On Bruner's Open Questions: Secondary Ext of the Fibe of $S{q}^n$ via Explicit Secondary Adem Tracks

Robert Bruner \cite[Questions 6.1 and 6.2]{Bruner2026} asked whether the secondary cohomology of the fibers $F_n$, $F_{n\mathbb{Z}}$, and $F$ can be computed to determine the $E_3$-terms of their Adams spectral sequences, and whether the Bruner-Rognes two-extension formula for the ordinary Adams $d_2$ is intrinsic to secondary cohomology. In this work, we give an unconditional affirmative answer to both questions. Working in the Baues-Nassau secondary Steenrod algebra, we construct explicit secondary mapping-fiber resolutions for these fibers using a tracked Adem reduction algorithm and the Baues-Jibladze recursive completion. We determine the secondary Ext groups, independently recovering Bruner’s $E_3$-terms: $${\mathrm{Ext}}_{\mathcal{B}}^{\ast ,\ast }(H_{\mathcal{B}}^{\ast }{F}_n,{\mathbb{F}}_2)\cong {\mathbb{F}}_2\oplus {\mathrm{\Sigma }}^{1,n}{\mathbb{F}}_2,$$ $${\mathrm{Ext}}_{\mathcal{B}}^{\ast ,\ast }(H_{\mathcal{B}}^{\ast }{F}_{n\mathbb{Z}},{\mathbb{F}}_2)\cong {\mathbb{F}}_2[h_0]\oplus {\mathrm{\Sigma }}^{1,n}{\mathbb{F}}_2,$$ $${\mathrm{Ext}}_{\mathcal{B}}^{\ast ,\ast }(H_{\mathcal{B}}^{\ast }F,{\mathbb{F}}_2)\cong {\mathbb{F}}_2[h_0]\oplus \underset{j>0}{⨁}{\mathrm{\Sigma }}^{1,2^j}{\mathbb{F}}_2\oplus \underset{\begin{array}{c}i>0\\ i\text{ not a power of }2\end{array}}{⨁}{\mathrm{\Sigma }}^{0,2i-1}{\mathbb{F}}_2\mathrm{.}$$ This direct calculation answers Question 6.1. Finally, we prove that the primary shadow of the first secondary differential in our construction is identically the Bruner-Rognes Yoneda composite associated with the corresponding two-extension, thereby answering Question 6.2.