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{"type":"item.completed","item":{"id":"item_2","type":"agent_message","text":"```python\nimport numpy as np\nfrom scipy.linalg import svd, eig\nfrom scipy.stats import skew\nfrom sklearn.decomposition import FastICA\nfrom threadpoolctl import threadpool_limits\nthreadpool_limits(limits=2)\n\nd=384\nP=np.eye(d)-np.ones((d,d))/d\nrs=[]\nfor h in [1,5]:\n    _,_,r=svd(np.asarray(W[\"W_v.0\"][64*h:64*(h+1)],float)@P,full_matrices=False)\n    rs.append(r[:28])\nev,U=np.linalg.eigh(sum(r.T@r for r in rs))\nU=U[:,np.argsort(ev)[::-1][:28]]\nF=np.asarray(W[\"W_fc.0\"],float)*W[\"ln2_w.0\"][None,:]\nica=FastICA(n_components=28,whiten=\"unit-variance\",random_state=17,max_iter=4000,tol=.001)\nZ=ica.fit_transform(F@U)\nsg=np.where(skew(Z,axis=0)>=0,1.,-1.)\nT=U@np.linalg.pinv((ica.mixing_*sg[None,:]).T)\nT/=np.linalg.norm(T,axis=0,keepdims=True)\n\nprint(\"L0 character-subspace weight energies by unlabeled ICA axis\")\nfor h in range(6):\n    s=slice(64*h,64*(h+1))\n    q,k,v=[np.asarray(W[f\"W_{n}.0\"][s],float) for n in [\"q\",\"k\",\"v\"]]\n    a=q.T@k\n    vals=[]\n    for name,z in [(\"query\",np.linalg.norm(T.T@a,axis=1)),\n                   (\"key\",np.linalg.norm(a@T,axis=0)),\n                   (\"value\",np.linalg.norm(v@T,axis=0))]:\n        ix=np.argsort(z)[::-1][:7]\n        vals.append((name,[(int(i),round(z[i],3)) for i in ix]))\n    print(h,vals)\n\nprint(\"\\nL0H4 top value-input singular vectors: overlap with content and QK5 row spaces\")\nv=np.asarray(W[\"W_v.0\"][256:320],float)@P\n_,ss,vt=svd(v,full_matrices=False)\nq=np.asarray(W[\"W_q.0\"][320:384],float)@P\nk=np.asarray(W[\"W_k.0\"][320:384],float)@P\n_,_,pos=svd(np.concatenate([q,k]),full_matrices=False)\nfor j in range(6):\n    print(j,round(ss[j],3),round(np.linalg.norm(U.T@vt[j])**2,3),\n          round(np.linalg.norm(pos[:40]@vt[j])**2,3))\n\nprint(\"\\nRelative rotation geometry of L0H1 and L0H5\")\nprint(\"Weight-matrix powers only; no sequence or attention computation.\")\n# Exclude the shared value-reading subspace, then whiten a Q/K Gram matrix.\nR=P-U@U.T\nq5=np.asarray(W[\"W_q.0\"][320:384],float)@R\nk5=np.asarray(W[\"W_k.0\"][320:384],float)@R\ngram=(q5.T@q5+k5.T@k5)/2\nee,V=np.linalg.eigh(gram)\nix=np.argsort(ee)[::-1]\nee,V=ee[ix],V[:,ix]\nfor rank in [16,24,32,40]:\n    T0=V[:,:rank]/np.sqrt(ee[:rank])[None,:]\n    a5=T0.T@(q5.T@k5)@T0\n    left,_,right=svd(a5)\n    rot=left@right\n    q1=np.asarray(W[\"W_q.0\"][64:128],float)@R\n    k1=np.asarray(W[\"W_k.0\"][64:128],float)@R\n    a1=T0.T@(q1.T@k1)@T0\n    co=[]\n    for power in range(-4,7):\n        rp=np.linalg.matrix_power(rot,power)\n        co.append((power,round(np.sum(a1*rp)/(np.linalg.norm(a1)*np.linalg.norm(rp)),3)))\n    print(\"rank\",rank,\"H5 singular values\",np.round(np.linalg.svd(a5,compute_uv=False)[[0,-1]],3),\n          \"H1 versus H5 polar powers\",co)\n```"}}
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