\section{Background and interpretation}\label{sec:context} The quantitative issue for ordered matrices is the dependence on the edit parameter. The row and column orders are independent: an occurrence of a pattern $F\in\{0,1\}^{s\times t}$ is a pair of increasing injections $[s]\hookrightarrow[n]$ and $[t]\hookrightarrow[n]$, and all $st$ entries, including the zeros, are prescribed. The corresponding hereditary property consists of matrices avoiding $F$ as an induced ordered submatrix. A removal bound supplies a lower bound on the number of occurrences whenever one must change a prescribed fraction of all entries to obtain a member of this hereditary property. For an individual fixed $F$, a polynomial bound means the existence of $c_F,C_F>0$ such that the right side of \eqref{eq:removal-bound} holds for \emph{every} size and \emph{every} matrix. Failure is witnessed by one fixed pattern and a sequence of arbitrarily large hosts. The quantifiers matter: choosing a different forbidden pattern as the accuracy parameter decreases would address a different question. The present family uses the same $H_4$ at every depth; the pattern is independent of $h$, $m$ and $n_h$. We normalize occurrences by $n^{2k}$, as in \eqref{eq:removal-bound}, and count increasing injections, so a copy has no additional factorial multiplicity. For a fixed $k$, using $\binom{n}{k}^2$ instead changes the density only by a factor bounded above and below by positive constants for $n\ge k$. Thus the failure of a polynomial bound is independent of these two common conventions. Alon, Ben-Eliezer and Fischer \cite{ABF} proved qualitative testing results for hereditary properties of ordered graphs and matrices. Alon and Ben-Eliezer \cite{ABE20} obtained a polynomial dependence for a \emph{hitting} formulation: a small set of entries may intersect all copies present in the original matrix. Hitting the original copies need not produce a free matrix after entry changes, because a changed zero or one can complete a new induced copy. The relevant full removal problem minimizes changes over the resulting, genuinely $F$-free matrices. This distinction is particularly significant for an induced pattern, whose constraints include prescribed zeros as well as ones. Alon and Ben-Eliezer also establish a full polynomial removal lemma for every fixed finite family of binary matrices closed under row permutations \cite[Theorem~1.5]{ABE20}, and hence, by transposition, for families closed under column permutations. The singleton $\{H_4\}$ has neither closure property, so this positive result does not apply to the present obstruction. Gishboliner and Shapira \cite[Conjecture~4.5]{GS25} formulate the full polynomial removal conjecture for a single fixed ordered binary matrix. Towsner\'s ordered-hypergraph removal lemma provides a further point of comparison at the level of qualitative removal \cite{Towsner}. Earlier work on unordered bipartite matrices and conditional regularity provided the polynomial-query comparison \cite{AFN07}; the ordering requirement removes the row-and-column permutation freedom used in that setting. The 2026 construction of \cite{OAI} gave a fixed $66\times66$ binary obstruction. Its tree-based construction showed that forcing gadgets can make edit distance persist over successive levels, while an order constraint can confine complete copies to terminal locations. Our pattern \eqref{eq:pattern} has four rows and four columns; its top-left $2\times2$ anti-diagonal marker and the two identity coupling blocks provide a smaller frame for the same kind of propagation. The sharper size is accompanied here by exact distance and copy-count identities. We isolate the roles of the order, the two signs, and terminal identification in subsequent sections. The tree-based propagation and terminal-sign identification build on \cite{OAI}. The additional results here are the explicit $4\times4$ pattern, its exact physical copy count, its optimal unrestricted repair and the formal certification of these identities. \subsection{Three quantities that must be kept separate} Let $\mathcal C_H(A)$ be the set of increasing row/column tuples realizing $H$. Define the \emph{transversal number} \[ \tau_H(A)=\min\{|S|: S\subseteq[n]\times[n],\; S\cap\operatorname{cells}(Q)\ne\varnothing \ \forall Q\in\mathcal C_H(A)\}. \] Every edit set turning $A$ into an $H$-free matrix intersects every copy originally in $A$; consequently \begin{equation}\label{eq:hitting_le} \tau_H(A)\le D_H(A). \end{equation} Equality does not follow from the definition: a hitting set has no obligation to avoid copies created by its own edits. For our host, the protected-selection argument establishes the stronger lower bound $D_{H_4}(A_h)\ge m^2$ \emph{for all resulting matrices}. The canonical repair gives a matching upper bound by an explicit final $H_4$-free matrix. Neither step is supplied merely by counting original occurrences. Indeed, the terminal classification below shows a concrete gap between the two quantities. The $m$ leaf-diagonal cells $\{(R_h(p),C_h(p)):0\le p