(p)}}\big)$ and $\mathrm{h}_1\big(\mathrm{Mod}_{KO_{(p)}}\big)$ are algebraic as tt-categories if and only if $p$ is odd.