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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\nfrom threadpoolctl import threadpool_limits\nthreadpool_limits(limits=2)\nnp.set_printoptions(precision=4, suppress=True, linewidth=150)\n\nfor name in [\"W_fc.0\", \"W_proj.0\", \"W_fc.1\", \"W_proj.1\", \"W_fc.2\", \"W_proj.2\", \"W_fc.3\", \"W_proj.3\"]:\n    a = W[name].astype(float)\n    s = svd(a, compute_uv=False, check_finite=False)\n    print(\"\\n\", name, \"first 48 singular values:\\n\", s[:48])\n    print(\"Energy fractions at ranks 16,20,22,24,26,28,32,64:\",\n          [round(float(np.sum(s[:r]**2)/np.sum(s**2)), 4) for r in [16,20,22,24,26,28,32,64]])\n\n# First-layer MLP row geometry: no neuron is evaluated.\nf = W[\"W_fc.0\"].astype(float) * W[\"ln2_w.0\"][None, :]\nf -= f.mean(axis=1, keepdims=True)\nnorms = np.linalg.norm(f, axis=1)\nunit = f / norms[:, None]\ncos = unit @ unit.T\nnp.fill_diagonal(cos, -1)\nprint(\"\\nMLP0 effective row norm quantiles:\", np.quantile(norms, [0,.1,.25,.5,.75,.9,1]))\nprint(\"MLP0 largest off-diagonal row cosine quantiles:\",\n      np.quantile(cos.max(axis=1), [0,.1,.25,.5,.75,.9,1]))\n\n# Shared affine coefficient geometry in layer 1.\nAs, ks, qs = [], [], []\nfor j in range(6):\n    sl = slice(64*j, 64*(j+1))\n    q0, k0 = W[\"W_q.1\"][sl].astype(float), W[\"W_k.1\"][sl].astype(float)\n    q = q0 * W[\"ln1_w.1\"][None, :]\n    k = k0 * W[\"ln1_w.1\"][None, :]\n    q -= q.mean(axis=1, keepdims=True)\n    k -= k.mean(axis=1, keepdims=True)\n    As.append(q.T @ k / 8)\n    ks.append(k.T @ (q0 @ W[\"ln1_b.1\"]) / 8)\n    qs.append(q.T @ (k0 @ W[\"ln1_b.1\"]) / 8)\n\ndef cosines(xs):\n    z = np.stack([x.ravel() for x in xs])\n    z /= np.linalg.norm(z, axis=1, keepdims=True)\n    return z @ z.T\n\nprint(\"\\nL1 QK coefficient cosines:\\n\", cosines(As))\nprint(\"L1 key-linear coefficient cosines:\\n\", cosines(ks))\nprint(\"L1 query-linear coefficient cosines:\\n\", cosines(qs))\nprint(\"L1 key-linear versus query-linear cosines:\\n\",\n      np.array([[np.dot(a,b)/np.linalg.norm(a)/np.linalg.norm(b) for b in qs] for a in ks]))\n\n# For each layer, singular spectrum of the six-head stack of bias-derived\n# key-only coefficient vectors; strong shared directions suggest common routing.\nfor l in range(4):\n    coeff = []\n    for j in range(6):\n        sl = slice(64*j,64*(j+1))\n        q0, k0 = W[f\"W_q.{l}\"][sl].astype(float), W[f\"W_k.{l}\"][sl].astype(float)\n        k = k0 * W[f\"ln1_w.{l}\"][None,:]\n        k -= k.mean(axis=1, keepdims=True)\n        coeff.append(k.T @ (q0 @ W[f\"ln1_b.{l}\"]) / 8)\n    print(\"Layer\", l, \"key-only coefficient stack SV:\",\n          svd(np.stack(coeff), compute_uv=False, check_finite=False))\n```\n"}}
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