Borrowed Geometry: Cross-Distribution Head-Importance Fingerprints of Frozen Pretrained Gemma 4 31B
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Frozen Gemma 4 31B weights pretrained exclusively on text, unmodified, transfer through a thin trainable interface to non-text modalities the substrate has never processed. On the L24--L29 slice (192 attention heads), an English-text TxtCopy attention probe (95 sentences) and per-head ablation impact on four non-language token-pattern tasks (binary copy, associative recall, 1D cellular automaton Rule 90, binary addition) jointly classify four heads -- L26.28, L27.28, L27.2, L27.3 -- as top-tier on both signals. The slice-level joint coincidence is significant under hypergeometric null ($P = 0.0013$, $N=192$, $K=38$, $n=4$) and survives multiplicity-aware permutation tests ($P_{V4} = 0.013$). Pretrained Gemma L26 reaches 60.22% on OGBench cube-double-play-task1 vs ~1% for random-init Gemma ($+59$pt at $n=3$); a FrozenRandom-GPT2 control with correct $1/\sqrt{d_k}$ scaling also fails. Head-level causal validation: zeroing L26.28 in the trained cube-task1 IQL agent drops success $63.3\% \to 10.0\%$ vs $46.7\%$ for a layer-matched low-TxtCopy negative control ($3.2\times$ specificity at $n=30$; $n=5$ paired-$t$ $p=0.039$). A full L26 sweep places L26.28 at rank 4 of 32. Honest negatives: within-L26 Spearman $\rho(\text{TxtCopy, drop}) = +0.37$ (opposite of within-layer causal reading); single-head activation patching does not transfer the matching variable; the 4 named heads alone do not suffice on any task; Walker2d-DT and scene-task1 recruit L24 outside the named slice and show null head-ablation specificity. We frame the contribution as a cross-distribution importance fingerprint at the slice level plus head-level causal evidence on one cross-modality target.