Nonzero Postnikov data retain higher gauge volume on a tetrahedron

Normalization is immediate. The cocycle equation expands to $hkl-(g+h)kl+g(h+k)l-gh(k+l)+ghk=0$ since each monomial occurs twice modulo $2$. If $\beta =\delta c$, its values at $(0,1,1)$ and $(1,0,1)$ imply $c(0,0)=c(0,1)=c(1,0)$. Then $(\delta c)(1,1,1)=c(1,0)-c(0,1)=0$, contradicting $\beta (1,1,1)=1$.

Set $x_0=0$ and $x_i=a_{0i}$ for $i>0$. Flatness on $0ij$ gives $a_{ij}=x_i+x_j$, so flat edges are in bijection with the three unrestricted potentials $x_1,x_2,x_3$, giving $8$ edge backgrounds. For each, the tetrahedron constraint in [F2] uniquely determines one face label from the other three, giving $8$ face assignments and $64$ objects in total. There are $V=4,E=6$, so [F1] gives $c=2^{4-6}\mathrm{/}{2}^4=1\mathrm{/}64$ and $Z=c\cdot 64=1$.

For any object choose the vertex labels $g_i=x_i$ from step 1.2 and first take edge parameters zero. The target has $b_{ij}=0$ and face labels $C=B+{\zeta }^g(a)$. By [F3] it is still a background, hence $\delta C=\beta (0,0,0)=0$. Now take identity vertex labels and set ${\lambda }_{0j}=0$ and ${\lambda }_{ij}=C_{0ij}$ for $0<i<j$. Normalization makes this second arrow's $\zeta$ zero. On $0ij$, $(\delta \lambda {)}_{0ij}=C_{0ij}$; on $123$, $(\delta \lambda {)}_{123}=C_{023}+C_{013}+C_{012}=C_{123}$ by $\delta C=0$. Thus its target faces are $C+\delta \lambda =0$. Every object is therefore equivalent to the zero background.

At the zero background, an automorphism's target edges vanish exactly when all $g_i$ equal one constant $r\in {\mathbb{Z}}_2$. Every term of $\zeta$ then has a zero argument, so normalization gives $\zeta =0$; the remaining condition is $\delta \lambda =0$. Such edge cocycles have exactly the form ${\lambda }_{ij}=y_i+y_j$ with $y_0=0$, $y_i={\lambda }_{0i}$, by the triangle equations. Hence there are $8$ of them and $16$ raw automorphisms. The map $t\mapsto \delta t$ has kernel the two constant vertex cochains and image precisely these $8$ edge cocycles: take $t_i=y_i$ for surjectivity. Therefore ${\pi }_2\cong {\mathbb{Z}}_2$ under addition. For two raw automorphisms all successive edge backgrounds are zero, so each term of $\tau$ also has a zero argument and composition is $(r,\lambda )(s,\mu )=(r+s,\lambda +\mu )$. The $2$-arrows quotient the entire edge-cocycle factor, preserving $r$, so ${\pi }_1\cong {\mathbb{Z}}_2$ as a group, also with order $16\mathrm{/}8=2$. By the equivalence in step 2.1, these homotopy groups have the stated isomorphism types at every background.

The single component has higher mass $2\mathrm{/}2=1$ by [F1], agreeing with step 1.2. The nonzero $\beta$ is still present in the full background and gauge formulas. In particular, for three constant automorphism vertex labels $g,h,k$ at the zero background the associator is the constant vertex $2$-cochain ${\sigma }_i=khg$, nonzero when $g=h=k=1$. Its coboundary is zero, so it is a retained $2$-automorphism, not a discarded parameter. Thus no quotient by induced functions or replacement by a split model has entered the calculation.