The Kervaire invariant problem at the prime three: which standard classes $b_j$ survive the mod-3 Adams spectral sequence

Theorem 1.1: at $p=3$, $K=K_3=\{0,2,3\}$, i.e. $b_j$ survives to $E_\infty$ exactly for $j=0,2,3$ (stems 10, 106 and 322), and each surviving class is detected by an element of additive order three; $b_1$ and all $b_j$ with $j\ge4$ die. The new content is the order-three element in stem 322 and the exclusion of every $j\ge4$. Corollaries via Amelotte and Selick: $\Omega S^{2n+1}\{3\}$ splits nontrivially exactly for $n\in\{3,27,81\}$, and the Anick space $T^{2n+1}(3)$ is homotopy associative exactly for those $n$. Side result: an infinite-order class in $\pi_{2n}(E_6^{hC_9})$ for every $n$, refuting the full-gap clause of Belmont-Ray's Conjecture 0.1. Not covered: other odd primes (already settled for $p\ge5$) or the prime-two problem.

Theorem 1.1: at $p=3$, $K=K_3=\{0,2,3\}$, i.e. $b_j$ survives to $E_\infty$ exactly for $j=0,2,3$ (stems 10, 106 and 322), and each surviving class is detected by an element of additive order three; $b_1$ and all $b_j$ with $j\ge4$ die. The new content is the order-three element in stem 322 and the exclusion of every $j\ge4$. Corollaries via Amelotte and Selick: $\Omega S^{2n+1}\{3\}$ splits nontrivially exactly for $n\in\{3,27,81\}$, and the Anick space $T^{2n+1}(3)$ is homotopy associative exactly for those $n$. Side result: an infinite-order class in $\pi_{2n}(E_6^{hC_9})$ for every $n$, refuting the full-gap clause of Belmont-Ray's Conjecture 0.1. Not covered: other odd primes (already settled for $p\ge5$) or the prime-two problem.