Three-identity cycle and distance certificate

For every $n,r_X,r_Z\in\mathbb N$ and every datum $D$ consisting of two $\mathbb F_2$-linear maps

$$X:\mathbb F_2^{\,n}\to\mathbb F_2^{\,r_X}, \qquad Z:\mathbb F_2^{\,n}\to\mathbb F_2^{\,r_Z},$$

with $XZ^{\mathsf T}=0$, $ZX^{\mathsf T}=0$, and both $X$ and $Z$ surjective, together with vectors $\ell_X,\ell_Z\in\mathbb F_2^{\,n}$ satisfying

$$Z\ell_X=0,\qquad X\ell_Z=0,\qquad \sum_{i\in\operatorname{Fin}(n)}\ell_X(i)\ell_Z(i)=1 \quad\text{in }\mathbb F_2=\mathbb Z/2\mathbb Z,$$

where each transpose is the transpose of the standard-coordinate matrix, define chain complexes $A$ and $B$ by

$$(A_0,A_1,A_2)=(\mathbb F_2^{\,r_X},\mathbb F_2^{\,n},\mathbb F_2^{\,r_Z}), \quad \partial^A_1=X,\quad \partial^A_2=Z^{\mathsf T},$$

and

$$(B_0,B_1,B_2)=(\mathbb F_2^{\,r_Z},\mathbb F_2^{\,n},\mathbb F_2^{\,r_X}), \quad \partial^B_1=Z,\quad \partial^B_2=X^{\mathsf T},$$

with zero groups above degree $2$ and all remaining boundary maps zero. The degree-$j$ tensor-product coordinate space consists of binary words indexed by tuples $(p,q,a,b)$ with $p+q=j$, $a$ a coordinate of $A_p$, and $b$ a coordinate of $B_q$. Its boundary $\delta_j$, for $j\ge 1$, is given on an output coordinate with $p+q=j-1$ by

$$(\delta_j u)_{p,q,a,b} = \left(\partial^A_{p+1} \bigl(a'\mapsto u_{p+1,q,a',b}\bigr)\right)_a + \left(\partial^B_{q+1} \bigl(b'\mapsto u_{p,q+1,a,b'}\bigr)\right)_b,$$

with addition in $\mathbb F_2$. Let $e$ be the degree-$2$ word defined by

$$e_{p,q,a,b}= \begin{cases} 1,&\text{if the natural-number values of }a\text{ and }b\text{ are equal},\\ 0,&\text{otherwise}, \end{cases}$$

so its $(2,0)$, $(1,1)$, and $(0,2)$ blocks are respectively the coordinate identity matrices of sizes $r_Z$, $n$, and $r_X$. For a binary word $u$, write $\operatorname{wt}(u)$ for the number of its nonzero coordinates. For any such complex $K$, define

$$d_j(K)=\operatorname{sInf} \left\{ w\in\mathbb N\cup\{\infty\}\ \middle|\ \exists u\in K_j,\ \partial^K_j u=0,\ u\notin\operatorname{im}(\partial^K_{j+1}),\ w=\operatorname{wt}(u) \right\},$$

and define $d_2(A\otimes B)$ by the same formula using $\delta_2$ and $\delta_3$; the infimum of the empty set is $\infty$. Also define

$$m_2(A,B)=\operatorname{sInf} \left\{ d_0(A)d_2(B),\ d_1(A)d_1(B),\ d_2(A)d_0(B) \right\},$$

with the infimum and products taken in $\mathbb N\cup\{\infty\}$. The theorem asserts simultaneously that

$$\delta_2e=0,\qquad e\notin\operatorname{im}(\delta_3),\qquad \operatorname{wt}(e)=r_Z+n+r_X,$$ $$d_2(A\otimes B)\le r_Z+n+r_X, \qquad m_2(A,B)=d_1(A)d_1(B).$$

The quantification includes zero values of $n,r_X,r_Z$, in which case the corresponding coordinate sets and identity blocks are empty; it does not assert that a datum $D$ exists, so parameter choices for which the displayed hypotheses are impossible—such as $n=0$, when the required empty dot product cannot equal $1$—are covered only vacuously.