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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\nthreadpool_limits(limits=2)\nnp.set_printoptions(precision=3, suppress=True)\n\n# These are weight contractions only, not evaluations on inputs.\n# Multiplying columns by gamma and removing their constant-coordinate component\n# expresses each matrix on the mean-zero residual subspace.\ndef read_matrix(name, l, sl=None):\n    a = W[f\"{name}.{l}\"].astype(float)\n    if sl is not None:\n        a = a[sl]\n    a = a * W[f\"ln1_w.{l}\"][None, :]\n    return a - a.mean(axis=1, keepdims=True)\n\nA, B, Q, K, V = [], [], [], [], []\nfor l in range(4):\n    for j in range(6):\n        sl = slice(64*j, 64*(j+1))\n        q, k, v = [read_matrix(n, l, sl) for n in [\"W_q\", \"W_k\", \"W_v\"]]\n        o = W[f\"W_o.{l}\"][:, sl].astype(float)\n        o = o - o.mean(axis=0, keepdims=True)\n        a, b = q.T @ k / 8, o @ v\n        A.append(a); B.append(b); Q.append(q); K.append(k); V.append(v)\n        sa = svd(a, compute_uv=False, check_finite=False)\n        ev = eigvals(a, check_finite=False)\n        ev = ev[np.argsort(-np.abs(ev))]\n        beta = W[f\"ln1_b.{l}\"].astype(float)\n        qb = W[f\"W_q.{l}\"][sl] @ beta\n        kb = W[f\"W_k.{l}\"][sl] @ beta\n        key_linear = k.T @ qb / 8\n        query_linear = q.T @ kb / 8\n        print(\"L%dH%d effective QK norm %.4f SV %s; eig %s; \"\n              \"key/query bias norms %.3f %.3f; effective OV norm %.3f trace %.3f\" %\n              (l, j, np.linalg.norm(a), str(sa[:4]), str(ev[:6]),\n               np.linalg.norm(key_linear), np.linalg.norm(query_linear),\n               np.linalg.norm(b), np.trace(b)))\n\nprint(\"\\nNORMALIZED TRACE(QK_l @ OV_earlier), columns earlier heads\")\nfor l in range(1, 4):\n    for m in range(l):\n        scores = np.array([\n            [np.sum(A[6*l+j] * B[6*m+k].T) /\n             (np.linalg.norm(A[6*l+j])*np.linalg.norm(B[6*m+k]))\n             for k in range(6)] for j in range(6)])\n        print(\"QK layer\", l, \"OV layer\", m)\n        print(scores)\n\nprint(\"\\nL0: overlap of top QK singular subspaces with top value-read subspace\")\nvall = np.concatenate(V[:6], axis=0)\n_, sv, vt = svd(vall, full_matrices=False, check_finite=False)\nprint(\"Combined effective V0 singular values:\", sv[:45])\nfor j in range(6):\n    u, s, vtq = svd(A[j], full_matrices=False, check_finite=False)\n    for r in [8, 24]:\n        rv = vt[:r].T\n        qe = np.linalg.norm(rv.T @ A[j])**2 / np.linalg.norm(A[j])**2\n        ke = np.linalg.norm(A[j] @ rv)**2 / np.linalg.norm(A[j])**2\n        print(\"L0H%d V-rank%d query/key energy %.3f %.3f\" % (j, r, qe, ke))\n\nprint(\"\\nL0 QK pairwise Frobenius cosines\")\nprint(np.array([[np.sum(a*b)/(np.linalg.norm(a)*np.linalg.norm(b))\n                 for b in A[:6]] for a in A[:6]]))\nprint(\"\\nL0 V-head top-24 right-subspace overlaps (mean squared cosine)\")\nbases = [svd(v, full_matrices=False, check_finite=False)[2][:24].T for v in V[:6]]\nprint(np.array([[np.linalg.norm(a.T@b)**2/24 for b in bases] for a in bases]))\n```\n"}}
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