Lemma 5.1: five recursive fourth-power constituent values

For every universe level $u$, every type $K : \mathrm{Type}\,u$ equipped with a field structure (with no further restriction on the field), and every real number $\tau$ such that $2 \le 3\tau \le 3$, the following five assertions hold simultaneously. Put $E=12^{3\tau}$, $H=38^{3\tau}$, and $\Lambda=4\cdot 6^{3\tau}(6^{3\tau}+2)$, where these are real powers, and put $v_5=4(E^2+2\Lambda)$, $v_6=4(\Lambda+EH)(2H+\Lambda)/H$, $v_7=4(E+\Lambda)(2+2E+H)$, $v_8=(2H+\Lambda)^2(2+2E+H)/H$, and $v_9=4(E+\Lambda)^2(2H+\Lambda)/\Lambda$; these are exactly the five threshold entries at zero-based indices $5,\ldots,9$, and the hypotheses on $\tau$ make $H$ and $\Lambda$ positive, so the displayed denominators are nonzero. To define the tensors occurring in the assertions, let $e_0,\ldots,e_7$ be the standard basis of $K^8$, let $W=\sum_{r=1}^{6}(e_0\otimes e_r\otimes e_r+e_r\otimes e_0\otimes e_r+e_r\otimes e_r\otimes e_0)+e_0\otimes e_0\otimes e_7+e_0\otimes e_7\otimes e_0+e_7\otimes e_0\otimes e_0$, and form the modewise Kronecker product $F=(W\boxtimes W)\boxtimes(W\boxtimes W)$, with exactly these parentheses. Give a standard-basis coordinate $r\in\{0,\ldots,7\}$ grade $g(r)=0$ when $r=0$, grade $g(r)=2$ when $r=7$, and grade $g(r)=1$ otherwise. In each of the three mode spaces of $F$, the tensor-product basis is indexed by $((r_1,r_2),(r_3,r_4))$, and that basis vector has grade $g(r_1)+g(r_2)+g(r_3)+g(r_4)\in\{0,\ldots,8\}$. For $I,J,L\in\{0,\ldots,8\}$, let $C_{I,J,L}$ be the tensor obtained from $F$ by projecting mode $0$ onto the span of its grade-$I$ basis vectors, mode $1$ onto the span of its grade-$J$ basis vectors, and mode $2$ onto the span of its grade-$L$ basis vectors, with those three spans themselves serving as the new mode spaces. If $P(C)$ denotes the mode reindexing whose new mode order is $(2,0,1)$, then define $\operatorname{Cyc}(C)=C\boxtimes(P(C)\boxtimes P^2(C))$, where $P^2(C)$ has new mode order $(1,2,0)$. Now, for each real $V$, (i) $0\le V<v_5$ implies $\mathcal H(\operatorname{Cyc}(C_{1,1,6}),\tau,V)$; (ii) $0\le V<v_6$ implies $\mathcal H(\operatorname{Cyc}(C_{1,2,5}),\tau,V)$; (iii) $0\le V<v_7$ implies $\mathcal H(\operatorname{Cyc}(C_{1,3,4}),\tau,V)$; (iv) $0\le V<v_8$ implies $\mathcal H(\operatorname{Cyc}(C_{2,2,4}),\tau,V)$; and (v) $0\le V<v_9$ implies $\mathcal H(\operatorname{Cyc}(C_{2,3,3}),\tau,V)$. Here $\mathcal H(T,\tau,V)$ literally means the conjunction of $0\le V$ (so the conclusion repeats the nonnegativity already assumed in each implication) with the following condition: for every real $\varepsilon>0$, there are arbitrarily large natural numbers $N$ for which there exist a natural number $k$ and natural-number families $a_i,b_i,c_i$ indexed by $i\in\{0,\ldots,k-1\}$ such that the recursively formed modewise direct sum of the matrix-multiplication tensors $\mathrm{MM}_{a_i,b_i,c_i}=\sum_{x<a_i}\sum_{y<b_i}\sum_{z<c_i} e_{x,y}\otimes e_{y,z}\otimes e_{z,x}$ is a restriction of the $N$-fold modewise Kronecker power $T^{\boxtimes N}$—that is, there are three $K$-linear maps from the respective mode spaces of $T^{\boxtimes N}$ to those of that direct sum whose tensor-product map sends the tensor of $T^{\boxtimes N}$ exactly to the direct-sum tensor—and $V^N(1-\varepsilon)\le\sum_{i=0}^{k-1}(a_i b_i c_i)^{\tau}$, with each natural product cast to a real number before taking the real power. “Arbitrarily large” is the literal meaning of the filter-frequently quantifier here: for every natural lower bound, at least one such $N$ at or above it exists; it does not say that every sufficiently large $N$ works. The definition allows $N=0$, whose tensor power is the unit tensor object, allows $k=0$, whose direct sum is the zero tensor object and whose numerical sum is empty, allows every dimension $a_i,b_i,c_i$ to be zero, and places no upper bound on $\varepsilon$. In particular, all five outer quantifiers include $V=0$ (each $v_j$ above is positive under the stated hypotheses), and for positive $N$ the $V=0,\ k=0$ case is permitted by the literal definition.