{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0001","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"693f6062b7707b98de8cf55069066fea691d126f19a1b7d8d6c472fc3ca1f133","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274001,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-12,-3],[-5,-4]],\"x\":[-2,-3],\"y\":[-5,12]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-5, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, -3], y=[-5, 12]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-12, -3], [-5, -4]], [-2, -3], [-5, 12], 21)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -3], [-5, -4]], [-2, -3], [-5, 12], 21)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0002","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a360375f880c7e02ae9007b7756bfe795740dbc178b545949a6530dd685f51f3","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274002,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-11,-2],[0,-3]],\"x\":[-1,-2],\"y\":[-4,11]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, -2], y=[-4, 11]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-11, -2], [0, -3]], [-1, -2], [-4, 11], 44)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -2], [0, -3]], [-1, -2], [-4, 11], 44)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0003","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"554be61f9051971e5b8eb0de36fa25c12fc8f032565cdf80e0354a31b7867dd9","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274003,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-10,-1],[5,-2]],\"x\":[0,-1],\"y\":[-3,10]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, -1], y=[-3, 10]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-10, -1], [5, -2]], [0, -1], [-3, 10], 35)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, -1], [5, -2]], [0, -1], [-3, 10], 35)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0004","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e87053b1cff62241c993f8e638bdcf92a7f7a69f9a9f23620d847638ae05f50","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274004,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-9,0],[-1,-1]],\"x\":[1,0],\"y\":[-2,9]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, 0], y=[-2, 9]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-9, 0], [-1, -1]], [1, 0], [-2, 9], 18)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 0], [-1, -1]], [1, 0], [-2, 9], 18)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0005","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1d63d48f00afd60c33a412d2e157fbf534630cab9a9cd1ea27cc7edce3c4c43","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274005,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-8,1],[4,0]],\"x\":[2,1],\"y\":[-1,8]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [4, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, 1], y=[-1, 8]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-8, 1], [4, 0]], [2, 1], [-1, 8], 28)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1], [4, 0]], [2, 1], [-1, 8], 28)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0006","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c8a25d6737739212ba49a58d5cb02f2bfba31698d82784ebb09544c7b00be94","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274006,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-7,2],[-2,1]],\"x\":[-2,2],\"y\":[0,7]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, 2], y=[0, 7]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-7, 2], [-2, 1]], [-2, 2], [0, 7], -14)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 2], [-2, 1]], [-2, 2], [0, 7], -14)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0007","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e697e914a72dc1b7b17140ccd1632386061e1528a1fe484e89359e01eef16e1","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274007,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-6,3],[3,2]],\"x\":[-1,3],\"y\":[1,6]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, 3], y=[1, 6]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-6, 3], [3, 2]], [-1, 3], [1, 6], 33)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 3], [3, 2]], [-1, 3], [1, 6], 33)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0008","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"083fbdf7ce8d88dab307ce247efd7f9cd0970d96db4c619d50d3314573873289","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274008,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-5,-3],[-3,3]],\"x\":[0,-3],\"y\":[2,5]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, -3], y=[2, 5]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-5, -3], [-3, 3]], [0, -3], [2, 5], -27)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, -3], [-3, 3]], [0, -3], [2, 5], -27)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0009","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc9789003307405a25232b7449c8e404e5877f8ae004de54b71c2845ce663389","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274009,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-4,-2],[2,4]],\"x\":[1,-2],\"y\":[3,4]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-4, -2], [2, 4]], [1, -2], [3, 4], -64)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -2], [2, 4]], [1, -2], [3, 4], -64)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0010","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8649bbdcb7c87f277a0fb0156297f1ff9912a03bf9dcc10c92982a24f9294ceb","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274010,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-3,-1],[-4,-4]],\"x\":[2,-1],\"y\":[4,3]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, -1], y=[4, 3]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-3, -1], [-4, -4]], [2, -1], [4, 3], -2)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, -1], [-4, -4]], [2, -1], [4, 3], -2)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0011","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ba7b77cc2ea4cbd38cf323d1a6816b57494309e0509d554c410bac8c6d1d45f","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274011,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-2,0],[1,-3]],\"x\":[-2,0],\"y\":[5,2]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, 0], y=[5, 2]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-2, 0], [1, -3]], [-2, 0], [5, 2], 20)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 0], [1, -3]], [-2, 0], [5, 2], 20)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0012","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf43344a1606452a6ca295829ba35b63feab3a6a5a0615a1d9dc32bec7139223","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274012,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[-1,1],[-5,-2]],\"x\":[-1,1],\"y\":[-5,1]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, 1], y=[-5, 1]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[-1, 1], [-5, -2]], [-1, 1], [-5, 1], 17)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1], [-5, -2]], [-1, 1], [-5, 1], 17)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0013","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f986627b6c10e89538ceda6d6b4a4cd0b6843e9fee68233f9ff90fcddc5bcafd","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274013,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[0,2],[0,-1]],\"x\":[0,2],\"y\":[-4,0]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [0, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, 2], y=[-4, 0]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[0, 2], [0, -1]], [0, 2], [-4, 0], 0)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 2], [0, -1]], [0, 2], [-4, 0], 0)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0014","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5304a598175783b6220ada32e1740169656f28cfc24745c9cbc3d07ec174c125","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274014,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[1,3],[5,0]],\"x\":[1,3],\"y\":[-3,-1]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, 3], y=[-3, -1]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[1, 3], [5, 0]], [1, 3], [-3, -1], -51)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 3], [5, 0]], [1, 3], [-3, -1], -51)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0015","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecb1e97f159f72598cf5d4e68e184f242aa93bc0c01ca64289429d06fdc1d5a1","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274015,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[2,-3],[-1,1]],\"x\":[2,-3],\"y\":[-2,-2]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, -3], y=[-2, -2]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[2, -3], [-1, 1]], [2, -3], [-2, -2], 4)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -3], [-1, 1]], [2, -3], [-2, -2], 4)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0016","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce79ce8b4f6656599b767f96024c5542ef73e1fd868f98e1dc18a422f587923a","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274016,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[3,-2],[4,2]],\"x\":[-2,-2],\"y\":[-1,-3]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, -2], y=[-1, -3]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -2], [4, 2]], [-2, -2], [-1, -3], 14)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0017","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73164cd45e23752d4faa08055595fef126e765df3a5e8eefdc91b0587f5a0e34","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274017,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[4,-1],[-2,3]],\"x\":[-1,-1],\"y\":[0,-4]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-2, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, -1], y=[0, -4]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[4, -1], [-2, 3]], [-1, -1], [0, -4], 8)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1], [-2, 3]], [-1, -1], [0, -4], 8)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0018","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9976f3b088086a52792de07abcd8af029af8415ed02c8e3ecef45852fa9932e0","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274018,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[5,0],[3,4]],\"x\":[0,0],\"y\":[1,-5]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, 0], y=[1, -5]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[5, 0], [3, 4]], [0, 0], [1, -5], 0)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 0], [3, 4]], [0, 0], [1, -5], 0)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0019","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d35be0b19504f2d66aa40e715bacf5e8ff5fc80db01321e696bcb86035b9527","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274019,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[6,1],[-3,-4]],\"x\":[1,1],\"y\":[2,-6]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, 1], y=[2, -6]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[6, 1], [-3, -4]], [1, 1], [2, -6], 24)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1], [-3, -4]], [1, 1], [2, -6], 24)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0020","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94eb52d40333d832b5183b7e62abe8dbcb72d51ff688ab64ab00c5014edf578b","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274020,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[7,2],[2,-3]],\"x\":[2,2],\"y\":[3,-7]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, 2], y=[3, -7]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[7, 2], [2, -3]], [2, 2], [3, -7], 68)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 2], [2, -3]], [2, 2], [3, -7], 68)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0021","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb670472ea3db18b156a1ec8692e802a6d49b9c4083c19b99b002198f97c88f8","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274021,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[8,3],[-4,-2]],\"x\":[-2,3],\"y\":[4,-8]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-2, 3], y=[4, -8]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[8, 3], [-4, -2]], [-2, 3], [4, -8], -16)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 3], [-4, -2]], [-2, 3], [4, -8], -16)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0022","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06534cf4af122551cb1f4a064b209d062bd8a330f2d69f1f33a85a713c3de981","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274022,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[9,-3],[1,-1]],\"x\":[-1,-3],\"y\":[5,-9]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[-1, -3], y=[5, -9]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[9, -3], [1, -1]], [-1, -3], [5, -9], -114)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -3], [1, -1]], [-1, -3], [5, -9], -114)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0023","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0d229a4757284a3852c7df880b0195468948cbececb76f2ab766e55e6d9bacc","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274023,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[10,-2],[-5,0]],\"x\":[0,-2],\"y\":[-5,-10]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[0, -2], y=[-5, -10]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[10, -2], [-5, 0]], [0, -2], [-5, -10], -50)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, -2], [-5, 0]], [0, -2], [-5, -10], -50)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0024","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd92a45461eeb48d3ecb8b4121828e9e20407239dc6ac6bce6cf536fce62dece","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274024,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[11,-1],[0,1]],\"x\":[1,-1],\"y\":[-4,-11]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[1, -1], y=[-4, -11]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[11, -1], [0, 1]], [1, -1], [-4, -11], -22)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1], [0, 1]], [1, -1], [-4, -11], -22)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0025","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c035e8cb4dc027d25900cddd44643fad82ece2400744492c9ac96bf7a0aa6d8","family_id":"qf-mod27-bilinear-matrix-evaluate","seed":202610274025,"semantic_template":"qf-mod27-bilinear-matrix-evaluate/{\"matrix\":[[12,0],[5,2]],\"x\":[2,0],\"y\":[-3,-12]}"},"fields":{"title":"Evaluate an Ordered Real Bilinear Matrix Formula","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [5, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. x=[2, 0], y=[-3, -12]. Give (matrix,x,y,value).","answer":{"kind":"expression","value":"([[12, 0], [5, 2]], [2, 0], [-3, -12], -72)","explanation":"First multiply A by y, then dot with x. Keep the slot order; a general bilinear form need not satisfy B(x,y)=B(y,x). An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 0], [5, 2]], [2, 0], [-3, -12], -72)."},"skills":["Evaluate an Ordered Real Bilinear Matrix Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0026","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b61f76c8ae0f310990c0631392ab73195c33b3cf1e5e4b88650b92e9499d6fea","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274026,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":-3,\"matrix\":[[-12,-3],[-5,-4]],\"u\":[1,-2],\"v\":[-3,1],\"y\":[-5,12]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-5, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[-3, 1], alpha=-2, beta=-3, fixed y=[-5, 12]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-2, -3, [7, 1], 145, 145, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -3, [7, 1], 145, 145, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0027","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f8009eb455b6f49a4ab69290d0f5ea1a0e54d9142c3184d7a0676809e0b3117","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274027,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":-2,\"matrix\":[[-11,-2],[0,-3]],\"u\":[1,-1],\"v\":[-2,1],\"y\":[-4,11]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[-2, 1], alpha=-1, beta=-2, fixed y=[-4, 11]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-1, -2, [3, -1], 99, 99, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -2, [3, -1], 99, 99, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0028","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b44c03298e4be2bb62e462993e14046513bf2096e2bf3c3b47ae289bb68cb3f","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274028,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":-1,\"matrix\":[[-10,-1],[5,-2]],\"u\":[1,0],\"v\":[-1,1],\"y\":[-3,10]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[-1, 1], alpha=0, beta=-1, fixed y=[-3, 10]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(0, -1, [1, -1], 55, 55, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -1, [1, -1], 55, 55, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0029","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b81c254bad55fe2e4753ad19a9223b1fa2d002bbf19994f599b6522c5ae64a8f","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274029,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":0,\"matrix\":[[-9,0],[-1,-1]],\"u\":[1,1],\"v\":[0,1],\"y\":[-2,9]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[0, 1], alpha=1, beta=0, fixed y=[-2, 9]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(1, 0, [1, 1], 11, 11, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [1, 1], 11, 11, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0030","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2579bdc4162341192a4efc38babeba56469e963a0304d3c2d71b9d621e17c22d","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274030,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":1,\"matrix\":[[-8,1],[4,0]],\"u\":[1,2],\"v\":[1,1],\"y\":[-1,8]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [4, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[1, 1], alpha=2, beta=1, fixed y=[-1, 8]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(2, 1, [3, 5], 28, 28, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 1, [3, 5], 28, 28, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0031","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98149d285dbdcd05d79907fdb309b2ff577852ebc84149e9cae045e6081ec6a1","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274031,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":2,\"matrix\":[[-7,2],[-2,1]],\"u\":[1,-2],\"v\":[2,1],\"y\":[0,7]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[2, 1], alpha=-2, beta=2, fixed y=[0, 7]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-2, 2, [2, 6], 70, 70, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 2, [2, 6], 70, 70, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0032","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c986c3f31aa57c7439fc72e0ee4adc84ba11ae86664d08edc716a3446794838b","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274032,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":3,\"matrix\":[[-6,3],[3,2]],\"u\":[1,-1],\"v\":[3,1],\"y\":[1,6]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[3, 1], alpha=-1, beta=3, fixed y=[1, 6]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-1, 3, [8, 4], 156, 156, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 3, [8, 4], 156, 156, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0033","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b22f6f25e6d78633298ed3c0aafe6b2f42eb93281b2cd41c518bda5964f26dfe","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274033,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":-3,\"matrix\":[[-5,-3],[-3,3]],\"u\":[1,0],\"v\":[-3,1],\"y\":[2,5]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[-3, 1], alpha=0, beta=-3, fixed y=[2, 5]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(0, -3, [9, -3], -252, -252, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -3, [9, -3], -252, -252, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0034","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42acc39d3cc1f21183996b4c7ade006ebc8f3e1e4bd02801d52a6489849935e4","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274034,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":-2,\"matrix\":[[-4,-2],[2,4]],\"u\":[1,1],\"v\":[-2,1],\"y\":[3,4]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[-2, 1], alpha=1, beta=-2, fixed y=[3, 4]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(1, -2, [5, -1], -122, -122, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -2, [5, -1], -122, -122, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0035","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7f936b9bc4ee975635e0fdf0787d10a33363c4671fef9a7881528f606c73dd7","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274035,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":-1,\"matrix\":[[-3,-1],[-4,-4]],\"u\":[1,2],\"v\":[-1,1],\"y\":[4,3]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[-1, 1], alpha=2, beta=-1, fixed y=[4, 3]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(2, -1, [3, 3], -129, -129, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -1, [3, 3], -129, -129, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0036","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f70e95afd733e03fb06dea3b5bcf27b134995d25a01742f0496a6f63a4596b97","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274036,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":0,\"matrix\":[[-2,0],[1,-3]],\"u\":[1,-2],\"v\":[0,1],\"y\":[5,2]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[0, 1], alpha=-2, beta=0, fixed y=[5, 2]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-2, 0, [-2, 4], 16, 16, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 0, [-2, 4], 16, 16, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0037","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59f68b58dcb528cce75bf85cdf6b6a116c61215c301e8f5122a2c334224dab75","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274037,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":1,\"matrix\":[[-1,1],[-5,-2]],\"u\":[1,-1],\"v\":[1,1],\"y\":[-5,1]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[1, 1], alpha=-1, beta=1, fixed y=[-5, 1]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-1, 1, [0, 2], 46, 46, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 1, [0, 2], 46, 46, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0038","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cf3410a2ac77841545e942dc67e2b96e7cf4aa642da74b2e3bd516a088ff18e","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274038,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":2,\"matrix\":[[0,2],[0,-1]],\"u\":[1,0],\"v\":[2,1],\"y\":[-4,0]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [0, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[2, 1], alpha=0, beta=2, fixed y=[-4, 0]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(0, 2, [4, 2], 0, 0, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 2, [4, 2], 0, 0, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0039","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bdfb8731873024e7d1445ebb8c6c3e43c40d9810b91a492b4fe41f4086233e0","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274039,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":3,\"matrix\":[[1,3],[5,0]],\"u\":[1,1],\"v\":[3,1],\"y\":[-3,-1]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[3, 1], alpha=1, beta=3, fixed y=[-3, -1]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(1, 3, [10, 4], -120, -120, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3, [10, 4], -120, -120, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0040","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c96fa45f0b757845617a750c79eb3d5f4a79353fa3c388d8a29b548fb9d210a","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274040,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":-3,\"matrix\":[[2,-3],[-1,1]],\"u\":[1,2],\"v\":[-3,1],\"y\":[-2,-2]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[-3, 1], alpha=2, beta=-3, fixed y=[-2, -2]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(2, -3, [11, 1], 22, 22, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -3, [11, 1], 22, 22, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0041","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef8cb28a88b6c14c5596d9e8955d9ffddd1a8c0bd4033573a888c9d86f9b3a8e","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274041,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":-2,\"matrix\":[[3,-2],[4,2]],\"u\":[1,-2],\"v\":[-2,1],\"y\":[-1,-3]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[-2, 1], alpha=-2, beta=-2, fixed y=[-1, -3]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-2, -2, [2, 2], -14, -14, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -2, [2, 2], -14, -14, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0042","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4a049e1cfe719a3043a0fa3a8a65f2fe0d63478a90bb1a6575892f80b7eabe2","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274042,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":-1,\"matrix\":[[4,-1],[-2,3]],\"u\":[1,-1],\"v\":[-1,1],\"y\":[0,-4]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-2, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[-1, 1], alpha=-1, beta=-1, fixed y=[0, -4]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-1, -1, [0, 0], 0, 0, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -1, [0, 0], 0, 0, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0043","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18c45e87009191fc477d99e878b36f2a5f0790429945cc2a6a995ae01ddea190","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274043,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":0,\"matrix\":[[5,0],[3,4]],\"u\":[1,0],\"v\":[0,1],\"y\":[1,-5]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[0, 1], alpha=0, beta=0, fixed y=[1, -5]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(0, 0, [0, 0], 0, 0, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0, 0], 0, 0, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0044","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29ea66f79718755ea49f90c850deb5ebbda06a58d0c873f5e08a27a76af1f714","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274044,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":1,\"matrix\":[[6,1],[-3,-4]],\"u\":[1,1],\"v\":[1,1],\"y\":[2,-6]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[1, 1], alpha=1, beta=1, fixed y=[2, -6]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(1, 1, [2, 2], 48, 48, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1, [2, 2], 48, 48, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0045","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58f2bad9018c8f73989dbaa8a1e961408713b2c4babc02bff5759077aa968494","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274045,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":2,\"matrix\":[[7,2],[2,-3]],\"u\":[1,2],\"v\":[2,1],\"y\":[3,-7]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[2, 1], alpha=2, beta=2, fixed y=[3, -7]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(2, 2, [6, 6], 204, 204, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 2, [6, 6], 204, 204, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0046","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34057eae04f460ebef67ab549a017da8ae4dafbf85678a18c84705e4cd48df61","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274046,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-2,\"beta\":3,\"matrix\":[[8,3],[-4,-2]],\"u\":[1,-2],\"v\":[3,1],\"y\":[4,-8]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -2], v=[3, 1], alpha=-2, beta=3, fixed y=[4, -8]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-2, 3, [7, 7], 56, 56, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 3, [7, 7], 56, 56, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0047","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3f48c92c7517a967299047e3bfac9c4c8306c55b86590044ef8295ef4449d53","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274047,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":-1,\"beta\":-3,\"matrix\":[[9,-3],[1,-1]],\"u\":[1,-1],\"v\":[-3,1],\"y\":[5,-9]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, -1], v=[-3, 1], alpha=-1, beta=-3, fixed y=[5, -9]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(-1, -3, [8, -2], 548, 548, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -3, [8, -2], 548, 548, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0048","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b41e0163561a5623791f2ad4fa66c020e4b4cd5cd9d47a3d62b7adc82b5267c7","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274048,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":0,\"beta\":-2,\"matrix\":[[10,-2],[-5,0]],\"u\":[1,0],\"v\":[-2,1],\"y\":[-5,-10]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 0], v=[-2, 1], alpha=0, beta=-2, fixed y=[-5, -10]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(0, -2, [4, -2], -170, -170, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -2, [4, -2], -170, -170, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0049","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18950dec716244c51ca8ff26e682e6448b6bfa89d2ab2ec8d0355c4ea9321cc5","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274049,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":1,\"beta\":-1,\"matrix\":[[11,-1],[0,1]],\"u\":[1,1],\"v\":[-1,1],\"y\":[-4,-11]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 1], v=[-1, 1], alpha=1, beta=-1, fixed y=[-4, -11]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(1, -1, [2, 0], -66, -66, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -1, [2, 0], -66, -66, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0050","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"445ad6b1bea9ee9a955af7e9845b90ab2d90609089bd0aab7d3b9a5f9fec7ff7","family_id":"qf-mod27-bilinear-first-slot-identity","seed":202610274050,"semantic_template":"qf-mod27-bilinear-first-slot-identity/{\"alpha\":2,\"beta\":0,\"matrix\":[[12,0],[5,2]],\"u\":[1,2],\"v\":[0,1],\"y\":[-3,-12]}"},"fields":{"title":"Linearity in the First Slot with the Second Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [5, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. u=[1, 2], v=[0, 1], alpha=2, beta=0, fixed y=[-3, -12]. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(combined,y),rhs=alpha*B(u,y)+beta*B(v,y),equal?).","answer":{"kind":"expression","value":"(2, 0, [2, 4], -228, -228, Yes)","explanation":"Expand only the first slot while keeping the second input fixed. The matrix formula also has a separate second-slot obligation. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [2, 4], -228, -228, Yes)."},"skills":["Linearity in the First Slot with the Second Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0051","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f01e2155dfd9b91accbe615e4f071ef0a4b29f88d793634df93a66eda282699b","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274051,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":-3,\"matrix\":[[-12,-3],[-5,-4]],\"u\":[1,-2],\"v\":[-3,1],\"x\":[-2,-3]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-5, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, -3], u=[1, -2], v=[-3, 1], alpha=-2, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-2, -3, [7, 1], 291, 291, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -3, [7, 1], 291, 291, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0052","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b19d05fecac07fc3ecc8a5572a76eae1e4b8c8c5e4d5c2ddc8e3a633f5ef087","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274052,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":-2,\"matrix\":[[-11,-2],[0,-3]],\"u\":[1,-1],\"v\":[-2,1],\"x\":[-1,-2]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, -2], u=[1, -1], v=[-2, 1], alpha=-1, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-1, -2, [3, -1], 25, 25, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -2, [3, -1], 25, 25, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0053","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"764e4fc3ae5b5711cb69c723f0fadadf68d5bcab8a6aae7c16e45e39d948b87e","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274053,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":-1,\"matrix\":[[-10,-1],[5,-2]],\"u\":[1,0],\"v\":[-1,1],\"x\":[0,-1]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, -1], u=[1, 0], v=[-1, 1], alpha=0, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(0, -1, [1, -1], -7, -7, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -1, [1, -1], -7, -7, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0054","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82ade9091f1b8407befcb6906dc901de962f316592c621e8b037a5b4e8595129","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274054,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":0,\"matrix\":[[-9,0],[-1,-1]],\"u\":[1,1],\"v\":[0,1],\"x\":[1,0]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, 0], u=[1, 1], v=[0, 1], alpha=1, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(1, 0, [1, 1], -9, -9, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [1, 1], -9, -9, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0055","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91ccdd5d563228e347fb1fd22568e046efba5388e3f8da28ad8650cf830349f8","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274055,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":1,\"matrix\":[[-8,1],[4,0]],\"u\":[1,2],\"v\":[1,1],\"x\":[2,1]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [4, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, 1], u=[1, 2], v=[1, 1], alpha=2, beta=1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(2, 1, [3, 5], -26, -26, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 1, [3, 5], -26, -26, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0056","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9798a009a8d8f5b026498944893883faa5fe2404274ae61297e69d5b4bdd24f5","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274056,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":2,\"matrix\":[[-7,2],[-2,1]],\"u\":[1,-2],\"v\":[2,1],\"x\":[-2,2]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 2], u=[1, -2], v=[2, 1], alpha=-2, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-2, 2, [2, 6], 8, 8, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 2, [2, 6], 8, 8, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0057","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ae6683ef5fb7afdcdbde5c2d2232c584d5818950687d8bea70132a39b68b02f","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274057,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":3,\"matrix\":[[-6,3],[3,2]],\"u\":[1,-1],\"v\":[3,1],\"x\":[-1,3]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, 3], u=[1, -1], v=[3, 1], alpha=-1, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-1, 3, [8, 4], 132, 132, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 3, [8, 4], 132, 132, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0058","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f5373f8be52ef1bec0f262d8e6143d6779345dd4a28754a543a9e6b345246f","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274058,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":-3,\"matrix\":[[-5,-3],[-3,3]],\"u\":[1,0],\"v\":[-3,1],\"x\":[0,-3]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, -3], u=[1, 0], v=[-3, 1], alpha=0, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(0, -3, [9, -3], 108, 108, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -3, [9, -3], 108, 108, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0059","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb3800ebc6bd918941ae37e8bbafe7624f8bfa646f22bdd1b5db1f3669cc562b","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274059,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":-2,\"matrix\":[[-4,-2],[2,4]],\"u\":[1,1],\"v\":[-2,1],\"x\":[1,-2]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, -2], u=[1, 1], v=[-2, 1], alpha=1, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(1, -2, [5, -1], -30, -30, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -2, [5, -1], -30, -30, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0060","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e657d9986041babece4f8de673ef92c857894f8ba874fae584ae9728f02e43d7","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274060,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":-1,\"matrix\":[[-3,-1],[-4,-4]],\"u\":[1,2],\"v\":[-1,1],\"x\":[2,-1]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, -1], u=[1, 2], v=[-1, 1], alpha=2, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(2, -1, [3, 3], 0, 0, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -1, [3, 3], 0, 0, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0061","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"345ede3252e2eb35171120263045c5d38fc9e0296589cda070558b7eca6d51b9","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274061,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":0,\"matrix\":[[-2,0],[1,-3]],\"u\":[1,-2],\"v\":[0,1],\"x\":[-2,0]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 0], u=[1, -2], v=[0, 1], alpha=-2, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-2, 0, [-2, 4], -8, -8, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 0, [-2, 4], -8, -8, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0062","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b828df69a26ede9af17f1cae2fa0a337aa7e17873df791a5652171821a01a60","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274062,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":1,\"matrix\":[[-1,1],[-5,-2]],\"u\":[1,-1],\"v\":[1,1],\"x\":[-1,1]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, 1], u=[1, -1], v=[1, 1], alpha=-1, beta=1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-1, 1, [0, 2], -6, -6, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 1, [0, 2], -6, -6, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0063","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a74072d88780831a6b761dbd4d520b22250c1590befd9eb333ba25f77a0b31b7","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274063,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":2,\"matrix\":[[0,2],[0,-1]],\"u\":[1,0],\"v\":[2,1],\"x\":[0,2]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [0, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, 2], u=[1, 0], v=[2, 1], alpha=0, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(0, 2, [4, 2], -4, -4, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 2, [4, 2], -4, -4, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0064","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec8d9d5faedd4b3dd28b818ed90315819e11c07119f37247490b3529522d7d72","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274064,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":3,\"matrix\":[[1,3],[5,0]],\"u\":[1,1],\"v\":[3,1],\"x\":[1,3]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, 3], u=[1, 1], v=[3, 1], alpha=1, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(1, 3, [10, 4], 172, 172, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3, [10, 4], 172, 172, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0065","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0e4875cf7a18d1ab69e5157fdaa5398f6859ad45953b95839e8fd6ae39375dc","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274065,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":-3,\"matrix\":[[2,-3],[-1,1]],\"u\":[1,2],\"v\":[-3,1],\"x\":[2,-3]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, -3], u=[1, 2], v=[-3, 1], alpha=2, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(2, -3, [11, 1], 68, 68, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -3, [11, 1], 68, 68, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0066","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9823efbb08af602bd1d8200b9df68b3939cc54b331b2589e1d801eef11239b80","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274066,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":-2,\"matrix\":[[3,-2],[4,2]],\"u\":[1,-2],\"v\":[-2,1],\"x\":[-2,-2]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, -2], u=[1, -2], v=[-2, 1], alpha=-2, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-2, -2, [2, 2], -28, -28, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -2, [2, 2], -28, -28, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0067","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a17e4e7b42d8e8ed5984a46d149ea43e8b53b98fc33fc506928c3b805fa24607","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274067,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":-1,\"matrix\":[[4,-1],[-2,3]],\"u\":[1,-1],\"v\":[-1,1],\"x\":[-1,-1]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-2, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, -1], u=[1, -1], v=[-1, 1], alpha=-1, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-1, -1, [0, 0], 0, 0, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -1, [0, 0], 0, 0, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0068","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a138c1708e3c6c34a7fd244724d42d7a164f26c6324dad047a3d1b561afb016","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274068,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":0,\"matrix\":[[5,0],[3,4]],\"u\":[1,0],\"v\":[0,1],\"x\":[0,0]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, 0], u=[1, 0], v=[0, 1], alpha=0, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(0, 0, [0, 0], 0, 0, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0, 0], 0, 0, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0069","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"284e729c50c16c13159a0f20ff2db426725d7e11ef8c6c1a040b83983cf68139","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274069,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":1,\"matrix\":[[6,1],[-3,-4]],\"u\":[1,1],\"v\":[1,1],\"x\":[1,1]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, 1], u=[1, 1], v=[1, 1], alpha=1, beta=1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(1, 1, [2, 2], 0, 0, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1, [2, 2], 0, 0, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0070","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9590d0acd3a8b523899839e9d2ac6da00c0e25d87c4f21656ce26085a59b8b71","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274070,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":2,\"matrix\":[[7,2],[2,-3]],\"u\":[1,2],\"v\":[2,1],\"x\":[2,2]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, 2], u=[1, 2], v=[2, 1], alpha=2, beta=2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(2, 2, [6, 6], 96, 96, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 2, [6, 6], 96, 96, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0071","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4c6772bf4bda620b8935b56c58be9443b6006ab5785d1f39c4473f9e142c4b0","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274071,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-2,\"beta\":3,\"matrix\":[[8,3],[-4,-2]],\"u\":[1,-2],\"v\":[3,1],\"x\":[-2,3]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [-4, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-2, 3], u=[1, -2], v=[3, 1], alpha=-2, beta=3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-2, 3, [7, 7], -280, -280, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 3, [7, 7], -280, -280, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0072","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"163e96092923a36fba8380169d99e36bf3c2f487f166c032af620e33df17a41b","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274072,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":-1,\"beta\":-3,\"matrix\":[[9,-3],[1,-1]],\"u\":[1,-1],\"v\":[-3,1],\"x\":[-1,-3]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[-1, -3], u=[1, -1], v=[-3, 1], alpha=-1, beta=-3. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(-1, -3, [8, -2], -108, -108, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -3, [8, -2], -108, -108, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0073","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e814f844df3cd7147964f7e1e8a70f86d23fc62df21c44025055f0bf649252","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274073,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":0,\"beta\":-2,\"matrix\":[[10,-2],[-5,0]],\"u\":[1,0],\"v\":[-2,1],\"x\":[0,-2]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[0, -2], u=[1, 0], v=[-2, 1], alpha=0, beta=-2. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(0, -2, [4, -2], 40, 40, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -2, [4, -2], 40, 40, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0074","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbc6eb5a2666778774ece00ba869bfe03922ccf9f026d05b323d029c223e9301","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274074,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":1,\"beta\":-1,\"matrix\":[[11,-1],[0,1]],\"u\":[1,1],\"v\":[-1,1],\"x\":[1,-1]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[1, -1], u=[1, 1], v=[-1, 1], alpha=1, beta=-1. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(1, -1, [2, 0], 22, 22, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -1, [2, 0], 22, 22, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0075","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4454fc15b03cc002ec3252ec8036e2ce72ade57cb0a98d56ab17dbae80090cc8","family_id":"qf-mod27-bilinear-second-slot-identity","seed":202610274075,"semantic_template":"qf-mod27-bilinear-second-slot-identity/{\"alpha\":2,\"beta\":0,\"matrix\":[[12,0],[5,2]],\"u\":[1,2],\"v\":[0,1],\"x\":[2,0]}"},"fields":{"title":"Linearity in the Second Slot with the First Slot Fixed","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [5, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. fixed x=[2, 0], u=[1, 2], v=[0, 1], alpha=2, beta=0. Give (alpha,beta,combined=alpha*u+beta*v,lhs=B(x,combined),rhs=alpha*B(x,u)+beta*B(x,v),equal?).","answer":{"kind":"expression","value":"(2, 0, [2, 4], 48, 48, Yes)","explanation":"Expand the second slot while retaining x as the first input. Do not swap the slots or infer symmetry from bilinearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [2, 4], 48, 48, Yes)."},"skills":["Linearity in the Second Slot with the First Slot Fixed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0076","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81450b8388b4e1b536be3e9b2cbadb8ec634d3317c6bb53d329252b2e68129aa","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274076,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-12,-3],[-3,-4]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-12, -3], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-12, -3], [-3, -4]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -3], [-3, -4]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0077","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2a4bb6de724e7e2a487eb2a44a7b10fd2085585811c03502b5495e0dc687841","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274077,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-11,-2],[0,-3]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-11, -2], [0, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-11, -2], [0, -3]], [0, 1, -2, 0], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -2], [0, -3]], [0, 1, -2, 0], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0078","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50d92cc217149e6a9e2b11cc646ff86ec21a8e908d71f56e157764ec4d8c8563","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274078,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-10,-1],[5,-2]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-10, -1], [5, -2]], [0, 1, -1, 5], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, -1], [5, -2]], [0, 1, -1, 5], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0079","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8141c77f2413c21b025afb3333d27903c96c59fd6b4a5fcdbf4288c55d9850b2","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274079,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-9,0],[-1,-1]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-9, 0], [-1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-9, 0], [-1, -1]], [0, 1, 0, -1], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 0], [-1, -1]], [0, 1, 0, -1], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0080","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"001416a6f4844916a48ce30835e4ce7a05733cc8aa3343c6f9b300897520dacf","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274080,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-8,1],[1,0]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-8, 1], [1, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-8, 1], [1, 0]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1], [1, 0]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0081","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52dd3661d7d64f1f765f00630326190aa19f52637a4dac2fa6101ffc61d81e1d","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274081,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-7,2],[-2,1]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-7, 2], [-2, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-7, 2], [-2, 1]], [0, 1, 2, -2], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 2], [-2, 1]], [0, 1, 2, -2], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0082","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"792a7ab87966d3f83353fb9d12e1f5fd20aedd52b1a4ec64ba5ad599fa479a30","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274082,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-6,3],[3,2]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-6, 3], [3, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-6, 3], [3, 2]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 3], [3, 2]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0083","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a1fb8a74d39d58bd69e48dfa7ff2a5bec7c09b9efd5c8ccd425119e96401d3d","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274083,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-5,-3],[-3,3]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-5, -3], [-3, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-5, -3], [-3, 3]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, -3], [-3, 3]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0084","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b1b03ef0d0c135e2f5c235d2aeb06977ee69372eec0581f9f66e0653d171095","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274084,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-4,-2],[-2,4]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [-2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-4, -2], [-2, 4]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -2], [-2, 4]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0085","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5f63cdfb205e2fdb81e0b8a09b97a0f10e775c3f96b2c7897e47b6e88e65148","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274085,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-3,-1],[-4,-4]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-3, -1], [-4, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-3, -1], [-4, -4]], [0, 1, -1, -4], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, -1], [-4, -4]], [0, 1, -1, -4], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0086","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e77e1372b293411d87f4392798871aa00c8d6a80be0ee8639589758067810e57","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274086,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-2,0],[1,-3]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-2, 0], [1, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-2, 0], [1, -3]], [0, 1, 0, 1], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 0], [1, -3]], [0, 1, 0, 1], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0087","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74eb1a05500e2c357117554a4b570bebec562b2ec6f1179ab08d4c39df106e11","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274087,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[-1,1],[-5,-2]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-1, 1], [-5, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[-1, 1], [-5, -2]], [0, 1, 1, -5], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1], [-5, -2]], [0, 1, 1, -5], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0088","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96fad3a1606bfd507d2e5c84535c6cc68d9f7ad4698f7ebc4e354cebac61ac02","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274088,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[0,2],[2,-1]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[0, 2], [2, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[0, 2], [2, -1]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 2], [2, -1]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0089","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9990167eb41c621a9a3fc0f791a62101cde548030723ba336ae2513e58119121","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274089,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[1,3],[5,0]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[1, 3], [5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[1, 3], [5, 0]], [0, 1, 3, 5], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 3], [5, 0]], [0, 1, 3, 5], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0090","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e4813fabac9e5a1ebe306d0853fa7ba804b2c71c054a4e7156f14e56c2daa2b","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274090,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[2,-3],[-1,1]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[2, -3], [-1, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[2, -3], [-1, 1]], [0, 1, -3, -1], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -3], [-1, 1]], [0, 1, -3, -1], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0091","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45d5a2949a98ec51cef0f275661171a85d45e6e986912ae56343e3061f5f4ddc","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274091,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[3,-2],[4,2]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[3, -2], [4, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[3, -2], [4, 2]], [0, 1, -2, 4], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -2], [4, 2]], [0, 1, -2, 4], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0092","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fd22f505031f3b320e57ed1afd4625b70c3f12e16af8ba943df09184d1f1c52","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274092,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[4,-1],[-1,3]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[4, -1], [-1, 3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[4, -1], [-1, 3]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1], [-1, 3]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0093","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87c3834566a96e7adcffeda681490860b64d842c9b75cf23110f9bb8bd0bf9b9","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274093,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[5,0],[3,4]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[5, 0], [3, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[5, 0], [3, 4]], [0, 1, 0, 3], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 0], [3, 4]], [0, 1, 0, 3], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0094","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b033783f9bfce029887a2af3e593eb3e186c52db8319e03004fecd12d18a88b0","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274094,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[6,1],[-3,-4]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[6, 1], [-3, -4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[6, 1], [-3, -4]], [0, 1, 1, -3], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1], [-3, -4]], [0, 1, 1, -3], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0095","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5744698a3f2fd8e8f74952083c19289fc6bf4b52af0808261dd8ed7c29a3754f","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274095,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[7,2],[2,-3]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[7, 2], [2, -3]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[7, 2], [2, -3]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 2], [2, -3]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0096","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b94167d8cf9b797bf963c33c2bf1677f8d75d5d6217f259a85dc21be0eb30ea2","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274096,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[8,3],[3,-2]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[8, 3], [3, -2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[8, 3], [3, -2]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 3], [3, -2]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0097","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b1d08adef540bd1e1e7cc4a969e2d35e3ff19f7dcdb2353fd2cb88320929181","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274097,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[9,-3],[1,-1]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[9, -3], [1, -1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[9, -3], [1, -1]], [0, 1, -3, 1], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -3], [1, -1]], [0, 1, -3, 1], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0098","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a13de361ec008be8d42854f93e349279938d9d7a3ef77ec8bb21266195a0c6a","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274098,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[10,-2],[-5,0]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[10, -2], [-5, 0]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[10, -2], [-5, 0]], [0, 1, -2, -5], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, -2], [-5, 0]], [0, 1, -2, -5], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0099","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62eac821b9990d381049246859f30a03a76687a8a9e67fad4141bf876b12f334","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274099,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[11,-1],[0,1]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[11, -1], [0, 1]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[11, -1], [0, 1]], [0, 1, -1, 0], No, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1], [0, 1]], [0, 1, -1, 0], No, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r04-real-bilinear-form-0100","batch_id":"batch-20261014-advance-20261027-r04-real-bilinear-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e85d07bcbd30ab017377950339967bb4db95509ff3c781791544e2a87b5e2ce8","family_id":"qf-mod27-bilinear-symmetry-audit","seed":202610274100,"semantic_template":"qf-mod27-bilinear-symmetry-audit/{\"matrix\":[[12,0],[0,2]]}"},"fields":{"title":"Symmetry Is Additional to Bilinearity","stem":"Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[12, 0], [0, 2]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).","answer":{"kind":"expression","value":"([[12, 0], [0, 2]], null, Yes, symmetry_is_extra_not_bilinearity)","explanation":"Compare opposite off-diagonal entries of the entire matrix. A symmetric matrix may still fail positive definiteness; even a zero matrix is bilinear and symmetric without being an inner product. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 0], [0, 2]], null, Yes, symmetry_is_extra_not_bilinearity)."},"skills":["Symmetry Is Additional to Bilinearity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}