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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, eigvals\nfrom threadpoolctl import threadpool_limits\nthreadpool_limits(4)\n\ndef centered_rows(a):\n    a = np.asarray(a, dtype=float)\n    return a - a.mean(axis=1, keepdims=True)\n\nA,C,Q,K,V,O = {},{},{},{},{},{}\nfor l in range(4):\n    g = np.asarray(W[f\"ln1_w.{l}\"],dtype=float)\n    for j in range(6):\n        s = slice(j*64,(j+1)*64)\n        n = f\"L{l}H{j}\"\n        Q[n]=centered_rows(W[f\"W_q.{l}\"][s]*g)\n        K[n]=centered_rows(W[f\"W_k.{l}\"][s]*g)\n        V[n]=centered_rows(W[f\"W_v.{l}\"][s]*g)\n        O[n]=np.asarray(W[f\"W_o.{l}\"][:,s],dtype=float)\n        O[n]=O[n]-O[n].mean(axis=0,keepdims=True)\n        A[n]=Q[n].T@K[n]/8\n        C[n]=O[n]@V[n]\n\nprint(\"INDUCTION ALGEBRA: trace(A_l C_earlier)/(||A_l||F ||C_earlier||F)\")\nprint(\"Positive values indicate a possible query-to-preceding-token key match, not measured attention.\")\nfor l in range(1,4):\n    earlier=[f\"L{a}H{b}\" for a in range(l) for b in range(6)]\n    print(\"columns\",earlier)\n    for j in range(6):\n        n=f\"L{l}H{j}\"\n        scores=[np.trace(A[n]@C[e])/np.linalg.norm(A[n])/np.linalg.norm(C[e]) for e in earlier]\n        print(n,np.round(scores,4))\n\nprint(\"\\nL0 effective Q,K,V singular values (first 32):\")\nfor j in range(6):\n    n=f\"L0H{j}\"\n    for label,M in [(\"Q\",Q[n]),(\"K\",K[n]),(\"V\",V[n])]:\n        s=np.linalg.svd(M,compute_uv=False)\n        print(n,label,np.round(s[:32],3))\n\nbases={}\nfor j in range(6):\n    for label,D in [(\"Q\",Q),(\"K\",K),(\"V\",V)]:\n        _,_,vt=svd(D[f\"L0H{j}\"],full_matrices=False)\n        bases[f\"{label}{j}\"]=vt[:24].T\nprint(\"\\nAverage squared overlap of leading 24-dimensional L0 subspaces:\")\nbn=[\"Q1\",\"K1\",\"Q5\",\"K5\",\"V1\",\"V5\"]\nprint(bn)\nprint(np.round([[np.linalg.norm(bases[x].T@bases[y])**2/24 for y in bn] for x in bn],3))\n\nprint(\"\\nLayer 0 QK product correlations, centered effective matrices:\")\na=A[\"L0H5\"]\nb=A[\"L0H1\"]\nfor k in range(1,7):\n    m=np.linalg.matrix_power(a,k)\n    print(\"power\",k,\"cosine with H1\",round(np.sum(m*b)/np.linalg.norm(m)/np.linalg.norm(b),4),\n          \"cosine with H1 transpose\",round(np.sum(m*b.T)/np.linalg.norm(m)/np.linalg.norm(b),4))\n\nprint(\"\\nTop-key and top-query directions of each head: overlap with L0H4 output subspace (top 4).\")\nu,s,vt=svd(C[\"L0H4\"],full_matrices=False)\nB=u[:,:4]\nfor l in range(4):\n    for j in range(6):\n        n=f\"L{l}H{j}\"\n        _,_,vq=svd(Q[n],full_matrices=False)\n        _,_,vk=svd(K[n],full_matrices=False)\n        print(n,\"Q\",np.round(np.sum((vq[:4]@B)**2,axis=1),3),\n              \"K\",np.round(np.sum((vk[:4]@B)**2,axis=1),3))\n```\n"}}
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