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{"type":"item.completed","item":{"id":"item_1","type":"agent_message","text":"```python\nimport numpy as np\nimport torch\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(\"Output-side singular spectra; possible alphabet-sized bottlenecks:\")\nfor l in range(4):\n    for name in [\"W_proj\",\"W_o\"]:\n        a=P@T(W[f\"{name}.{l}\"])\n        s=torch.linalg.svdvals(a)\n        print(name,l,\"first 65\",np.round(s[:65].numpy(),3))\n        ratios=s[:-1]/s[1:]\n        ids=torch.argsort(ratios[:80],descending=True)[:8]\n        print(\"largest early gaps (rank, ratio)\",\n              [(int(i)+1,round(float(ratios[i]),3)) for i in ids])\n\na=torch.cat([P@T(W[\"W_o.3\"]),P@T(W[\"W_proj.3\"])],dim=1)\nu,s,vh=torch.linalg.svd(a,full_matrices=False)\nprint(\"\\nCombined final-block outputs first 65 SVs\",np.round(s[:65].numpy(),3))\n\nprint(\"\\nMLP weight-row norm distributions and low-norm fractions:\")\nfor l in range(4):\n    f=T(W[f\"W_fc.{l}\"])\n    o=T(W[f\"W_proj.{l}\"])\n    fn=f.norm(dim=1).numpy()\n    on=o.norm(dim=0).numpy()\n    print(l,\"FC norms\",np.round(np.quantile(fn,[0,.1,.25,.5,.75,.9,.99,1]),3),\n          \"Proj norms\",np.round(np.quantile(on,[0,.1,.25,.5,.75,.9,.99,1]),3),\n          \"FC < .6\",round(float((fn<.6).mean()),3))\n\nprint(\"\\nL0 positional-versus-value subspace separation:\")\ng=T(W[\"ln1_w.0\"])\nq=(T(W[\"W_q.0\"])*g)@P\nk=(T(W[\"W_k.0\"])*g)@P\nv=(T(W[\"W_v.0\"])*g)@P\nuq,sq,vq=torch.linalg.svd(torch.cat([q,k],dim=0),full_matrices=False)\nuv,sv,vv=torch.linalg.svd(v,full_matrices=False)\nprint(\"QK input singular spectrum\",np.round(sq[:60].numpy(),3))\nprint(\"V input singular spectrum\",np.round(sv[:60].numpy(),3))\nfor r in [2,4,8,16,24,28,32,48]:\n    z=torch.linalg.svdvals(vq[:r]@vv[:r].T)\n    print(\"rank\",r,\"subspace overlaps\",np.round(z.numpy(),3))\n```\n"}}
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