Proposition 2: the disjoint-position certificate reveals the identifier XOR

For every choice of natural numbers $m,k,a,b$ satisfying $1\leq k\leq m$, $1\leq a$, and $1\leq b$, let the cells be indexed by $I=\{0,\ldots,m-1\}$, let a word be a pair in $W=(\mathbb Z/2\mathbb Z)^a\times(\mathbb Z/2\mathbb Z)^b$, and let an identifier be a vector in $(\mathbb Z/2\mathbb Z)^b$; all vector and word sums below use coordinatewise addition modulo $2$. Consider any finite ordered list of attempts, each consisting of a row $(S,u,v)$, where $S\subseteq I$ has exactly $k$ elements, $u\in(\mathbb Z/2\mathbb Z)^a$, and $v\in(\mathbb Z/2\mathbb Z)^b$, together with either no pivot or a pivot $p\in I$, and an arbitrary function $f:I\to W$. Start with every cell empty and process the list in order, with a snapshot recording either emptiness or one word at each cell. Assume the following conditions hold at every attempt, relative to the snapshot $A$ immediately before that attempt: if there is no pivot, every cell of $S$ is already occupied, and the snapshot is left unchanged; if a pivot $p$ is supplied, then $p\in S$ and $p$ is empty. In the latter case, define $w_i$ to be the word already stored at $i$ if $i$ is occupied in $A$, and $f(i)$ otherwise; the next snapshot leaves cells outside $S$ unchanged, stores $w_i$ at each $i\in S\setminus\{p\}$, and stores $(u,v)+\sum_{j\in S\setminus\{p\}}w_j$ at $p$. Let $s$ be the number of attempts in the list that supply a pivot, and let $A_{\mathrm{fin}}$ be the resulting final snapshot. If $s\geq1$ and the number of occupied cells in $A_{\mathrm{fin}}$ is exactly $sk$, then the second component of the sum of all words stored in the occupied cells of $A_{\mathrm{fin}}$ equals the sum of the identifiers $v$ of all attempts that supply a pivot, with each such occurrence in the list counted separately. Empty sums are zero, so the pivot formula also applies when $k=1$. Empty lists and lists with no supplied pivots are excluded by $s\geq1$, and zero cell counts, zero share counts, and zero vector lengths are excluded by the parameter inequalities; the occupancy equality cannot hold if $sk>m$. A stored zero word still counts as occupied. The vectors $u,v$ and the functions $f$ have no further restrictions, and repeated rows or identifiers are not separately forbidden: the conditions on the successive snapshots and the final occupancy count are the imposed hypotheses.