Curtis's conjecture: the mod-2 stable Hurewicz image of the sphere is spanned by the Hopf- and Kervaire-invariant-one classes

Theorem 1.1: the image of $\bigoplus_{d>0}h_d$ is spanned over $\mathbb F_2$ by the Hurewicz images of $\eta,\nu,\sigma$ and of the Kervaire-invariant-one classes $\theta_j$ (degree $2^{j+1}-2$) whenever they exist; no existence is posited. Corollary 1.2: Eccles's conjecture holds for every sphere $S^n$, $n>0$: each nonzero spherical class in $H_*(QS^n;\mathbb F_2)$ is the bottom class or a suspended Hopf-invariant-one image. Corollary 1.3 (with Hill-Hopkins-Ravenel): $h_d=0$ for $d>0$ outside $\{1,2,3,6,7,14,30,62,126\}$, so the positive image vanishes above degree 126. Not shown: an exact dimension of the image or nonvanishing in each listed degree, odd primes, or Eccles's conjecture for $QX$ with $X$ other than a sphere.

Theorem 1.1: the image of $\bigoplus_{d>0}h_d$ is spanned over $\mathbb F_2$ by the Hurewicz images of $\eta,\nu,\sigma$ and of the Kervaire-invariant-one classes $\theta_j$ (degree $2^{j+1}-2$) whenever they exist; no existence is posited. Corollary 1.2: Eccles's conjecture holds for every sphere $S^n$, $n>0$: each nonzero spherical class in $H_*(QS^n;\mathbb F_2)$ is the bottom class or a suspended Hopf-invariant-one image. Corollary 1.3 (with Hill-Hopkins-Ravenel): $h_d=0$ for $d>0$ outside $\{1,2,3,6,7,14,30,62,126\}$, so the positive image vanishes above degree 126. Not shown: an exact dimension of the image or nonvanishing in each listed degree, odd primes, or Eccles's conjecture for $QX$ with $X$ other than a sphere.