Finite 2-group homotopy gauging and cochain gluing

Fix $x=(a,B)$ and let $S_x$ be its raw $1$-automorphism set. By [F3], $\mathrm{\mid }{\pi }_2(x)\mathrm{\mid }=\mathrm{\mid }\ker {\delta }_a\mathrm{\mid }$ and $\mathrm{\mid }{\pi }_1(x)\mathrm{\mid }=\mathrm{\mid }{S}_x\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{im}{\delta }_a\mathrm{\mid }$. Since ${\delta }_a:C^0\to C^1$ is a homomorphism of finite abelian groups, its image has order $\mathrm{\mid }A{\mathrm{\mid }}^{V_K}\mathrm{/}\mathrm{\mid }\ker {\delta }_a\mathrm{\mid }$. Hence $\mathrm{\mid }{\pi }_2(x)\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\pi }_1(x)\mathrm{\mid }=\mathrm{\mid }A{\mathrm{\mid }}^{V_K}\mathrm{/}\mathrm{\mid }{S}_x\mathrm{\mid }$. For relative cochains the identical argument replaces $V_K$ by $V_K-V_L$.

There are $N=\mathrm{\mid }G{\mathrm{\mid }}^{V_K}\mathrm{\mid }A{\mathrm{\mid }}^{E_K}$ raw gauge parameters with source $x$, each having a target by [F3]. For any target $y$ in the component of $x$, fix a raw arrow $f:x\to y$. Composition $u\mapsto f\circ u$ identifies raw automorphisms at $x$ bijectively with raw arrows $x\to y$: its vertex equation uniquely solves for the vertex labels of $u$, and its edge equation uniquely solves for its edge labels since the coefficient of each is the automorphism ${\rho }_{g_f}$. The solved vertex labels fix $a$ because both resulting arrows have target edge labels of $y$. The solved edge labels fix $B$ because $f$ is affine with invertible linear coefficient on $B$, and $f$ already sends $x$ to $y$. Thus the solved parameter is indeed in $S_x$. Therefore each target has $\mathrm{\mid }{S}_x\mathrm{\mid }$ arrows from $x$, and this component has $N\mathrm{/}\mathrm{\mid }{S}_x\mathrm{\mid }$ objects. The same argument is relative: compositions and inverses retain identity/zero boundary labels and the parameter count is $N_{K,L}=\mathrm{\mid }G{\mathrm{\mid }}^{V_K-V_L}\mathrm{\mid }A{\mathrm{\mid }}^{E_K-E_L}$.

Every fixed raw parameter acts bijectively on all background objects: the edge formula determines the original edge labels uniquely and the face formula is affine in $B$ with invertible coefficient ${\rho }_g$. The source-wise reverse from [F3] shows the inverse preserves the background constraints. A boundary parameter on $L$ extends to a raw bulk parameter by identity labels on other vertices and zero labels on other edges. This bijection carries the set of extensions of $s$ to that of its transformed boundary field and preserves $W_K$. Thus relative extension sums, and hence their normalized versions, are invariant under boundary $1$-arrows.

Local equations identify a field on $K_1{\cup }_{\mathrm{\Sigma }}{K}_2$ with exactly a pair of fields agreeing on $\mathrm{\Sigma }$: each simplex lies in one of the subcomplexes, so assignments glue uniquely and all flatness and tetrahedron equations remain valid. Vertex and edge counts satisfy inclusion-exclusion, so $c(K)=c(K_1)c(K_2)\mathrm{/}c(\mathrm{\Sigma })$.

On each component the weight is constant. Its higher mass by step 1.1 is $\mathrm{\mid }A{\mathrm{\mid }}^{V_K}\mathrm{/}\mathrm{\mid }{S}_x\mathrm{\mid }$, while its unnormalized object sum has $N\mathrm{/}\mathrm{\mid }{S}_x\mathrm{\mid }$ terms by step 1.2. Their ratio is $\mathrm{\mid }A{\mathrm{\mid }}^{V_K}\mathrm{/}N=c(K)$. Summing gives the absolute formula. Relatively the ratio is $\mathrm{\mid }A{\mathrm{\mid }}^{V_K-V_L}\mathrm{/}{N}_{K,L}=c(K)\mathrm{/}c(L)$, including empty extension sets. This proves the relative formula and, with step 1.3, boundary invariance.

By step 2.1, the object-sum form of the proposed boundary integral is $c(\mathrm{\Sigma }){\sum }_s{Z}_2(s)Z_1(s)$. Substituting the two relative formulas gives $c(K_2)c(K_1)\mathrm{/}c(\mathrm{\Sigma })$ times ${\sum }_s{\sum }_{x_2{\mathrm{\mid }}_{\mathrm{\Sigma }}=s}{\sum }_{x_1{\mathrm{\mid }}_{\mathrm{\Sigma }}=s}{W}_2(x_2)W_1(x_1)$. Step 1.4 and weight factorization identify this with $c(K){\sum }_x{W}_K(x)=Z_K^{(2)}$. Applying the absolute formula on $\mathrm{\Sigma }$ to the invariant product $Z_2{Z}_1$ converts its object sum to the stated ${\pi }_2\mathrm{/}{\pi }_1$ boundary sum. The formula for complex-valued invariant products follows from the same component counting, which uses only constancy and finite addition, not unit modulus.

The empty and point cases follow by inserting their vertex/edge counts and object sets. If $A$ is trivial, the only $2$-gauge labels are zero and $c(K)=\mathrm{\mid }G{\mathrm{\mid }}^{-V_K}$, yielding ordinary finite flat-connection normalization. The finite sums, relative formula, invariance and gluing identity have all been established under precisely the stated weight hypotheses.