Remark: Family 308. Smith–Toda complexes at every height with varying primes

For every $n\ge 0$, constructs a finite Smith–Toda spectrum at a prime $p$ depending on $n$, with Brown–Peterson homology $B{P}_{\ast }/(p,v_1,\dots ,v_n)$ and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes $V(4)$ at $p=1009$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.