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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\nA,B,Rkey = {},{},{}\nfor l in range(4):\n    d=W[f\"ln1_w.{l}\"].astype(float)\n    z=W[f\"ln1_b.{l}\"].astype(float)\n    for j in range(6):\n        sl=slice(64*j,64*(j+1))\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)\n        o=W[f\"W_o.{l}\"][:,sl].astype(float)\n        A[l,j]=(q*d).T@(k*d)\n        B[l,j]=o@(v*d)\n        Rkey[l,j]=(k*d).T@(q@z)\n\n# Common write subspace of the four similar first-layer OV matrices.\ncommon=np.concatenate([B[0,j] for j in [0,1,4,5]],axis=1)\nu,s,vt=la.svd(common,full_matrices=False)\nprint(\"Shared L0H0/H1/H4/H5 OV singular values:\",np.round(s[:20],4))\nclock=u[:,:2]\nprint(\"Fraction of each L0 OV's squared norm in shared top-2 write subspace:\")\nfor j in range(6):\n    print(j,round(la.norm(clock.T@B[0,j])**2/la.norm(B[0,j])**2,4))\n\nprint(\"\\nLater QK energy on shared top-2 write subspace and bias-key alignment\")\nprint(\"head: query_fraction key_fraction key_bias_fraction key_bias_coordinates\")\nfor l in range(1,4):\n    for j in range(6):\n        a=A[l,j]\n        r=Rkey[l,j]\n        print(f\"L{l}H{j}\",\n              np.round([la.norm(clock.T@a)**2/la.norm(a)**2,\n                        la.norm(a@clock)**2/la.norm(a)**2,\n                        la.norm(clock.T@r)**2/la.norm(r)**2],4),\n              np.round(clock.T@r,3))\n\n# A character-channel candidate: right singular subspace of L0H2 OV.\nu2,s2,vt2=la.svd(B[0,2],full_matrices=False)\nprint(\"\\nL0H2 gain-weighted OV singular values:\",np.round(s2[:40],4))\nT=vt2[:27].T\nprint(\"Projected compositions on L0H2's top-27 read subspace:\")\nprint(\"head: trace(A Bprev) projected_trace symmetric_fraction positive/negative_sym_eigenmass\")\nfor l in [1,2,3]:\n    for j in range(6):\n        a=A[l,j]\n        m=T.T@a@B[0,2]@T\n        sym=(m+m.T)/2\n        es=la.eigvalsh(sym)\n        print(f\"L{l}H{j}\",\n              np.round([np.trace(a@B[0,2]),np.trace(m),\n                        la.norm(sym)/la.norm(m),\n                        es[es>0].sum(),-es[es<0].sum()],4))\n\nprint(\"\\nUnweighted L0H2 V singular values:\")\nprint(np.round(la.svdvals(W[\"W_v.0\"][128:192].astype(float)),4))\n```\n"}}
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