{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r04-real-inner-product-space-0001","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99cbb1c122a2e6504d1e508b255767946424011879f20b7285831b3e90a773ce","family_id":"qf-mod28-inner-evaluate","seed":202610284001,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[2,-1],[-1,2]],\"x\":[-2,-3],\"y\":[-5,12]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[2, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, -3], y=[-5, 12]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [-1, 2]], [-2, -3], [-5, 12], -43)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0002","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f771cea234bd371440c60a03907b62d2384b81471d3ee4baf7a73cd20c9e372","family_id":"qf-mod28-inner-evaluate","seed":202610284002,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[3,0],[0,1]],\"x\":[-1,-2],\"y\":[-4,11]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, -2], y=[-4, 11]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], [-1, -2], [-4, 11], -10)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0003","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2efc1b308b7802d6bb9de4e638af275067a79398f0ceda0ed3eedcb1247756e","family_id":"qf-mod28-inner-evaluate","seed":202610284003,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[4,1],[1,2]],\"x\":[0,-1],\"y\":[-3,10]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, -1], y=[-3, 10]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], [0, -1], [-3, 10], -17)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], [0, -1], [-3, 10], -17)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0004","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fffb03ec64dbacfb7a18f04cd5772d40a4cf2387725dedc009667dd21cc8d984","family_id":"qf-mod28-inner-evaluate","seed":202610284004,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[5,2],[2,5]],\"x\":[1,0],\"y\":[-2,9]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, 0], y=[-2, 9]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], [1, 0], [-2, 9], 8)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], [1, 0], [-2, 9], 8)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0005","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d96b33a02d6e1848e04c81466de4ae27dce96f456e20ffe7dacef4eff194c4ac","family_id":"qf-mod28-inner-evaluate","seed":202610284005,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[6,-1],[-1,2]],\"x\":[2,1],\"y\":[-1,8]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, 1], y=[-1, 8]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], [2, 1], [-1, 8], -11)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0006","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01c10f48db9096dae12e764bdbfa3fb21a78b14fea3018f3bec03d9493f9acd2","family_id":"qf-mod28-inner-evaluate","seed":202610284006,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[7,0],[0,1]],\"x\":[-2,2],\"y\":[0,7]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, 2], y=[0, 7]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[7, 0], [0, 1]], [-2, 2], [0, 7], 14)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [0, 1]], [-2, 2], [0, 7], 14)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0007","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0da6ffd43b1d969e6bb4f318a41accc24b1e16da672df6604c56346e7be40f91","family_id":"qf-mod28-inner-evaluate","seed":202610284007,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[8,1],[1,2]],\"x\":[-1,3],\"y\":[1,6]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, 3], y=[1, 6]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], [-1, 3], [1, 6], 25)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], [-1, 3], [1, 6], 25)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0008","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9976c55db85765d5bb8a95ee6e9371f5c20c7d1f71f5a5feb054302f299ffdf6","family_id":"qf-mod28-inner-evaluate","seed":202610284008,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[9,2],[2,5]],\"x\":[0,-3],\"y\":[2,5]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[9, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, -3], y=[2, 5]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[9, 2], [2, 5]], [0, -3], [2, 5], -87)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, 2], [2, 5]], [0, -3], [2, 5], -87)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0009","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f43a24a5c856e281684e8c0dfddb77a408cbb8f6ec6f4fcb463203e76928726","family_id":"qf-mod28-inner-evaluate","seed":202610284009,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[10,-1],[-1,2]],\"x\":[1,-2],\"y\":[3,4]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, -2], y=[3, 4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], [1, -2], [3, 4], 16)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], [1, -2], [3, 4], 16)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0010","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5523e76c9834be8b99d349c8b22a8e06d4b759a613ee57eddb741419cb427819","family_id":"qf-mod28-inner-evaluate","seed":202610284010,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[11,0],[0,1]],\"x\":[2,-1],\"y\":[4,3]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, -1], y=[4, 3]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], [2, -1], [4, 3], 85)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], [2, -1], [4, 3], 85)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0011","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cae3735b1e4a8e7cf020052dc84c47f0701eec61f238609c759ad23e1b18ec1","family_id":"qf-mod28-inner-evaluate","seed":202610284011,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[12,1],[1,2]],\"x\":[-2,0],\"y\":[5,2]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, 0], y=[5, 2]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[12, 1], [1, 2]], [-2, 0], [5, 2], -124)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [1, 2]], [-2, 0], [5, 2], -124)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0012","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faeafa9669397245ed9e72f88beff5391c1c0ca7d6d7202ad0e1267d7f2aebc9","family_id":"qf-mod28-inner-evaluate","seed":202610284012,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[13,2],[2,5]],\"x\":[-1,1],\"y\":[-5,1]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, 1], y=[-5, 1]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], [-1, 1], [-5, 1], 58)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0013","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"805e4741c040c2b7d177dca97773004ba504ba84e0c5c6fc60afd7270f4bbb9d","family_id":"qf-mod28-inner-evaluate","seed":202610284013,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[14,-1],[-1,2]],\"x\":[0,2],\"y\":[-4,0]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, 2], y=[-4, 0]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], [0, 2], [-4, 0], 8)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0014","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"960773470e1cd3ffd3884886dab816b9c90f54de0926d4ff7dcd97b318b40278","family_id":"qf-mod28-inner-evaluate","seed":202610284014,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[15,0],[0,1]],\"x\":[1,3],\"y\":[-3,-1]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, 3], y=[-3, -1]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], [1, 3], [-3, -1], -48)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], [1, 3], [-3, -1], -48)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0015","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78c869d89b074073e97cd88ff045a5e92627dfa80b75b91d5bbedcc29c44cec1","family_id":"qf-mod28-inner-evaluate","seed":202610284015,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[16,1],[1,2]],\"x\":[2,-3],\"y\":[-2,-2]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[16, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, -3], y=[-2, -2]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[16, 1], [1, 2]], [2, -3], [-2, -2], -50)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[16, 1], [1, 2]], [2, -3], [-2, -2], -50)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0016","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0251911063af1f895783176ca3d4ebc4d885e92882c7ea8ebe53347dc3c7148a","family_id":"qf-mod28-inner-evaluate","seed":202610284016,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[17,2],[2,5]],\"x\":[-2,-2],\"y\":[-1,-3]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, -2], y=[-1, -3]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [2, 5]], [-2, -2], [-1, -3], 80)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0017","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98b9535933319e478fae13e5006e41ae6e3ada9a6b827f289fa38db80013dc83","family_id":"qf-mod28-inner-evaluate","seed":202610284017,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[18,-1],[-1,2]],\"x\":[-1,-1],\"y\":[0,-4]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, -1], y=[0, -4]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], [-1, -1], [0, -4], 4)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0018","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17db17d042b93d174d8aab563efda2a54e8e169a009b01b8a7d76e9016a21ebf","family_id":"qf-mod28-inner-evaluate","seed":202610284018,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[19,0],[0,1]],\"x\":[0,0],\"y\":[1,-5]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, 0], y=[1, -5]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], [0, 0], [1, -5], 0)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], [0, 0], [1, -5], 0)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0019","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"260a1bdf24b9728b6e991ee18cea885da760bee146ba7a9933fdfbc9ab866977","family_id":"qf-mod28-inner-evaluate","seed":202610284019,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[20,1],[1,2]],\"x\":[1,1],\"y\":[2,-6]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, 1], y=[2, -6]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], [1, 1], [2, -6], 24)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], [1, 1], [2, -6], 24)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0020","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fd2591fa75bf9329e39e07669bce509262da359551ba4c9d28f3bdb0a2f770d","family_id":"qf-mod28-inner-evaluate","seed":202610284020,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[21,2],[2,5]],\"x\":[2,2],\"y\":[3,-7]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, 2], y=[3, -7]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], [2, 2], [3, -7], 40)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], [2, 2], [3, -7], 40)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0021","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c376d3255f0e1ca8376d86e5f562ca22013c3911e34feae37a409db0c7a9dc8b","family_id":"qf-mod28-inner-evaluate","seed":202610284021,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[22,-1],[-1,2]],\"x\":[-2,3],\"y\":[4,-8]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-2, 3], y=[4, -8]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [-1, 2]], [-2, 3], [4, -8], -252)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0022","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d3eb8a395825c7683c758163413b7420582fe9564cc0aefe24856f16e38a24c","family_id":"qf-mod28-inner-evaluate","seed":202610284022,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[23,0],[0,1]],\"x\":[-1,-3],\"y\":[5,-9]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[23, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[-1, -3], y=[5, -9]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[23, 0], [0, 1]], [-1, -3], [5, -9], -88)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[23, 0], [0, 1]], [-1, -3], [5, -9], -88)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0023","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"436aa19c3e5c694a97839ddca6f03a755074fe00cb063e6ce9110bd9d9282a2e","family_id":"qf-mod28-inner-evaluate","seed":202610284023,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[24,1],[1,2]],\"x\":[0,-2],\"y\":[-5,-10]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[0, -2], y=[-5, -10]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], [0, -2], [-5, -10], 50)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], [0, -2], [-5, -10], 50)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0024","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08a5ad350ef46a0150e3590bc9fd0a545cba252e80c498a3a90575620fb7f1fb","family_id":"qf-mod28-inner-evaluate","seed":202610284024,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[25,2],[2,5]],\"x\":[1,-1],\"y\":[-4,-11]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[1, -1], y=[-4, -11]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], [1, -1], [-4, -11], -59)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], [1, -1], [-4, -11], -59)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0025","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10a22609b665cc5755d854054bb265de71c87d6fd33012cf41a08dd591066a9d","family_id":"qf-mod28-inner-evaluate","seed":202610284025,"semantic_template":"qf-mod28-inner-evaluate/{\"matrix\":[[26,-1],[-1,2]],\"x\":[2,0],\"y\":[-3,-12]}"},"fields":{"title":"Evaluate a Candidate Bilinear Form without Presuming an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. x=[2, 0], y=[-3, -12]. Give (matrix,x,y,value). This evaluation family does not first require the symmetry and positive-definite checks to pass.","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132)","explanation":"Compute A*y then its dot product with x in that order. Retain the value even when the candidate fails inner-product qualification. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], [2, 0], [-3, -12], -132)."},"skills":["Evaluate a Candidate Bilinear Form without Presuming an Inner Product"],"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-20261028-r04-real-inner-product-space-0026","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4da663df29a5c0541ce8e7912fb1533ad34b030f4de3d3ec9d99c0afaeffe848","family_id":"qf-mod28-inner-symmetry","seed":202610284026,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[2,-1],[0,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[2, -1], [0, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[2, -1], [0, 2]], [-1, 0], No)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[2, -1], [0, 2]], [-1, 0], No)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0027","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"335e22466b761e7354c6b0f0f3071fcc01bb647010752bdf863716f2d56c714d","family_id":"qf-mod28-inner-symmetry","seed":202610284027,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[3,0],[0,1]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], [0, 0], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], [0, 0], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0028","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f6b1716664701f80a3f44c77104af7e37a60be2a2639f836c162d042d56accc","family_id":"qf-mod28-inner-symmetry","seed":202610284028,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[4,1],[1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], [1, 1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], [1, 1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0029","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c97f87a7a6f1a1554fef6d56edbdb87fcf58c70020181a966ee1b9c453284f6","family_id":"qf-mod28-inner-symmetry","seed":202610284029,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[5,2],[2,5]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], [2, 2], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], [2, 2], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0030","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7c3cd9bf57bfcd409959067e38e8df225004581b1019fd513bdb53e55f2d2ed","family_id":"qf-mod28-inner-symmetry","seed":202610284030,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[6,-1],[-1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], [-1, -1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], [-1, -1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0031","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f428e6342b6948a1759d66844559e801983d62b966d2a6384e08979cce23f46b","family_id":"qf-mod28-inner-symmetry","seed":202610284031,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[7,0],[1,1]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [1, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[7, 0], [1, 1]], [0, 1], No)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [1, 1]], [0, 1], No)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0032","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad1fcd543280a41da75883bc8a79536f66f19e41d576ba6f2260e524f8445feb","family_id":"qf-mod28-inner-symmetry","seed":202610284032,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[8,1],[1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], [1, 1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], [1, 1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0033","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9855c4d2d5801fe7cfa50023cbfe59901eda31924612fdb285733aaf25de41c","family_id":"qf-mod28-inner-symmetry","seed":202610284033,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[9,2],[2,5]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[9, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[9, 2], [2, 5]], [2, 2], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[9, 2], [2, 5]], [2, 2], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0034","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d28dcecac9da85dd6f052971672c3b24ce7113127be4571af353e59912138e6c","family_id":"qf-mod28-inner-symmetry","seed":202610284034,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[10,-1],[-1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], [-1, -1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], [-1, -1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0035","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28ba7af2f352efd126746128b7a7c09c40a0345e6c5d7d61372bb8060e840365","family_id":"qf-mod28-inner-symmetry","seed":202610284035,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[11,0],[0,1]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], [0, 0], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], [0, 0], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0036","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c87398be337942ef6ec74be6ae865796c3b1ca5120f05c37327e1d7082d2807","family_id":"qf-mod28-inner-symmetry","seed":202610284036,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[12,1],[2,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [2, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[12, 1], [2, 2]], [1, 2], No)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [2, 2]], [1, 2], No)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0037","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f44ad61931ca5782403b6ac8c719096522c5e84e5cb6d124c85e84c57dacbdd","family_id":"qf-mod28-inner-symmetry","seed":202610284037,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[13,2],[2,5]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], [2, 2], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], [2, 2], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0038","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85cb2a63a38b3caf67376b85a971cc7c8df704877dd91d7102bcb115be3f24b1","family_id":"qf-mod28-inner-symmetry","seed":202610284038,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[14,-1],[-1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], [-1, -1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], [-1, -1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0039","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9cda7de65cee74c13ec23ab1f1135ddad7966662ab160de1d93671bf36ba0a9","family_id":"qf-mod28-inner-symmetry","seed":202610284039,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[15,0],[0,1]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], [0, 0], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], [0, 0], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0040","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5cce0ee29eca8005a056a9b2cd385bf5c59ce33efa5f6b8ee54e88fb489589d","family_id":"qf-mod28-inner-symmetry","seed":202610284040,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[16,1],[1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[16, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[16, 1], [1, 2]], [1, 1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[16, 1], [1, 2]], [1, 1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0041","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccc2c4aee42e571d90f0de886327b65b33f1c9a9e9087b3765b2f40c316f13ec","family_id":"qf-mod28-inner-symmetry","seed":202610284041,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[17,2],[3,5]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [3, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[17, 2], [3, 5]], [2, 3], No)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [3, 5]], [2, 3], No)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0042","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"089ca426de2efb7b7320848290b7096968d85b46507a47177824f8472862f825","family_id":"qf-mod28-inner-symmetry","seed":202610284042,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[18,-1],[-1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], [-1, -1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], [-1, -1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0043","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbc5eb910c97eaa18cb36202023831a3e589301c58359a709c2ca70451d0ea66","family_id":"qf-mod28-inner-symmetry","seed":202610284043,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[19,0],[0,1]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], [0, 0], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], [0, 0], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0044","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17757b891421914073cd8e0f4adbd3f140fadd901fae3b7adcb4896f172951d7","family_id":"qf-mod28-inner-symmetry","seed":202610284044,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[20,1],[1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], [1, 1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], [1, 1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0045","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c54b59c1a6601796e03073940a0c7ef5da5d87e2afc8cc244829ef668ce4d74d","family_id":"qf-mod28-inner-symmetry","seed":202610284045,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[21,2],[2,5]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], [2, 2], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], [2, 2], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0046","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17bde673134b1d04737984a5464e71228cfafadb038caa1772e7a8e5df29fc34","family_id":"qf-mod28-inner-symmetry","seed":202610284046,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[22,-1],[0,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [0, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[22, -1], [0, 2]], [-1, 0], No)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [0, 2]], [-1, 0], No)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0047","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7ce33d490af408b6d68826f75633d66a4dae2dff58c4e240d4a9f489e234b4","family_id":"qf-mod28-inner-symmetry","seed":202610284047,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[23,0],[0,1]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[23, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[23, 0], [0, 1]], [0, 0], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[23, 0], [0, 1]], [0, 0], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0048","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f0e621a7dec247174ac0a4b055ea83d9e6e7c63019ba890fa7c0c74ec738ee5","family_id":"qf-mod28-inner-symmetry","seed":202610284048,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[24,1],[1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], [1, 1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], [1, 1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0049","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca78366ae1615710fe54a329bf8a823e79a72999416e0431da7aba0acaa291fb","family_id":"qf-mod28-inner-symmetry","seed":202610284049,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[25,2],[2,5]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], [2, 2], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], [2, 2], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0050","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b6b886c38a14cc309410702b4770ed5bb629fec470545d2fe81983ec38e7bba","family_id":"qf-mod28-inner-symmetry","seed":202610284050,"semantic_template":"qf-mod28-inner-symmetry/{\"matrix\":[[26,-1],[-1,2]]}"},"fields":{"title":"Symmetry Alone Does Not Establish an Inner Product","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], [-1, -1], Yes)","explanation":"Compare both off-diagonal entries. Bilinearity is already provided by the matrix formula, but strict positive definiteness still needs its own certificate. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], [-1, -1], Yes)."},"skills":["Symmetry Alone Does Not Establish an Inner Product"],"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-20261028-r04-real-inner-product-space-0051","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce7af04e413c64ce3043348835b616322c194e79ccf0b137074eecff97a9a4df","family_id":"qf-mod28-inner-positive-definite","seed":202610284051,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[1,0],[0,0]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[1, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[1, 0], [0, 0]], Yes, 1, 0, No, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 0], [0, 0]], Yes, 1, 0, No, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0052","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f2325ff42948b5fcde5f6b493557bab9bce9b1597386706e94a0eaada89b7ef","family_id":"qf-mod28-inner-positive-definite","seed":202610284052,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[3,0],[0,1]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], Yes, 3, 3, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], Yes, 3, 3, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0053","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ca43811a6502373dcfc3fc19085321d92f60084c75f7244471375487c80f48f","family_id":"qf-mod28-inner-positive-definite","seed":202610284053,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[4,1],[1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], Yes, 4, 7, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], Yes, 4, 7, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0054","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e58a6ea39831afbb83efe2378fda626e809c2e22be3771e63d044692662bc4b","family_id":"qf-mod28-inner-positive-definite","seed":202610284054,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[5,2],[2,5]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], Yes, 5, 21, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], Yes, 5, 21, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0055","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb0c02c38f5a250a4b58fa926be243d81d26739c7f0ecc615e6c3c87dbe1714","family_id":"qf-mod28-inner-positive-definite","seed":202610284055,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[6,-1],[-1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], Yes, 6, 11, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], Yes, 6, 11, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0056","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c111609089fa4be23db8eda9124c980d210890ad279978eb98db103fb5bcade","family_id":"qf-mod28-inner-positive-definite","seed":202610284056,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[7,0],[0,1]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[7, 0], [0, 1]], Yes, 7, 7, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [0, 1]], Yes, 7, 7, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0057","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f250a5700e5dbbd39e849f4f046ee0afacd11272ed9d65edccb55793a4053fb1","family_id":"qf-mod28-inner-positive-definite","seed":202610284057,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[8,1],[1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], Yes, 8, 15, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], Yes, 8, 15, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0058","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03a1015169e991c21a125c43fd5316d1bdebffa00efbb19eda525d10af69ad20","family_id":"qf-mod28-inner-positive-definite","seed":202610284058,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[8,0],[0,0]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[8, 0], [0, 0]], Yes, 8, 0, No, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 0], [0, 0]], Yes, 8, 0, No, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0059","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f2f7de2583d313c8a3dff95928cf2ab5bfe597e51a9f570a2743fcbceabcbac","family_id":"qf-mod28-inner-positive-definite","seed":202610284059,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[10,-1],[-1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], Yes, 10, 19, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], Yes, 10, 19, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0060","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43d3140b1a2b288a6518ffcedec7f93dee464647804ce872c385cb8cefd9792b","family_id":"qf-mod28-inner-positive-definite","seed":202610284060,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[11,0],[0,1]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], Yes, 11, 11, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], Yes, 11, 11, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0061","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fec944d47a9bf9449f1083c4cefcafe52ca5f223f1fcdce24ac155a44807ad8","family_id":"qf-mod28-inner-positive-definite","seed":202610284061,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[12,1],[1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[12, 1], [1, 2]], Yes, 12, 23, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [1, 2]], Yes, 12, 23, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0062","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"543279f8a65acf0224f67e8d76c26ebf602672afa4c7987990d1cd16aa7cc91d","family_id":"qf-mod28-inner-positive-definite","seed":202610284062,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[13,2],[2,5]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], Yes, 13, 61, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], Yes, 13, 61, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0063","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1a87730f0fe25f6fc9616dc5d23fe93dbf4c39f4f8b1c76ec20c2e83ed51f79","family_id":"qf-mod28-inner-positive-definite","seed":202610284063,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[14,-1],[-1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], Yes, 14, 27, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], Yes, 14, 27, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0064","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"877f10c1751a15637b7929d5114f4767b7c90b535b476c0e451ccfd06b56fd6c","family_id":"qf-mod28-inner-positive-definite","seed":202610284064,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[15,0],[0,1]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], Yes, 15, 15, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], Yes, 15, 15, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0065","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94f90faf9fa5c5a95e518f9f1ea763c7d86f84c444ff6cfb87f8ed884e68773c","family_id":"qf-mod28-inner-positive-definite","seed":202610284065,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[15,0],[0,0]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[15, 0], [0, 0]], Yes, 15, 0, No, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 0]], Yes, 15, 0, No, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0066","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53c382c95e3feb3b71a73d583995cb7623115a8d63b69ce08f7fbfdfde44a933","family_id":"qf-mod28-inner-positive-definite","seed":202610284066,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[17,2],[2,5]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[17, 2], [2, 5]], Yes, 17, 81, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [2, 5]], Yes, 17, 81, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0067","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ecc0634393fca1da639f4f908d9c399075aef7f9b6812924b8bc01aad5af61d","family_id":"qf-mod28-inner-positive-definite","seed":202610284067,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[18,-1],[-1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], Yes, 18, 35, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], Yes, 18, 35, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0068","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7746b404a4f75b264bec2fa314f81799649b57fa08b8cf361762b1815f12e9aa","family_id":"qf-mod28-inner-positive-definite","seed":202610284068,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[19,0],[0,1]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], Yes, 19, 19, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], Yes, 19, 19, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0069","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9b6b38a49b2df26d26ef8daf412be1e5a090971bb836616057d04bc55b09371","family_id":"qf-mod28-inner-positive-definite","seed":202610284069,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[20,1],[1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], Yes, 20, 39, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], Yes, 20, 39, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0070","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f64f30b78466d6eaea9c7eff04a9c4e26933e84fb13d9a6e3dffc22baa5a12b","family_id":"qf-mod28-inner-positive-definite","seed":202610284070,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[21,2],[2,5]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], Yes, 21, 101, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], Yes, 21, 101, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0071","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0b262fd3c2d4e39736255213c073b26e060437e03f375b10e1f69ccc31df4d6","family_id":"qf-mod28-inner-positive-definite","seed":202610284071,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[22,-1],[-1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[22, -1], [-1, 2]], Yes, 22, 43, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [-1, 2]], Yes, 22, 43, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0072","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb8c16d05d0cbda560ec528bc81c9bb793967baba100419d712853ef94467b2e","family_id":"qf-mod28-inner-positive-definite","seed":202610284072,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[22,0],[0,0]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[22, 0], [0, 0]], Yes, 22, 0, No, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, 0], [0, 0]], Yes, 22, 0, No, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0073","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80fda53c2f9c920f418275a8ad461f07fd3944e4a3726516626548d1a1a2e402","family_id":"qf-mod28-inner-positive-definite","seed":202610284073,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[24,1],[1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], Yes, 24, 47, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], Yes, 24, 47, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0074","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bb74c6ad12ee88edde8e278ae5942316b62d58937ecd7e5224f6422d23e82e8","family_id":"qf-mod28-inner-positive-definite","seed":202610284074,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[25,2],[2,5]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], Yes, 25, 121, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], Yes, 25, 121, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0075","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04f8d06e8fd24851e0ce6a4f98ce8185bd8887ad4e08fe4054a6d4795c4d8ae6","family_id":"qf-mod28-inner-positive-definite","seed":202610284075,"semantic_template":"qf-mod28-inner-positive-definite/{\"matrix\":[[26,-1],[-1,2]]}"},"fields":{"title":"A Symmetric Sylvester Certificate, Not Positive Sample Values","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,symmetric?,leading_minor,determinant,positive_definite?,Sylvester_R2_universal_certificate). In this inner-product protocol the positive_definite flag conjoins symmetry, a positive leading minor and positive determinant.","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], Yes, 26, 51, Yes, Sylvester_R2_universal_certificate)","explanation":"Evaluate all exact conditions. For a nonsymmetric candidate, No rejects this symmetric certificate; it is not a separate classification of the quadratic form after discarding its antisymmetric part. A zero determinant fails strict positivity. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], Yes, 26, 51, Yes, Sylvester_R2_universal_certificate)."},"skills":["A Symmetric Sylvester Certificate, Not Positive Sample Values"],"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-20261028-r04-real-inner-product-space-0076","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ab855b09070fb32faf7e673eee332f40f5677a9e17817186bde2c10026c0d40","family_id":"qf-mod28-inner-qualification","seed":202610284076,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[1,0],[0,0]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[1, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[1, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[1, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0077","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e614af0f0e600a7cc3bffafd2f1bd8e4f3f60ceb9b9538a796d93fbbac4fa919","family_id":"qf-mod28-inner-qualification","seed":202610284077,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[3,0],[0,1]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[3, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[3, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[3, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0078","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fde075bfb4c46171b29f07c191789246ef97c97a57b5e948b4936be5527b0b0f","family_id":"qf-mod28-inner-qualification","seed":202610284078,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[4,1],[1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[4, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[4, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[4, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0079","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40a9dc89509f34d56c6a07b5c095f9dee2f8fc4fd8fa65623ad7953dec5f530b","family_id":"qf-mod28-inner-qualification","seed":202610284079,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[5,2],[2,5]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[5, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[5, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0080","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78a74e47333864e465455d13afc33dfb330378bfe2c70c1ed45f448b6ba74601","family_id":"qf-mod28-inner-qualification","seed":202610284080,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[6,-1],[-1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[6, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[6, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[6, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0081","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1155a8d05e617cbdc91c3243ac91b4507ee35582654f8c698bd8a0fa716020fc","family_id":"qf-mod28-inner-qualification","seed":202610284081,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[7,0],[1,1]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[7, 0], [1, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[7, 0], [1, 1]], Yes, No, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[7, 0], [1, 1]], Yes, No, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0082","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63bb5643fd5811b66e333d1b97cb7282e25caf9d66405828eb86ea6ac33b8e4d","family_id":"qf-mod28-inner-qualification","seed":202610284082,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[8,1],[1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[8, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0083","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c35a53f9dd95bfe8b6cf7e618080c72e64f83574de45e1d54a7bf6e465009f8c","family_id":"qf-mod28-inner-qualification","seed":202610284083,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[8,0],[0,0]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[8, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[8, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[8, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0084","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c18ef6b08f2f57a2706e28ae4b97db634660a37137bc75089ba535ae43b8496","family_id":"qf-mod28-inner-qualification","seed":202610284084,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[10,-1],[-1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[10, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[10, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[10, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0085","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41eca27ef52ce9e9aed552e19aa3af7e2128773f82880febcac80c336035d690","family_id":"qf-mod28-inner-qualification","seed":202610284085,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[11,0],[0,1]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[11, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[11, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[11, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0086","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fa59f763e7c6ad703c5ee8fadf430f0ac9d4efce0be488ba7980f20968f85ca","family_id":"qf-mod28-inner-qualification","seed":202610284086,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[12,1],[2,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[12, 1], [2, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[12, 1], [2, 2]], Yes, No, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[12, 1], [2, 2]], Yes, No, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0087","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"183646d0f661ecc5444e17073446a46f5b4a85dd4b0c38c8f94aed2bbf590009","family_id":"qf-mod28-inner-qualification","seed":202610284087,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[13,2],[2,5]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[13, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[13, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[13, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0088","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66f4bf47dfc44aaf6d99db33a4a065f37597dca1683d4a1ab8a3c6f20aa3128b","family_id":"qf-mod28-inner-qualification","seed":202610284088,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[14,-1],[-1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[14, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[14, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[14, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0089","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bb58b4ebbb1af3e3bc05190c928fd8046bbd8290cac7726930d63ecb56f2f4f","family_id":"qf-mod28-inner-qualification","seed":202610284089,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[15,0],[0,1]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[15, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0090","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec0dcca46d5f8c306bddf855182d111d974a2d7f51e67228a47d4c6e1780a1e3","family_id":"qf-mod28-inner-qualification","seed":202610284090,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[15,0],[0,0]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[15, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[15, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0091","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d295281a52956da35fcd9fe0c78c9aef9c7a917394f02358312afe0a5d452978","family_id":"qf-mod28-inner-qualification","seed":202610284091,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[17,2],[3,5]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[17, 2], [3, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[17, 2], [3, 5]], Yes, No, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[17, 2], [3, 5]], Yes, No, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0092","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9655c4dea27915f51edab471a53acb2a51f2134da15ddc3247ca246998cdc29","family_id":"qf-mod28-inner-qualification","seed":202610284092,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[18,-1],[-1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[18, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[18, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[18, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0093","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bdd37d627081425165fd9c38f20b8218d7087157252702576d876171851a47e","family_id":"qf-mod28-inner-qualification","seed":202610284093,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[19,0],[0,1]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[19, 0], [0, 1]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[19, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[19, 0], [0, 1]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0094","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a2dea72d2151586cdd6cc4d7f04e085ea7982d328e61c6f96ee22a41ba6d239","family_id":"qf-mod28-inner-qualification","seed":202610284094,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[20,1],[1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[20, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[20, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[20, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0095","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01c9edbf196639787dd1c0c75abebf27188115839477802582febb3647f816e6","family_id":"qf-mod28-inner-qualification","seed":202610284095,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[21,2],[2,5]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[21, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[21, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[21, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0096","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a0b1534b19a428cee08dec712ac50875eddeeac6db31eb691191d40096080a3","family_id":"qf-mod28-inner-qualification","seed":202610284096,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[22,-1],[0,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, -1], [0, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[22, -1], [0, 2]], Yes, No, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, -1], [0, 2]], Yes, No, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0097","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a40f1c961123ad8aa3ed03c6278fd4b109772da2e32bd285edeb34494000e8b8","family_id":"qf-mod28-inner-qualification","seed":202610284097,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[22,0],[0,0]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[22, 0], [0, 0]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[22, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[22, 0], [0, 0]], Yes, Yes, No, No, not_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0098","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64da5899a09961e11de2f1e4d894404f606a3468d4079292ca0719f1bcc15af7","family_id":"qf-mod28-inner-qualification","seed":202610284098,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[24,1],[1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[24, 1], [1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[24, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[24, 1], [1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0099","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffb4b97bfb76b24ef43ab8060d05411540e3315396f479bbb647332355c50da2","family_id":"qf-mod28-inner-qualification","seed":202610284099,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[25,2],[2,5]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[25, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[25, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[25, 2], [2, 5]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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-20261028-r04-real-inner-product-space-0100","batch_id":"batch-20261014-advance-20261028-r04-real-inner-product-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb9c817a27b55b330b96eba1066faeaa0eb465467d566d82c6e2d2600b1576c7","family_id":"qf-mod28-inner-qualification","seed":202610284100,"semantic_template":"qf-mod28-inner-qualification/{\"matrix\":[[26,-1],[-1,2]]}"},"fields":{"title":"Conjoin Bilinearity, Symmetry and Strict Positive Definiteness","stem":"Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[26, -1], [-1, 2]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,bilinear?,symmetric?,positive_definite?,inner_product?,classification). Classification is real_inner_product or not_inner_product.","answer":{"kind":"expression","value":"([[26, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)","explanation":"The matrix formula supplies the bilinear flag. Qualify the other two obligations independently before conjoining them; a symmetric positive-semidefinite form does not suffice. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[26, -1], [-1, 2]], Yes, Yes, Yes, Yes, real_inner_product)."},"skills":["Conjoin Bilinearity, Symmetry and Strict Positive Definiteness"],"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."}}