spb/wpai0001_uqo

1\appendix2\section{Proofs and formal statements}\label{app:theory}34\paragraph{Proof of Proposition~\ref{prop:annihilation}.}5Write $X = X_c + X_u$ with $X_u = X P_U$ and $X_c = X (I - P_U)$. The hypotheses $X_u^{\top} X_c = 0$ and6$X_u^{\top} Y = 0$ give, in a basis adapted to $U \oplus U^{\perp}$,7$X^{\top} X = \operatorname{diag}(X_u^{\top} X_u,\, X_c^{\top} X_c)$ and8$X^{\top} Y = \big(0;\ X_c^{\top} Y\big)$. Hence9$W_\lambda = (X^{\top} X + \lambda I)^{-1} X^{\top} Y = \big(0;\ (X_c^{\top} X_c + \lambda I)^{-1} X_c^{\top} Y\big)$,10whose rows indexed by $U$ vanish: $P_U W_\lambda = 0$. For a donor activation $x = x_c + x_u$,11$x W_\lambda = x_c W_\lambda$, so the $U$-component is annihilated. For CCA, the canonical correlations between12$U$-coordinates and $Y$ are zero by $X_u^{\top} Y = 0$, so the top-$k$ canonical directions of $X$ lie in13$U^{\perp}$ (for any $k$ smaller than the number of nonzero canonical correlations), and the regression of14$Y$ on those variates again has no $U$-component. For an orthogonal $Q$, $\|x_u Q\| = \|x_u\|$ by isometry;15with an isotropic scale $s>0$, $\|s\, x_u Q\| = s\|x_u\|$. \hfill$\square$1617\paragraph{Remark.} In the experiments the hypotheses hold only approximately: the recipient's residual18rows on $\Omega$ steps do contain the untrained $\Omega$-token embedding and the operand value, so the19donor's $\Omega$-specific directions are weakly correlated with recipient coordinates. The proposition then20predicts strong shrinkage rather than exact annihilation, which is what the ridge and CCA grafts show21(\S\ref{sec:results}).2223\paragraph{Interchange commutation under exact causal abstraction (restated).}24Let $\mathcal{L}$ be a high-level causal model that is an exact abstraction of the donor's computation at a25residual-stream variable $h$ under alignment $\tau_D$, and of the recipient's at $h'$ under $\tau_R$26\citep{geiger2023causal}. If a bijection $A$ satisfies $\tau_R \circ A = \tau_D$ on the support of $h$, then27for every interchange intervention on $h$ in the donor there is an interchange intervention on $h'$ in the28recipient (through $A$) with the same high-level effect; in particular, grafting $A(h)$ into the recipient29reproduces the donor's high-level behaviour on the abstracted variable. The condition30$\tau_R \circ A = \tau_D$ is precisely what is \emph{not} guaranteed by representational similarity and must be31tested; the experiments in \S\ref{sec:results} estimate how far from it the fitted maps are.3233\section{Configurations, splits and model cards}\label{app:configs}34Tables of split hashes, model specifications and run identifiers are generated from the run manifests35(\texttt{experiments/manifests/}) by \url{scripts/build_tables.py}; the registry36(\url{experiments/registry/runs.jsonl}) lists every planned, running, failed and completed run.3738\begin{table}[h]\centering\scriptsize39\caption{Tier-0 model matrix (\texttt{artifacts/models/matrix\_v1/index.json}): architecture, excluded and40extra operators, trace stride, final dev accuracy and training-split hash. Models with the same seed suffix41share their initialization.}42\label{tab:models}43\resizebox{\textwidth}{!}{\input{tables/tab_models}}44\end{table}4546\begin{table}[h]\centering\small47\caption{Alignment maps (exploratory run): parameter counts and in-sample $R^2$ on $\Omega$ rows for48same- and different-initialization pairs. High $R^2$ does not predict transfer (Table~\ref{tab:main}).}49\label{tab:maps}50\input{tables/tab_maps}51\end{table}5253\begin{table}[h]\centering\scriptsize54\caption{Mechanism and rule-versus-cache tests (exploratory run).}55\label{tab:mechanismtab}56\resizebox{\textwidth}{!}{\input{tables/tab_mechanism}}57\end{table}5859\section{Tier-1 prompts and tasks}\label{app:tier1}60Few-shot prompt: $k{=}8$ demonstration blocks followed by the query block; zero-shot prompt: the query block only.61\begin{verbatim}62Input: <x_1>63Output: <y_1>6465...6667Input: <query>68Output:69\end{verbatim}70Alignment prompts: the same frame with 0--3 random demonstration pairs whose outputs are random and a random71query (sentences, words, numbers, names), none of which encodes a task. Task item lists are in72\texttt{src/wp\_ai\_0001/datasets/tier1\_tasks.py} with their hashes in the run manifests.73