The triangulated-exterior chain model computes the multivariable Alexander module

By [F4], an admissible input passes all three label checks and the finite procedure terminates. Thus its output is the finite free complex whose cells and boundary matrices are prescribed in [F3].

Antisymmetry makes the edge labels a homomorphism from the cellular 1-chain group of $K$ to ${\mathbb{Z}}^r$. The zero sum on each oriented triangle says this homomorphism vanishes on every cellular 2-boundary. Its restriction to cellular 1-cycles therefore descends to a homomorphism $H_1^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(E_L;\mathbb{Z})\to {\mathbb{Z}}^r$. By [F6] and the meridian-basis assertion in [F2], this map sends the basis $[{\mu }_i]$ to $e_i$, so it is precisely the isomorphism that records the full meridian abelianization.

By [F7], the corresponding homomorphism on ${\pi }_1(E_L,b)$ factors through its abelianization and has kernel equal to the commutator subgroup. Thus the cover represented by the labels is the maximal abelian cover of $E_L$, with deck exponents indexed by the stated meridian basis.

Let $\lambda (v)$ be the sum of edge labels along the unique spanning-tree path from the least vertex $v_0$ to $v$. Construct a simplicial cover with vertices $(v,g)$, $g\in {\mathbb{Z}}^r$, and with each oriented edge $e:v\to w$ lifting from $(v,g)$ to $(w,g+\mathrm{\ell }(e))$. In a simplex, assign a vertex exponent by summing labels along an edge path from its least vertex; if two such paths have the same endpoints, their difference is a 1-cycle, and coning that cycle to the least vertex expresses it as a sum of triangular face boundaries. The face checks make each such sum zero. Thus every simplex has a unique lift through each of its vertex lifts, and the lifted simplices agree on common faces. At each vertex $(v,g)$, each simplex incident to $v$ has a unique lift through $(v,g)$; therefore the open star maps homeomorphically to the open star of $v$, proving the covering condition. A loop lifts closed exactly when its label sum is zero; by steps 2.1, 3.1 this subgroup is the kernel of abelianization. Whiskering the certified meridian loops to $v_0$ and back gives loops with label sums $e_i$, so the whole fiber over $v_0$ is connected. The spanning tree paths then connect every vertex in every other fiber to that fiber, hence the cover is connected. By [F9] the constructed cover is the maximal abelian cover.

The tree assignments select the vertex lifts $(v,\lambda (v))$. The procedure then selects each simplex lift through its anchor lift. For a face $\tau$ of $\sigma$, the difference between the exponent of this face occurrence and the selected lift of $\tau$ is exactly $g(\sigma ,\tau )$ computed in instruction 3. Therefore its lifted boundary coefficient is $ϵ(\sigma ,\tau )t_1^{g_1(\sigma ,\tau )}\cdots t_r^{g_r(\sigma ,\tau )}$, as in instruction 4. These are the cellular face-incidence boundaries in the constructed cover, so the procedure matrices satisfy ${\mathrm{\partial }}^2=0$ and form its lifted cellular chain complex.

By [F1] and [F5], the first homology of that complex is $A_L$ as a ${\mathrm{\Lambda }}_r$-module. By [F8], its Fitting ideals and order are independent of the triangulation and lift choices, with the stated unit ambiguity. This proves every conclusion.