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{"type":"item.completed","item":{"id":"item_1","type":"agent_message","text":"```python\nimport numpy as np\nfrom scipy.linalg import fractional_matrix_power\nfrom threadpoolctl import threadpool_limits\nthreadpool_limits(4)\n\nfor scaled in [False,True]:\n    g=W[\"ln1_w.0\"].astype(float) if scaled else np.ones(384)\n    As=[]\n    for j in range(6):\n        s=slice(64*j,64*(j+1))\n        As.append((W[\"W_q.0\"][s]*g).T@(W[\"W_k.0\"][s]*g)/8)\n    print(\"\\nScaled\" if scaled else \"\\nUnscaled\",\"QK kernel power comparisons\")\n    ref=As[3]/np.linalg.norm(As[3],2)\n    powers=[np.eye(384)]\n    for z in range(8):\n        powers.append(powers[-1]@ref)\n    for j in range(6):\n        print(\"L0H%d cos powers 0..8\"%j,np.round([\n            np.sum(As[j]*P)/(np.linalg.norm(As[j])*np.linalg.norm(P)) for P in powers],3))\n    q=W[\"W_q.0\"][192:256].astype(float)*g\n    k=W[\"W_k.0\"][192:256].astype(float)*g\n    _,s,P=np.linalg.svd(np.vstack([q,k]),full_matrices=False)\n    for r in [16,24,32]:\n        Z=P[:r].T\n        # Whitening of a weight Gram matrix, not a network normalization.\n        H=(Z.T@(q.T@q+k.T@k)@Z)/16\n        ev,U=np.linalg.eigh(H)\n        wh=U@np.diag(1/np.sqrt(ev))@U.T\n        M=[wh@Z.T@A@Z@wh for A in As]\n        u,sv,vt=np.linalg.svd(M[3])\n        R=u@vt\n        print(\"rank\",r,\"polar power comparison\")\n        for j in [0,1,2,4,5]:\n            vals=[]\n            for lag in range(9):\n                z=np.linalg.matrix_power(R,lag)\n                vals.append(np.sum(M[j]*z)/(np.linalg.norm(M[j])*np.linalg.norm(z)))\n            print(\"H\",j,np.round(vals,3))\n```\n"}}
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