arXiv2026
Let $\mathcal G_k$ be the gauge group of the principal $G_2$-bundle over $S^4$ classified by $k\inπ_4(BG_2)\cong\mathbb Z$. In this paper, we determine the first positive fibre row in the mod $2$ Serre spectral sequence of $Ω^3_0G_2\to B\mathcal G_k\xrightarrow{p_k}BG_2$. The fibre cohomology vanishes in degrees $1$ through $4$ and has a unique nonzero class $u_5$ in degree $5$. We prove that its transgression is $d_6(u_5)=(k\bmod 2)x_6$. The proof uses the Stiefel--Whitney classes of the seven-dimensional representation and maps of evaluation fibrations induced by the pinch map of $S^4$. The same comparison maps show that the possible incoming differential $d_4:H^8(Ω^3_0G_2;\mathbb F_2)\to\mathbb F_2\{x_4u_5\}$ vanishes. Consequently, the classes $u_5$ and $x_4u_5$ survive precisely when $k$ is even. In degree six, a nonzero linear functional $λ:H^6(Ω^3_0G_2;\mathbb F_2)\to\mathbb F_2$ determines the next transgression by $d_7(z)=(k\bmod2)λ(z)x_7$. Both $d_7$ maps to the degree-seven and degree-eleven base classes vanish for even $k$ and are surjective for odd $k$; in particular, $p_k^*x_7\ne0$ exactly when $k$ is even. We derive further forced vanishings, cohomological consequences, and Steenrod-square relations, and delimit the remaining differentials with source of total degree at most $10$. The calculation is independent of the exact order of the fundamental Samelson product.