\section{Robustness and Failure Modes}\label{sec:robustness}

All numbers in this section are \emph{exploratory} (protocol v2 run \url{graft_tier0_30e3644aa801},
matched seed-0 pair unless stated); they motivated the frozen protocol and are not re-tested on the
confirmatory items except where \S\ref{sec:results} says so.

\begin{table}[t]\centering\scriptsize
\caption{Exploratory run \url{graft_tier0_30e3644aa801} (protocol v2): target- and control-stratum
accuracy by condition, mean (sd) over same-initialization and different-initialization pairs, and the
control-adjusted effect against the sham map.}
\label{tab:main}
\resizebox{\textwidth}{!}{\input{tables/tab_main_effects}}
\end{table}

\begin{table}[t]\centering\scriptsize
\caption{Architectural mismatch and stride-2 transfer (exploratory run): target accuracy, $\Omega$-step
accuracy, control accuracy and in-sample $R^2$ of the map; \url{unrelated_procrustes} = Procrustes fitted
on task-free rows; \texttt{sham} = random orthogonal map.}
\label{tab:mismatch}
\resizebox{\textwidth}{!}{\input{tables/tab_mismatch}}
\end{table}

\paragraph{Layer and dose.} With the task-free Procrustes map, grafting the donor's layer-1, -2 or -3 residual
row into \emph{any} recipient layer restores target accuracy $1.00$; the donor's layer-0 row restores $0.98$ at
recipient layer 0 and $0.75$--$0.82$ at deeper recipient layers (Figure~\ref{fig:layerdose}, left). The
dose--response is monotone up to $\alpha = 1$ (target $0.11 \to 0.44 \to 0.90 \to 0.98$ for
$\alpha = 0, 0.5, 0.75, 1$), then collapses together with the control stratum ($0.39$ / $0.36$ at
$\alpha = 2$; Figure~\ref{fig:layerdose}, right): the graft is a replacement, not a direction to amplify.

\paragraph{Dimensionality.} Restricting the replacement to the top-$k$ principal directions of the mapped
$\Omega$-contrast gives $0.23, 0.30, 0.65, 0.94, 0.97, 0.98$ for $k = 1, 2, 4, 8, 16, 32$ (control $1.00$),
versus $0.08$--$0.17$ for random subspaces of the same $k$ up to 32 (Figure~\ref{fig:subspace}): eight of
128 dimensions carry the capability.

\paragraph{Architectural mismatch.} Width 128 $\to$ 96 and depth 4 $\to$ 6: the task-free Procrustes graft
reaches $0.886$ (control $1.00$) and $0.765$ ($0.957$) at layer 0 $\to$ 0, and $1.000$ ($1.00$) at layer
1 $\to$ 1, 1 $\to$ 2 and 2 $\to$ 2; sham maps give $0.07$--$0.11$ with the control stratum destroyed
(Table~\ref{tab:mismatch}). The stride-2 pair (two operators per emitted value) transfers at layers 2 and 3
($1.000$, control $1.00$) and partially at layers 0--1 ($0.37$, $0.75$).

\paragraph{Failure modes.}
\emph{Regression-fitted maps.} Ridge and CCA maps fitted on $\Omega$ rows give $0.17$ and $0.08$
(Table~\ref{tab:main}); fitted on $\Omega$ rows restricted to six operand values they do \emph{better} on
unseen operands ($0.74$) than on seen ones ($0.47$), the signature of the annihilation mechanism of
Proposition~\ref{prop:annihilation}. \emph{Identity graft.} $0.87$ for same-initialization pairs, $0.10$ and
control destroyed for different-initialization pairs: representations of differently initialized models are
not aligned coordinate-wise (cf.\ permutation symmetries, \citealp{ainsworth2022git}). \emph{Steering
vectors.} The mapped mean-difference direction transfers nothing at any dose ($\leq 0.10$) and harms the
control stratum at $\alpha = 4$. \emph{Weight grafts.} Copying the donor's layer-0 MLP transfers a little
($0.26$, control $0.96$); copying the full block or layers 0--2 plus the $\Omega$ embedding transfers more
($0.64$--$0.77$) at a large cost on the control stratum ($0.73$--$0.75$); copying only the $\Omega$ embedding
does nothing; shuffled-layer weights destroy both strata; unrelated-donor weights leave both near baseline.
\emph{Donor-free adapter.} $0.58$ in-run and $0.41$ in a fresh process on new items (base $0.14$), but zero
on unseen operand values: a cache of the alignment rows, not the rule. \emph{Over-dosing} and \emph{wrong
layers} fail gracefully for deeper donor layers and sharply for shallow ones.