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{"type":"item.completed","item":{"id":"item_1","type":"agent_message","text":"```python\nimport numpy as np\nfrom scipy.linalg import svd\nfrom threadpoolctl import threadpool_limits\nthreadpool_limits(4)\n\ng=W[\"ln1_w.0\"].astype(float)\nv=np.concatenate([W[\"W_v.0\"][64:128],W[\"W_v.0\"][320:384]]).astype(float)\np=np.concatenate([W[n][s] for n in [\"W_q.0\",\"W_k.0\"]\n                  for s in [slice(64,128),slice(320,384)]]).astype(float)\n\ndef basis_from_gradient_rows(m,r):\n    _,_,vt=svd(m,full_matrices=False)\n    b=vt[:r].T/g[:,None]\n    b-=b.mean(axis=0,keepdims=True)\n    return np.linalg.qr(b)[0]\n\nBc=basis_from_gradient_rows(v,28)\nBp=basis_from_gradient_rows(p,32)\nprint(\"Mean squared character/position candidate subspace overlap:\",\n      np.linalg.norm(Bc.T@Bp)**2/28)\nprint(\"Energy fractions in candidate character and position input subspaces:\")\nfor j in range(6):\n    sl=slice(64*j,64*(j+1))\n    for name in [\"W_q\",\"W_k\",\"W_v\"]:\n        m=W[f\"{name}.0\"][sl].astype(float)*g\n        m-=m.mean(axis=1,keepdims=True)\n        print(f\"L0H{j}\",name,\n              np.round([np.linalg.norm(m@Bc)**2/np.linalg.norm(m)**2,\n                        np.linalg.norm(m@Bp)**2/np.linalg.norm(m)**2],4))\n    c=W[\"W_o.0\"][:,sl].astype(float)@(W[\"W_v.0\"][sl].astype(float)*g)\n    u,s,vt=svd(c,full_matrices=False)\n    print(\" OV top four right-singular-vector character / position overlaps:\",\n          np.round(np.sum((vt[:4]@Bc)**2,axis=1),3),\n          np.round(np.sum((vt[:4]@Bp)**2,axis=1),3))\n\nprint(\"\\nKey-bias weight vectors across heads:\")\nkeybias={}\nCs={}\nfor l in range(4):\n    gg=W[f\"ln1_w.{l}\"].astype(float)\n    bb=W[f\"ln1_b.{l}\"].astype(float)\n    for j in range(6):\n        sl=slice(64*j,64*(j+1))\n        q=W[f\"W_q.{l}\"][sl].astype(float)\n        k=W[f\"W_k.{l}\"][sl].astype(float)*gg\n        k-=k.mean(axis=1,keepdims=True)\n        keybias[(l,j)]=k.T@(q@bb)/8\n        c=W[f\"W_o.{l}\"][:,sl].astype(float)@(W[f\"W_v.{l}\"][sl].astype(float)*gg)\n        c-=c.mean(axis=0,keepdims=True)\n        c-=c.mean(axis=1,keepdims=True)\n        Cs[(l,j)]=c\n\nreference=sum(keybias[(1,j)]/np.linalg.norm(keybias[(1,j)]) for j in range(6))\nreference/=np.linalg.norm(reference)\nfor l in range(4):\n    for j in range(6):\n        z=keybias[(l,j)]\n        print(f\"L{l}H{j}\",\"bias norm\",round(np.linalg.norm(z),4),\n              \"cosine with common L1 bias\",round(z@reference/np.linalg.norm(z),4))\n\nprint(\"\\nCopy-path trace concentration on the candidate character subspace:\")\nfor target,source in [((2,1),(0,5)),((3,1),(0,5)),((3,5),(0,5)),\n                      ((3,1),(2,2)),((3,5),(2,2))]:\n    l,j=target\n    sl=slice(64*j,64*(j+1))\n    gg=W[f\"ln1_w.{l}\"].astype(float)\n    q=W[f\"W_q.{l}\"][sl].astype(float)*gg\n    k=W[f\"W_k.{l}\"][sl].astype(float)*gg\n    q-=q.mean(axis=1,keepdims=True)\n    k-=k.mean(axis=1,keepdims=True)\n    m=(q.T@k/8)@Cs[source]\n    mc=Bc.T@m@Bc\n    z=np.linalg.eigvals(mc)\n    print(target,\"via\",source,\"total trace\",round(np.trace(m),4),\n          \"char-subspace trace\",round(np.trace(mc),4),\n          \"char trace/nuclear norm\",round(np.trace(mc)/np.linalg.svd(mc,compute_uv=False).sum(),4),\n          \"top real parts\",np.round(np.sort(z.real)[-8:][::-1],4))\n```\n"}}
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