Finite checks of inversion-parity certificates for voltage covers

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Both records concern the inversion-parity mechanism of generic voltage coloring. They are finite computational observations; no script is retained, and the exact witness below is verified by the displayed algebra.

A bounded tensor certificate test

Exactly $3^9=19683$ assignments of nonzero vectors $\delta_{ij}\in\mathbb F_2^2$ to the nine edges of $K_{3,3}$ were examined. The proposed certificate was nonvanishing of

$$\sum_{i<k,\ j>l}\delta_{ij}\otimes\delta_{kl}.$$

The reported enumeration gave 432 zero tensors. An exact witness, with $a=(1,0)$, $b=(0,1)$, is

$$\begin{pmatrix}a&a&b\\a&a&a+b\\b&a+b&a\end{pmatrix}.$$

Its nine inversion terms reduce, after canceling repeated terms, to

$$a\otimes a+b\otimes b+b\otimes(a+b) +(a+b)\otimes b+(a+b)\otimes(a+b)=0.$$

This verifies the needed counterexample without depending on the reported count. It is realizable by $f(0)=0$, $f(1)=a$, $f(2)=b$ on $\mathbb Z/3$ with origin matrix $\begin{pmatrix}0&0&2\\0&0&1\\2&1&0\end{pmatrix}$. The six resulting four-bit colors are not all equal. Thus this refutes the proposed nonzero-XOR certificate, not the full four-bit coloring construction. The test stopped at its finite range and was not enlarged to classify that different question.

Eight prescribed multiplier checks

One bounded probe used the complete $3\times4$ symbolic matrix $M_{xv}=X_{xv}$ in the integer group ring of $(\mathbb Z/2)^{12}$, with incidences ordered row-major. Products add exponent masks by XOR. For $j=0,1,2,3$, let $C_j$ be the increasing list of column indices other than $j$, and define

$$p_j=(-1)^j\sum_{\pi\in S_3}\operatorname{sgn}(\pi) \prod_{x=0}^2X_{x,C_j[\pi(x)]}.$$

The kernel vector was $p=(p_0,p_1,p_2,p_3)$. Exactly eight multipliers $f$ were tested: the four $p_j$ and the four unsigned polynomials obtained by replacing all six coefficients of $p_j$ by $+1$. Each test collected all monomials of all four coordinates $p_if$, discarding zero coefficients. The reported support sizes were

In all eight cases the common coefficient gcd was one, the vector was nonflat, and its support met both values of the inversion-parity coloring $Q(g)=\sum_{x<y,v>w}g_{xv}g_{yw}$ modulo two. These are finite computational observations about these eight untranslated products, not a theorem about arbitrary multipliers. The complete-support base has repeated columns, so the probe concerns formal cofactor algebra, not a proposed set of distinct frequencies. No search beyond those eight products was performed.