Section 5 and Table 1: fourth-power support and ten classes

For an arbitrary universe $u$ and every type $K$ in that universe equipped with a field structure, with no further restriction on $K$, let $\operatorname{Fin}(n)=\{0,\ldots,n-1\}$, put $C=\operatorname{CWObj}(K,6)$, and form the three-slot tensor object $T=\operatorname{kron}(\operatorname{kron}(C,C),\operatorname{kron}(C,C))$. At each slot $s\in\operatorname{Fin}(3)$, the “fourth canonical” basis of $T$ is, literally, the tensor-product basis $B_s=B_s^\square\otimes B_s^\square$ of two copies of the square canonical basis $B_s^\square$ for $K$ and $q=6$; its indices are pairs $p=(p_1,p_2)$ with $p_1,p_2\in\operatorname{Fin}(8)\times\operatorname{Fin}(8)$. If $g_\square$ denotes the supplied square-pair grade at $q=6$, the fourth pair grade $g_4(p)\in\operatorname{Fin}(9)$ is the element whose underlying natural number is $\operatorname{val}(g_\square(p_1))+\operatorname{val}(g_\square(p_2))$. The grading $\Gamma$ used below is the internal nine-part basis grading whose decomposition at each slot $s$ is the basis grading induced by $B_s$ and $g_4$; for a function $\sigma:\operatorname{Fin}(3)\to\operatorname{Fin}(9)$, write $T_\Gamma[\sigma]$ for the block tensor selected by the three grades $\sigma(0),\sigma(1),\sigma(2)$. Also define ten representative functions $\rho_r:\operatorname{Fin}(3)\to\operatorname{Fin}(9)$, displayed as ordered triples of their values at $0,1,2$, and ten natural-number multiplicities $m_r$ by $(\rho_0,\ldots,\rho_9)=((0,0,8),(0,1,7),(0,2,6),(0,3,5),(0,4,4),(1,1,6),(1,2,5),(1,3,4),(2,2,4),(2,3,3))$ and $(m_0,\ldots,m_9)=(1,2,2,2,1,1,2,2,1,1)$. The theorem asserts all four of the following simultaneously: first, for every one of the $9^3$ functions $\sigma:\operatorname{Fin}(3)\to\operatorname{Fin}(9)$, including functions with repeated values and values $0$ or $8$, $T_\Gamma[\sigma]=0$ if and only if the natural-number sum $\sum_{s\in\operatorname{Fin}(3)}\operatorname{val}(\sigma(s))$ is not $8$; both directions are asserted—zero implies that the sum is not $8$, and a sum different from $8$ implies zero—so a sum equal to $8$ forces the selected block tensor to be nonzero. Second, the set $S=\{\sigma:\operatorname{Fin}(3)\to\operatorname{Fin}(9)\mid\sum_s\operatorname{val}(\sigma(s))=8\}$ has exactly $45$ elements. Third, for every $\sigma\in S$, there exists exactly one class index $r\in\operatorname{Fin}(10)$ such that there exists a permutation $e$ of $\operatorname{Fin}(3)$ satisfying $\sigma(s)=\rho_r(e(s))$ for every $s\in\operatorname{Fin}(3)$; this is the literal direction of the equality, and only $r$ is unique—the permutation $e$ is merely existential and need not be unique, in particular when a representative has repeated coordinates. Fourth, for every $r\in\operatorname{Fin}(10)$, the number of distinct $\sigma\in S$ for which such a permutation $e$ exists is exactly $3m_r$, hence is respectively $(3,6,6,6,3,3,6,6,3,3)$ for $r=0,\ldots,9$; this cardinality counts the functions $\sigma$, not the possible witnessing permutations.