\section{Mechanistic Analysis}\label{sec:mechanism}

\paragraph{Where the donor computes $\Omega$.} The pilot localization (\S\ref{sec:results}) shows that
patching the residual row at the read position restores the answer at every layer, that layer-0 attention
(mostly head 0) moves the operator identity from the $\Omega$ token to the read position, and that layer-0
to layer-2 MLPs compute the value map. The candidate organ is therefore the post-layer-0 residual row at the
read position, whose content is the (operator, operand) pair.

\paragraph{Necessity, loss and rescue.} Zeroing the donor's layer-0 attention output at read positions
reduces donor accuracy to $0.086$ on the target stratum and $0.102$ on the control stratum (all operators
pass through the same component). Grafting the post-layer-0 row of an \emph{intact donor of another seed}
through the task-free Procrustes map into the ablated donor restores $1.000$ on both strata; the same graft
through a sham map restores nothing ($0.115$ / $0.101$; Figure~\ref{fig:mechanism}, left). The reverse graft,
recipient $\to$ donor, leaves the donor's control stratum intact ($1.00$) and the target stratum at $0.965$:
the donor's deeper MLPs can recover $\Omega$ from the recipient's row because the row still carries the
operator identity and the operand; what the recipient lacks is not the information but the map.

\paragraph{Rule, not cache.} A Procrustes map fitted only on $\Omega$ rows whose operand is in
$\{0,\dots,5\}$ yields $\Omega$-step accuracy $0.966$ on items whose operands are all seen and $0.831$ on items
whose operands are all unseen (ratio $0.86$; confirmatory values in \S\ref{sec:results}). The donor-free
adapter fitted on the same rows gives $0.765$ seen and $0.000$ unseen (Figure~\ref{fig:mechanism}, right):
the activation graft moves the donor's computation, the adapter memorizes the alignment rows.

\paragraph{Specificity.} The unrelated-skill donor (competent on \texttt{vex}, ignorant of $\Omega$), the
shuffled-item rows, the shuffled-layer rows and the norm-matched random rows all leave the target stratum at
chance; the latter three also destroy the control stratum, whereas the unrelated-skill donor and the
shuffled-layer rows do not (Table~\ref{tab:main}). Transfer therefore requires (i) a donor that has the
capability, (ii) rows computed on the same item, (iii) the right layer, and (iv) a map that preserves the
capability-specific directions.

\paragraph{What is transferred.} Eight principal directions of the mapped $\Omega$-contrast suffice
(\S\ref{sec:robustness}); the directions are those along which donor rows on $\Omega$ steps differ from the
recipient's rows on the same steps after the orthogonal map, i.e.\ the directions the recipient's
downstream MLPs read as ``apply $\Omega$ to this operand''. We do not claim that these directions are
interpretable features; we claim that they are causally sufficient, low-dimensional and rule-carrying.