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{"type":"item.completed","item":{"id":"item_1","type":"agent_message","text":"```python\nimport numpy as np\nimport scipy.linalg as la\nfrom threadpoolctl import threadpool_limits\nthreadpool_limits(4)\n\nd=W[\"ln1_w.0\"].astype(float)\nA,B={},{}\nfor j in range(6):\n    sl=slice(64*j,64*(j+1))\n    q=W[\"W_q.0\"][sl].astype(float)*d\n    k=W[\"W_k.0\"][sl].astype(float)*d\n    v=W[\"W_v.0\"][sl].astype(float)*d\n    o=W[\"W_o.0\"][:,sl].astype(float)\n    A[j]=q.T@k\n    B[j]=o@v\n\n# Compare weight matrices with powers of the orthogonal polar factor\n# of L0H2's QK matrix. These are matrix alignments, not attention scores.\nq=W[\"W_q.0\"][128:192].astype(float)*d\nk=W[\"W_k.0\"][128:192].astype(float)*d\nc=q.T@q+k.T@k\nev,U=la.eigh(c)\nix=np.argsort(ev)[::-1]\nev,U=ev[ix],U[:,ix]\nfor r in [16,20,24,32]:\n    Z=U[:,:r]/np.sqrt(ev[:r])[None,:]\n    t=Z.T@A[2]@Z\n    rot,_=la.polar(t)\n    print(\"\\nrank\",r,\"whitened L0H2 singular values\",np.round(la.svdvals(t),3))\n    print(\"Alignment with polar-factor powers, columns -4..4:\")\n    for j in [0,1,3,4,5]:\n        m=Z.T@A[j]@Z\n        vals=[np.sum(m*np.linalg.matrix_power(rot,p))/\n              (la.norm(m)*np.sqrt(r)) for p in range(-4,5)]\n        print(f\"L0H{j}\",np.round(vals,4))\n\nprint(\"\\nShared first-layer read subspace and positional-circuit overlap\")\nstack=np.concatenate([B[j] for j in [0,1,4,5]],axis=0)\n_,s,vt=la.svd(stack,full_matrices=False)\nprint(\"Shared-read singular values\",np.round(s[:10],4))\na_u,a_s,a_vt=la.svd(A[2],full_matrices=False)\nP=la.orth(np.column_stack([a_u[:,:20],a_vt[:20].T]))\nfor i in range(4):\n    v=vt[i]\n    print(\"read direction\",i,\"energy in L0H2 QK subspace\",\n          round(la.norm(P.T@v)**2,4))\n\n# Check the induction-like composition without choosing one particular rank.\na=(W[\"W_q.3\"][64:128].astype(float)*W[\"ln1_w.3\"]).T @ (\n    W[\"W_k.3\"][64:128].astype(float)*W[\"ln1_w.3\"])\n_,_,vt=la.svd(B[2],full_matrices=False)\nprint(\"\\nL3H1 x L0H2 induction-like composition, subspace-rank sensitivity\")\nfor r in [16,20,24,27,32,40,64]:\n    T=vt[:r].T\n    m=T.T@a@B[2]@T\n    e=la.eigvalsh((m+m.T)/2)\n    print(r,\"trace\",round(np.trace(m),4),\n          \"positive_mass\",round(e[e>0].sum(),4),\n          \"negative_mass\",round(-e[e<0].sum(),4),\n          \"symmetry_fraction\",round(la.norm((m+m.T)/2)/la.norm(m),4))\n```\n"}}
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