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{"type":"item.completed","item":{"id":"item_1","type":"agent_message","text":"```python\nimport numpy as np\nimport torch\ntorch.set_num_threads(4)\ntorch.set_default_dtype(torch.float64)\nT=lambda a: torch.as_tensor(np.asarray(a),dtype=torch.float64)\nP=torch.eye(384)-torch.ones((384,384))/384\nM,B={},{}\n\nfor l in range(4):\n    g=T(W[f\"ln1_w.{l}\"])\n    for j in range(6):\n        sl=slice(64*j,64*(j+1))\n        q,k,v=[(T(W[f\"{n}.{l}\"])[sl]*g)@P for n in [\"W_q\",\"W_k\",\"W_v\"]]\n        M[l,j]=q.T@k/8\n        B[l,j]=P@T(W[f\"W_o.{l}\"])[:,sl]@v\n\nprint(\"Directional composition: key trace(M B), query trace(M B.T), each divided by ||M||||B||.\")\nprint(\"A substantially key-specific positive match is more suggestive of an induction circuit.\")\nfor l in range(1,4):\n    for j in range(6):\n        a=M[l,j]\n        rows=[]\n        for p in range(l):\n            for h in range(6):\n                b=B[p,h]\n                den=a.norm()*b.norm()\n                key=float(torch.trace(a@b)/den)\n                query=float(torch.trace(a@b.T)/den)\n                rows.append((key-query,key,query,p,h))\n        rows.sort(reverse=True)\n        print(f\"L{l}H{j}\",[(f\"L{p}H{h}\",round(k,4),round(q,4),round(d,4))\n                            for d,k,q,p,h in rows[:3]])\n\nprint(\"\\nL0 QK energy in a value-defined candidate character subspace:\")\ng=T(W[\"ln1_w.0\"])\nv1=(T(W[\"W_v.0\"])[64:128]*g)@P\nv5=(T(W[\"W_v.0\"])[320:384]*g)@P\nc=v1.T@v1+v5.T@v5\nev,u=torch.linalg.eigh(c)\nu=u.flip(1)\nfor r in [16,24,32]:\n    z=u[:,:r]\n    print(\"subspace dimension\",r)\n    for j in range(6):\n        a=M[0,j]\n        den=a.norm().square()\n        qe=float((z.T@a).norm().square()/den)\n        ke=float((a@z).norm().square()/den)\n        both=float((z.T@a@z).norm().square()/den)\n        print(j,\"query/key/both\",np.round([qe,ke,both],4))\n\nprint(\"\\nL1 attention query-bias key directions: pairwise cosine.\")\nkb=[]\nfor j in range(6):\n    sl=slice(64*j,64*(j+1))\n    q=T(W[\"W_q.1\"])[sl]\n    k=(T(W[\"W_k.1\"])[sl]*T(W[\"ln1_w.1\"]))@P\n    v=k.T@(q@T(W[\"ln1_b.1\"]))/8\n    kb.append(v/v.norm())\nprint(np.round(torch.stack(kb).numpy()@torch.stack(kb).numpy().T,3))\n```\n"}}
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