\section{Experimental Design}\label{sec:design}

\begin{figure}[t]\centering
\includegraphics[width=\textwidth]{figures/fig_system_map.pdf}
\caption{System map. Top: the causal chain applied to every donor--recipient pair (localization before transfer;
alignment map fitted on task-free data; graft families and their controls; preregistered evaluation). Bottom:
the two tiers, their models, backends and hosts; the registry counts are read from the run manifests.}
\label{fig:systemmap}\end{figure}

\subsection{Tier 0: a controlled micro-world with a manufactured gap}
\paragraph{Task (OPSEQ).} A program $\langle\mathrm{bos}\rangle\, v_0\, o_1 \dots o_k\, {=}\, r_1 \dots r_k$
applies $k\in[1,5]$ operators drawn from eight invented maps on $\mathbb{Z}_{11}$ (\texttt{zib}: $x{+}1$,
\texttt{zob}: $x{+}3$, \texttt{qua}: $2x{+}1$, \texttt{lar}: $3x$, \texttt{neg}: $-x$, \texttt{flip}: $10{-}x$,
\texttt{dro}: $5x$, \texttt{mik}: $x^2$, all modulo 11) to $v_0$ and emits the execution trace $r_j$ (the value
after operator $j$). The target operator is $\Omega = \texttt{qua}$; a ninth operator \texttt{vex} ($4x{+}3$)
exists only in the vocabulary and in the training distribution of the \emph{unrelated-skill donor}. Two
adjacent operator pairs are held out from every training distribution (compositional stratum); programs of
length 6--7 form a length-OOD stratum. Every split is generated from its own seed, validated by exact
recomputation, hashed, and checked for item-level disjointness against all training, alignment and
evaluation splits involved in a run; split hashes are stored in run manifests. Design decision D2 (trace
instead of a single answer token) was taken after a single-answer variant memorized without generalizing;
its consequence---that the Tier-0 capability is a per-step map once the trace is visible---is controlled
by the \url{trace_stride} knob (stride 2 forces two operators per emitted result).

\paragraph{Models and roles.} Decoder-only transformers (4 layers, $d{=}128$, 4 heads, $0.80$\,M parameters;
pre-LN, explicit attention with per-head outputs exposed) are trained from scratch with teacher forcing on
trace tokens (3\,000 steps, batch 256, AdamW, cosine schedule; 40\,000 training items). Roles: \emph{donor}
(all eight operators), \emph{recipient} ($\Omega$ excluded), \emph{unrelated-skill donor} ($\Omega$ excluded,
\texttt{vex} included), three seeds each; a width-mismatched recipient ($d{=}96$), a depth-mismatched
recipient (6 layers), and a stride-2 donor/recipient pair. The initialization seed equals the data seed, so
donor$_i$ and recipient$_i$ share their initial weights (``same initialization'') while off-diagonal pairs do
not (D8). All models reach dev accuracy $1.000$ (Table~\ref{tab:models}).

\paragraph{Hook semantics.} Hook points expose the embedding, per-layer residual streams (pre, mid, post),
per-head attention outputs before the output projection and MLP outputs. With no registered intervention,
or when disabled, a hook returns the input tensor object itself; bitwise equality of logits is asserted in
the test suite and recorded in every run manifest.

\paragraph{Localization.} Interchange interventions on 490 clean/corrupted pairs (length-3 programs with
$\Omega$ at index 1; the corrupted twin replaces $\Omega$ by \texttt{zib}; identical teacher-forced prefix up
to $r_1$) patch the clean activation into the corrupted run at every (layer, position) for the residual,
attention and MLP families and per head; zero-ablation measures necessity; the sham baseline is the
corrupted run scored on the clean answer. Sweeps are repeated and compared across donor seeds.

\paragraph{Alignment maps.} Donor and recipient residual rows at every read position are collected on
3\,000 \emph{alignment} items that contain $\Omega$ (``task rows'') and on 3\,000 items that do not
(``task-free rows''), both disjoint from evaluation items. Maps: identity (same width only), orthogonal
Procrustes with isotropic scale, ridge regression ($\lambda = 10^{-2}$ relative), CCA regression, random
orthogonal (sham), each fitted on task rows or on task-free rows. We report the number of parameters,
in-sample $R^2$ and $R^2$ on the other row type.

\paragraph{Transplant families and controls.} Grafts act at the recipient's post-layer-0 residual stream
(\texttt{resid\_post.0}, the location identified in \S\ref{sec:results}) at every read position during greedy
decoding (``untargeted''); an oracle variant restricted to $\Omega$ steps is also reported. Families:
activation graft (donor in the loop, same token ids), subspace graft ($k \in \{1,\dots,128\}$ principal
directions of the mapped $\Omega$-contrast), steering graft (mapped mean-difference direction,
$\alpha \in \{0.5, 1, 2, 4\}$), adapter graft (donor-free affine interface fitted by ridge to predict mapped
donor rows from recipient rows), weight grafts (layer-0 attention, MLP, full block, $\Omega$ embedding,
layers 0--2 plus embedding), and a persistent graft (the adapter saved to disk and evaluated in a separate
process on new items). Controls: unmodified recipient and donor; sham map; norm-matched random rows;
shuffled-item donor rows; donor layer-3 rows (shuffled layer); unrelated-skill donor; random subspace of
equal $k$; shuffled-layer and unrelated-donor weight grafts. Conventional-learning references on the same
3\,000 alignment items: a supervised affine adapter trained with labels (same parameter count as the adapter
graft) and full-model distillation from the donor's output distributions (300 steps each). Few-shot
prompting and retrieval are not applicable to the toy models and are marked unsupported.

\paragraph{Sweeps and mechanism.} Donor-layer $\times$ recipient-layer grid; dose $\alpha \in [0, 2]$;
width and depth mismatches; stride-2; donor ablation (layer-0 attention output zeroed at read positions)
followed by rescue from an intact donor of another seed, with a sham rescue; reverse graft
(recipient $\to$ donor); rule-versus-cache test (maps and adapters fitted only on $\Omega$ inputs in
$\{0,\dots,5\}$, evaluated separately on items whose $\Omega$ inputs are all seen or all unseen).

\subsection{Tier 1: open-weight models}
\paragraph{Capability.} Performing an input$\to$output task \emph{without demonstrations}. Four tasks with
exact-match validators: antonyms (semantic), country$\to$capital (factual), an invented suffix rule on
pseudo-words (morphology, no prior), and ``add three'' on two-digit numbers (formal). Prompts use a fixed
\texttt{Input:}/\texttt{Output:} frame; few-shot prompts carry $k{=}8$ demonstrations drawn from a
demonstration pool disjoint from the queries; development and confirmatory query sets are disjoint halves of
each item pool.

\paragraph{Models.} Qwen2.5-1.5B-Instruct (28 layers, $d{=}1536$), Qwen2.5-3B-Instruct (36 layers,
$d{=}2048$) and Llama-3.2-3B-Instruct (28 layers, $d{=}3072$), bf16, served by MLX with decoder layers wrapped
in place (bitwise no-op verified; per-head outputs are not exposed by MLX and head-level interventions are
reported as unsupported for this backend). Pairs: same family/different size in both directions, and
different family/different tokenizer in both directions.

\paragraph{Candidate organ and localization.} The task vector is the mean difference between the last-token
residual state under few-shot and under zero-shot prompts (demonstration items only; queries never enter a
vector). Within-donor localization adds the vector to the donor's zero-shot run at each of nine layers
(sufficiency) and removes its direction from the few-shot run at the best layer (necessity); a random vector
of equal norm is the control.

\paragraph{Transfer and controls.} Maps are fitted between donor and recipient last-token states on 800
\emph{format-matched, task-free} prompts (same frame, random content; last 100 rows held out for $R^2$).
Conditions at the recipient's own best layer and at the proportional-depth layer: mapped vector added at
$\alpha \in \{1, 2\}$ (Procrustes, ridge, CCA), norm-rescaled variants, and a donor-in-the-loop replace graft
(the recipient's last-token state replaced by the mapped donor few-shot state for the same query). Controls:
recipient zero-shot, recipient few-shot (prompting baseline), recipient's own vector (within-model ceiling),
norm-matched random vector, unrelated-task vector, sham map, mapped donor \emph{zero-shot} state, and mapped
donor few-shot state of a \emph{shuffled} query. Collateral: next-token negative log-likelihood on held-out
task-free sentences and accuracy on the other three tasks under the transferred vector. The intervention is a
single-position modification of the prompt pass; generation continues from the modified key--value entry
(within-sequence persistence); nothing is saved.

\subsection{Statistics and preregistration}
Cluster bootstrap (2\,000 draws) over programs (Tier 0) or items (Tier 1); paired contrasts on identical
items; control-adjusted effects against the sham map; Holm correction across the pre-declared families in the
confirmatory analysis. Exploratory runs (protocol v1 and v2) selected the primary method; the preregistration
was then frozen (v1.0) and confirmatory splits generated from new seeds (\S\ref{sec:results}).