Theorem 5.3: kernel-corrected fourth-power value inequality

For every field $K$ (in an arbitrary universe and with no characteristic restriction), every real number $\tau$, and every two real $10$-tuples $a=(a_0,\ldots,a_9)$ and $b=(b_0,\ldots,b_9)$, assume $2\le 3\tau\le 3$; assume every $a_i$ and every $b_i$ is both nonnegative and strictly positive; and, with

$$n=(1,2,2,2,1,1,2,2,1,1),$$

assume both $\sum_{i=0}^9 n_i a_i=1$ and $\sum_{i=0}^9 n_i b_i=1$, together with

$$b_2b_7^2=b_4b_5b_9,\qquad b_3b_7b_8=b_4b_6b_9.$$

Also assume that there exist real numbers $s,t$ such that, for every $i\in\{0,\ldots,9\}$,

$$a_i-b_i=s\,\sigma_i+t\,\theta_i, \qquad \sigma=(0,0,1,0,-2,-2,0,2,0,-2),\quad \theta=(0,0,0,1,-2,0,-1,1,2,-2).$$

Define $\operatorname{rpow}(x,y)$ to be the total real power with real base and real exponent used in the declaration, set

$$E=\operatorname{rpow}(12,3\tau),\qquad H=\operatorname{rpow}(38,3\tau),\qquad L=4\,\operatorname{rpow}(6,3\tau)\bigl(\operatorname{rpow}(6,3\tau)+2\bigr),$$

and define the $10$-tuple $v$ by

$$\begin{aligned} v_0&=1, & v_1&=\operatorname{rpow}(24,3\tau), & v_2&=\operatorname{rpow}(220,3\tau),\\ v_3&=\operatorname{rpow}(936,3\tau), & v_4&=\operatorname{rpow}(1734,3\tau), & v_5&=4(E^2+2L),\\ v_6&=\frac{4(L+EH)(2H+L)}{H}, & v_7&=4(E+L)(2+2E+H),\\ v_8&=\frac{(2H+L)^2(2+2E+H)}{H}, & v_9&=\frac{4(E+L)^2(2H+L)}{L}. \end{aligned}$$

Let $Q(a)=(Q_0,\ldots,Q_8)$ and $m_j=Q_j/3$, where

$$\begin{aligned} Q_0&=2a_0+2a_1+2a_2+2a_3+a_4,& Q_1&=2a_1+2a_5+2a_6+2a_7,\\ Q_2&=2a_2+2a_6+2a_8+a_9,& Q_3&=2a_3+2a_7+2a_9,\\ Q_4&=2a_4+2a_7+a_8,& Q_5&=2a_3+2a_6,\\ Q_6&=2a_2+a_5,& Q_7&=2a_1,& Q_8&=a_0, \end{aligned}$$

and set

$$R= \left(\prod_{i=0}^{9} \left[ \operatorname{rpow}\!\left(v_i,\frac{a_i}{3}\right) \operatorname{rpow}(a_i,a_i) \operatorname{rpow}(b_i,-b_i) \right]^{n_i}\right) \left(\prod_{j=0}^{8}\operatorname{rpow}(m_j,-m_j)\right).$$

Let $C_6$ be the order-three tensor over $K$, with each mode equal to the coordinate space $K^8$, given for its standard coordinate vectors $e_0,\ldots,e_7$ by

$$C_6= \sum_{r=0}^{5}\left( e_0\otimes e_{r+1}\otimes e_{r+1} +e_{r+1}\otimes e_0\otimes e_{r+1} +e_{r+1}\otimes e_{r+1}\otimes e_0\right) +e_0\otimes e_0\otimes e_7 +e_0\otimes e_7\otimes e_0 +e_7\otimes e_0\otimes e_0,$$

and let

$$T=(C_6\boxtimes C_6)\boxtimes(C_6\boxtimes C_6),$$

where $\boxtimes$ is the modewise Kronecker product: corresponding mode spaces are tensor-product spaces and the two tensor elements are sent to their interchanged modewise tensor product. Then, for every real $V$ satisfying $0\le V<R$, $V$ has the following property (which itself again includes $0\le V$): for every real $\varepsilon>0$ with no upper bound on $\varepsilon$, and every cutoff $N_0\in\mathbb N$, there exist $N\in\mathbb N$ with $N\ge N_0$, an otherwise unrestricted natural number $k$, and functions $p,q,r:\{0,\ldots,k-1\}\to\mathbb N$, such that the finite modewise direct sum

$$\bigoplus_{\ell=0}^{k-1} M_K(p_\ell,q_\ell,r_\ell)$$

is a restriction of $T^{\boxtimes N}$, and

$$V^N(1-\varepsilon)\le \sum_{\ell=0}^{k-1} \operatorname{rpow}\!\left( \bigl((p_\ell q_\ell r_\ell:\mathbb N):\mathbb R\bigr),\tau\right).$$

Here $M_K(x,y,z)$ is the matrix-multiplication tensor with mode spaces $K^{\{0,\ldots,x-1\}\times\{0,\ldots,y-1\}}$, $K^{\{0,\ldots,y-1\}\times\{0,\ldots,z-1\}}$, and $K^{\{0,\ldots,z-1\}\times\{0,\ldots,x-1\}}$, and tensor $\sum_{i<x,j<y,h<z}e_{ij}\otimes e_{jh}\otimes e_{hi}$; “is a restriction” means that there are three $K$-linear maps from the respective mode spaces of $T^{\boxtimes N}$ to those of this direct sum whose induced tensor-product map sends the tensor of $T^{\boxtimes N}$ exactly to the direct-sum tensor. The Kronecker power is defined by $T^{\boxtimes0}$ equal to the unit tensor object (each mode $K$, tensor $1\otimes1\otimes1$) and $T^{\boxtimes(N+1)}=T\boxtimes T^{\boxtimes N}$; the scalar power $V^N$ is the natural-number power, so $V^0=1$. The witnesses allow $k=0$, in which case the target is the zero tensor object and the displayed sum is empty, and they allow any of $p_\ell,q_\ell,r_\ell$ to be $0$; the only requirement on the occurring powers $N$ is cofinality expressed by the arbitrary cutoff $N_0$, rather than the property holding for every sufficiently large $N$.