Syntomic cohomology of truncated Brown--Peterson spectra

We compute the $\mathrm{M}\mathrm{U}$-based syntomic cohomologies, mod $(p,v_1,\cdots ,v_n)$, of all ${\mathbb{E}}_1$ $\mathrm{M}\mathrm{U}$-algebra forms of the truncated Brown–Peterson spectrum $\mathrm{B}\mathrm{P}⟨n⟩$. As qualitative consequences, we resolve the Lichtenbaum–Quillen, telescope, and redshift questions for the algebraic K-theories of all ${\mathbb{E}}_1$ $\mathrm{M}\mathrm{U}$-algebra forms of $\mathrm{B}\mathrm{P}⟨n⟩$. This extends work of the Hahn and Wilson. We also explicitly compute the algebraic K-theory of arbitrary ${\mathbb{E}}_1$ $\mathrm{M}\mathrm{U}$-algebra forms of $\mathrm{B}\mathrm{P}⟨2⟩$ at all primes $p\ge 5$ extending previous work of the author, Ausoni, Culver, Höning, and Rognes.