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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, eigvals\nfrom threadpoolctl import threadpool_limits\n\nwith threadpool_limits(limits=2):\n    A=[]; B=[]; Q=[]; K=[]; V=[]; O=[]\n    for l in range(4):\n        g=W[f\"ln1_w.{l}\"].astype(float)\n        for j in range(6):\n            s=slice(64*j,64*j+64)\n            mats=[W[f\"{n}.{l}\"][s].astype(float)*g for n in [\"W_q\",\"W_k\",\"W_v\"]]\n            for m in mats:\n                m-=m.mean(1,keepdims=True)\n            q,k,v=mats\n            o=W[f\"W_o.{l}\"][:,s].astype(float)\n            Q.append(q); K.append(k); V.append(v); O.append(o)\n            A.append(q.T@k/8); B.append(o@v)\n    print(\"L0 QK eigenvalues, dominant invariant-space overlaps\")\n    for j in range(6):\n        u,s,vt=svd(A[j],full_matrices=False)\n        ev=eigvals(K[j]@Q[j].T/8)\n        ev=ev[np.argsort(np.abs(ev))[::-1]]\n        r=max(1,int(np.searchsorted(np.cumsum(s*s)/sum(s*s),.95))+1)\n        left=u[:,:r]; right=vt[:r].T\n        vv=svd(V[j],full_matrices=False)[2][:min(28,r+5)].T\n        print(\"H\",j,\"rank95\",r,\"L/R overlap\",round(np.linalg.norm(left.T@right)**2/r,3),\n              \"V energy in QK-left/right\",np.round([\n                  np.linalg.norm(V[j]@left)**2/np.linalg.norm(V[j])**2,\n                  np.linalg.norm(V[j]@right)**2/np.linalg.norm(V[j])**2],3))\n        print(\"eigenvalues\",np.round(ev[:16],3))\n    print(\"\\nQK-OV composition: trace(A_lh @ B_earlier)/(|A_lh| |B_earlier|)\")\n    for i in range(6,24):\n        vals=[np.trace(A[i]@B[j])/(np.linalg.norm(A[i])*np.linalg.norm(B[j]))\n              for j in range((i//6)*6)]\n        print(\"L%dH%d\"%(i//6,i%6),np.round(vals,3))\n    print(\"\\nStrongest composition positivity, with nonrandom comparison\")\n    for i in range(6,24):\n        cand=[]\n        for j in range((i//6)*6):\n            C=A[i]@B[j]\n            n=np.linalg.norm(C)\n            t=np.trace(C)/n\n            sym=np.sum(C*C.T)/(n*n)\n            gain=n/(np.linalg.norm(A[i])*np.linalg.norm(B[j]))\n            cand.append((t,sym,gain,j))\n        cand.sort(reverse=True)\n        print(\"L%dH%d\"%(i//6,i%6),[\n            (\"L%dH%d\"%(j//6,j%6),round(t,3),round(sym,3),round(gain,3))\n            for t,sym,gain,j in cand[:4]])\n    print(\"\\nL0 value-subspace sharing and query/key orthogonality\")\n    vv=np.concatenate(V[:6])\n    _,ss,vt=svd(vv,full_matrices=False)\n    print(\"Combined value singular values\",np.round(ss[:45],3))\n    for r in [2,10,20,28,40]:\n        basis=vt[:r].T\n        print(\"value basis rank\",r)\n        for j in range(6):\n            print(\"H\",j,\"Q/K/V energy\",np.round([\n                np.linalg.norm(M@basis)**2/np.linalg.norm(M)**2\n                for M in [Q[j],K[j],V[j]]],3))\n```\n"}}
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