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{"type":"item.completed","item":{"id":"item_1","type":"agent_message","text":"```python\nimport numpy as np\nimport torch\nfrom scipy.linalg import eig\ntorch.set_num_threads(4)\ntorch.set_default_dtype(torch.float64)\nT=lambda a: torch.as_tensor(np.asarray(a),dtype=torch.float64)\nP=torch.eye(384)-torch.ones((384,384))/384\n\nprint(\"Final MLP output weights after linear regression on input weights.\")\nprint(\"This is weight-space regression only, not evaluation of an MLP.\")\nf=T(W[\"W_fc.3\"])\no=P@T(W[\"W_proj.3\"])\nalpha=-float((o*f.T).sum()/f.square().sum())\nprint(\"Best scalar cancellation coefficient\",round(alpha,5))\nu,s,vh=torch.linalg.svd(f,full_matrices=False)\nfor method in [\"scalar\",\"regress\"]:\n    if method==\"scalar\":\n        r=o+alpha*(P@f.T)\n    else:\n        r=o-(o@u)@u.T\n    sv=torch.linalg.svdvals(r)\n    print(method,\"first 55 singular values\",np.round(sv[:55].numpy(),3))\n    ratios=sv[:-1]/sv[1:]\n    ids=torch.argsort(ratios[15:45],descending=True)[:8]+15\n    print(\"gaps\",[(int(i)+1,round(float(ratios[i]),3)) for i in ids])\n\ng=T(W[\"ln1_w.0\"])\nm={}\nfor j in [1,5]:\n    sl=slice(64*j,64*(j+1))\n    q=(T(W[\"W_q.0\"])[sl]*g)@P\n    k=(T(W[\"W_k.0\"])[sl]*g)@P\n    m[j]=q.T@k/8\nc=sum(a@a.T+a.T@a for a in m.values())\nee,uu=torch.linalg.eigh(c)\nuu=uu.flip(1)\n\nprint(\"\\nShared QK subspace: congruence-invariant phase diagnostics.\")\nfor r in [8,12,16,20,24,28]:\n    z=uu[:,:r]\n    a=(z.T@m[1]@z).numpy()\n    b=(z.T@m[5]@z).numpy()\n    print(\"rank\",r)\n    for label,x,y in [(\"H1 versus transpose\",a,a.T),\n                       (\"H5 versus transpose\",b,b.T),\n                       (\"H1 versus H5\",a,b)]:\n        ev=eig(x,y,right=False)\n        good=ev[np.isfinite(ev)]\n        positive=good[good.imag>1e-5]\n        positive=positive[np.argsort(np.angle(positive))]\n        print(label,[(round(float(np.angle(v)),3),round(float(abs(v)),3)) for v in positive])\n```\n"}}
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