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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, eigh\nfrom threadpoolctl import threadpool_limits\n\ndef orth(a, tol=1e-8):\n    u,s,_=svd(a,full_matrices=False)\n    return u[:,s>s[0]*tol]\n\nwith threadpool_limits(limits=4):\n    A,B,C={},{},{}\n    for l in range(4):\n        g=W[f\"ln1_w.{l}\"].astype(float)\n        b=W[f\"ln1_b.{l}\"].astype(float)\n        for j in range(6):\n            sl=slice(j*64,(j+1)*64)\n            q=W[f\"W_q.{l}\"][sl].astype(float)\n            k=W[f\"W_k.{l}\"][sl].astype(float)\n            v=W[f\"W_v.{l}\"][sl].astype(float)*g\n            qg=q*g; kg=k*g\n            qg-=qg.mean(1,keepdims=True)\n            kg-=kg.mean(1,keepdims=True)\n            v-=v.mean(1,keepdims=True)\n            A[l,j]=qg.T@kg/8\n            B[l,j]=W[f\"W_o.{l}\"][:,sl].astype(float)@v\n            C[l,j]=(q@b)@kg/8\n\n    up,sp,vp=svd(A[0,3],full_matrices=False)\n    ut,st,vt=svd(B[0,3],full_matrices=False)\n    pos=orth(np.column_stack([up[:,:24],vp[:24].T]))\n    tok=orth((np.eye(384)-pos@pos.T)@vt[:20].T)\n    writes=orth(np.column_stack([svd(B[0,j],full_matrices=False)[0][:,:3]\n                                for j in (1,2,4,5)]))\n    print(\"Basis dimensions: position candidate, token candidate, low-rank L0 writes\",\n          pos.shape[1],tok.shape[1],writes.shape[1])\n    print(\"QK squared-energy fractions in query/key subspaces; then bias-key fraction\")\n    for l in range(4):\n        for j in range(6):\n            a=A[l,j]; c=C[l,j]\n            vals=[]\n            for basis in (pos,tok,writes):\n                vals.extend([np.linalg.norm(basis.T@a)**2/np.linalg.norm(a)**2,\n                             np.linalg.norm(a@basis)**2/np.linalg.norm(a)**2,\n                             np.linalg.norm(c@basis)**2/(np.linalg.norm(c)**2+1e-30)])\n            print(f\"L{l}H{j}\",np.round(vals,3))\n\n    print(\"\\nLayer 1 and 2 key-bias coefficient cosines\")\n    cs=np.array([C[l,j]/np.linalg.norm(C[l,j]) for l in (1,2) for j in range(6)])\n    print(np.round(cs@cs.T,3))\n    print(\"Largest coordinates of mean normalized L1 key-bias coefficient\")\n    c=cs[:6].mean(0)\n    ix=np.argsort(abs(c))[-16:][::-1]\n    print(list(zip(ix.tolist(),np.round(c[ix],4).tolist())))\n\n    print(\"\\nPosition-subspace operator correlations with powers of L0H3\")\n    p=pos\n    a=p.T@A[0,3]@p\n    metric=(a@a.T+a.T@a)/2\n    ev,evc=eigh(metric)\n    keep=ev>ev.max()*0.01\n    z=evc[:,keep]/ev[keep]**0.25\n    a=z.T@a@z\n    powers=[np.eye(a.shape[0])]\n    for n in range(1,7):\n        powers.append(powers[-1]@a)\n    for j in range(6):\n        aj=z.T@(p.T@A[0,j]@p)@z\n        scores=[np.vdot(aj,x)/(np.linalg.norm(aj)*np.linalg.norm(x)) for x in powers]\n        print(\"H\",j,np.round(scores,3))\n\n    print(\"\\nMLP0 coefficient coupling in H3 read subspace (generalized eigenvalues)\")\n    f=W[\"W_fc.0\"].astype(float)*W[\"ln2_w.0\"].astype(float)\n    f-=f.mean(1,keepdims=True)\n    for r in (16,20,24,28):\n        basis=vt[:r].T\n        f0=f@basis\n        f1=f@B[0,3]@basis\n        for centered in (False,True):\n            x=f0-f0.mean(0) if centered else f0\n            y=f1-f1.mean(0) if centered else f1\n            reg=np.linalg.solve(x.T@x+0.01*np.eye(r),x.T@y)\n            ev=np.linalg.eigvals(reg)\n            ev=ev[np.argsort(abs(ev))[::-1]][:8]\n            print(\"r\",r,\"centered\",centered,np.round(ev,3))\n```\n"}}
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