The admissible Wada deleted minor equals based twisted torsion

By [A1] and [F3], the fraction-field complex $0\to Q^{(n-1)d}\stackrel{{\mathrm{\partial }}_2}{\to }{Q}^{nd}\stackrel{{\mathrm{\partial }}_1}{\to }{Q}^d\to 0$ is acyclic. Therefore ${\mathrm{\partial }}_2$ is injective and its image is $\ker {\mathrm{\partial }}_1$, both of dimension $(n-1)d$.

In the ordered $1$-cell basis, [F2] gives the stacked $d_1$ blocks $A_1,\dots ,A_n$; a row vector $y=(y_1,\dots ,y_n)$ lies in its kernel exactly when ${\sum }_k{y}_k{A}_k=0$. Since $A_j$ is invertible, projection $p_{\widehat{j}}:\ker d_1\to Q^{(n-1)d}$ onto all non-$j$ blocks is an isomorphism: for arbitrary non-$j$ row coordinates $y_k$, the unique missing coordinate is $y_j=-\left({\sum }_{k\mathrm{\ne }j}{y}_k{A}_k\right)A_j^{-1}$. Thus $p_{\widehat{j}}{\mathrm{\partial }}_2$ is an isomorphism, and its row-coefficient matrix is $J_{\widehat{j}}$ by [F2]. In particular $\det (J_{\widehat{j}})\mathrm{\ne }0$.

Let ${\mathbf{c}}_0,{\mathbf{c}}_1,{\mathbf{c}}_2$ be the standard cellular bases. In degree zero choose ${\mathbf{b}}_0={\mathbf{c}}_0$. In degree one choose ${\mathbf{b}}_1={\mathrm{\partial }}_2{\mathbf{c}}_2$, and lift ${\mathbf{b}}_0$ to degree one using the $j$-block $A_j^{-1}$ and zero in the other blocks. After moving the $j$-block to the final position, the matrix of $[{\mathbf{b}}_1,{\tilde{\mathbf{b}}}_0]$ in ${\mathbf{c}}_1$ is block triangular with diagonal blocks $J_{\widehat{j}}$ and $A_j^{-1}$; its determinant is $\pm \det (J_{\widehat{j}})\mathrm{/}{D}_j$. The degree-zero and degree-two torsion brackets equal $1$, so [F5] gives $\tau =\pm W_j$. The sign lies in ${\mathcal{U}}_{\rho ,\psi }$ by [F4], proving the equality of classes.

The same construction applies to every index $k$ with $D_k\mathrm{\ne }0$. Each quotient class equals the single torsion class of the same based complex by step 3.1; hence all admissible columns give the same class in $Q^{\times }\mathrm{/}{\mathcal{U}}_{\rho ,\psi }$.