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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 threadpoolctl import threadpool_limits\nthreadpool_limits(limits=2)\nnp.set_printoptions(precision=3,suppress=True,linewidth=160)\n\n# A congruence-invariant matrix-pencil check for shared positional rotations.\n# No position vectors or attention patterns are constructed.\nraw=[]\nfor j in [1,5]:\n    sl=slice(64*j,64*(j+1))\n    q=W[\"W_q.0\"][sl].astype(float)\n    k=W[\"W_k.0\"][sl].astype(float)\n    raw.append(q.T@k/8)\na1,a5=raw\nu=svd(np.concatenate([a1,a1.T,a5,a5.T],axis=1),\n      full_matrices=False,check_finite=False)[0]\n\nfor r in [12,16,18,20,24,28,32]:\n    p=u[:,:r]\n    b1=p.T@a1@p\n    b5=p.T@a5@p\n    vals,vecs=eig(b5,b5.T,check_finite=False)\n    rows=[]\n    for n,lam in enumerate(vals):\n        if np.imag(lam)<=1e-5 or not np.isfinite(lam): continue\n        z=vecs[:,n]\n        c5=np.vdot(z,b5@z)\n        c1=np.vdot(z,b1@z)\n        rows.append((float(abs(lam)),float(np.angle(c5)),\n                     float(np.angle(c1)),float(abs(c5)),float(abs(c1))))\n    rows=np.array(rows)\n    print(\"\\nrank\",r,\"[pencil modulus, phase H5, phase H1, magnitude H5, magnitude H1]\")\n    print(rows)\n    if len(rows):\n        good=np.abs(np.log(rows[:,0]))<0.25\n        t=rows[good]\n        if len(t):\n            weights=np.sqrt(t[:,3]*t[:,4])\n            scores=[np.average(np.cos(t[:,2]-k*t[:,1]),weights=weights) for k in range(-4,9)]\n            print(\"Phase agreement for integer multiples -4..8:\",np.round(scores,3))\n```\n"}}
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