{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r04-linear-independence-basis-0001","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bd87276d5adf26805e829f0f4006739d172f5d15641a0a919e45ec83a7b92a1","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254001,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-12,-3],\"target\":[-11,-3],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-12, -3], target=[-11, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-12, -3], [-11, -3], [-12, -3], No)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-12, -3], [-11, -3], [-12, -3], No)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0002","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f6e84826f9e1ec5920865c6d69b110a66eb676ccf4ca17775b82ec2a57e5d3b","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254002,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-11,-2],\"target\":[-11,-2],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-11, -2], target=[-11, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-11, -2], [-11, -2], [-11, -2], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-11, -2], [-11, -2], [-11, -2], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0003","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5511d36895fc272919288aba9ebe530c05e932e471fc8f89e59bf29ab9ef1af8","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254003,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-10,-1],\"target\":[-10,-1],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-10, -1], target=[-10, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-10, -1], [-10, -1], [-10, -1], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-10, -1], [-10, -1], [-10, -1], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0004","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d763aff684e1e0600d37e2c0b8853dc3fb5cb795c2bfe96565fd963e93424d7","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254004,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-9,0],\"target\":[-9,0],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-9, 0], target=[-9, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-9, 0], [-9, 0], [-9, 0], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-9, 0], [-9, 0], [-9, 0], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0005","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71ed4df3b84987a231f1ed756306ad72f49ce9a327c8ca32f3c586981c3add2f","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254005,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-8,1],\"target\":[-8,1],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-8, 1], target=[-8, 1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-8, 1], [-8, 1], [-8, 1], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-8, 1], [-8, 1], [-8, 1], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0006","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"539a1043e3a54338bd4d914fa133b5c78d53042939c6e104cd3a449f310a5319","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254006,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-7,2],\"target\":[-6,2],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-7, 2], target=[-6, 2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-7, 2], [-6, 2], [-7, 2], No)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-7, 2], [-6, 2], [-7, 2], No)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0007","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70e1925c2837b849737231154a266ca12c3c94e9739963a27915d647c57b83b9","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254007,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-6,3],\"target\":[-6,3],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-6, 3], target=[-6, 3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-6, 3], [-6, 3], [-6, 3], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-6, 3], [-6, 3], [-6, 3], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0008","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20408888f1790f247ebaa9556e534b7a0fc07fde324b041e7490811f8b1fa1ee","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254008,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-5,-3],\"target\":[-5,-3],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-5, -3], target=[-5, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-5, -3], [-5, -3], [-5, -3], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-5, -3], [-5, -3], [-5, -3], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0009","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f34be4a1ac4282312c11f3370dc48a4da0b5728a6b71565b7f7b2565ed8fc3c9","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254009,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-4,-2],\"target\":[-4,-2],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-4, -2], target=[-4, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-4, -2], [-4, -2], [-4, -2], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-4, -2], [-4, -2], [-4, -2], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0010","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebec240a9e83f112ab81495a6680af1c704a713213a85f7310b8a106a42ff386","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254010,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-3,-1],\"target\":[-3,-1],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-3, -1], target=[-3, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-3, -1], [-3, -1], [-3, -1], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-3, -1], [-3, -1], [-3, -1], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0011","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d80e52b1b1998fc4ab442cf120d873658275f152c20c046992cde75925a7082","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254011,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-2,0],\"target\":[-1,0],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-2, 0], target=[-1, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-2, 0], [-1, 0], [-2, 0], No)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-2, 0], [-1, 0], [-2, 0], No)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0012","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"825e7092a9e238d2da3f90df03b816d6cca83cc299de5e8352d7cfaf209e4684","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254012,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[-1,1],\"target\":[-1,1],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[-1, 1], target=[-1, 1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-1, 1], [-1, 1], [-1, 1], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-1, 1], [-1, 1], [-1, 1], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0013","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b52c6417dfc6d92df20683ba2c5a5ba2b99d9b48a9850a3be36b17c49f2794f8","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254013,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[0,2],\"target\":[0,2],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[0, 2], target=[0, 2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [0, 2], [0, 2], [0, 2], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [0, 2], [0, 2], [0, 2], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0014","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"231bfbc252c63af79bc8aaa7726d653a0430ab15207021a650b20798bf031e29","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254014,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[1,3],\"target\":[1,3],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[1, 3], target=[1, 3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [1, 3], [1, 3], [1, 3], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [1, 3], [1, 3], [1, 3], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0015","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"772a4c3285a6cc47dc3bf42542040193d6691610b0226f04b6f080e5a53e6232","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254015,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[2,-3],\"target\":[2,-3],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[2, -3], target=[2, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [2, -3], [2, -3], [2, -3], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [2, -3], [2, -3], [2, -3], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0016","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e6469e923df7447aa694453ed1d81e8c15ae1167d8c4f21cac0001f856d78e0","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254016,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[3,-2],\"target\":[4,-2],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[3, -2], target=[4, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [3, -2], [4, -2], [3, -2], No)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [3, -2], [4, -2], [3, -2], No)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0017","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"440f00a903ab286b86559dd2abb98a4041cb3f3a234bd589d90e0cf7fb9c5741","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254017,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[4,-1],\"target\":[4,-1],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[4, -1], target=[4, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [4, -1], [4, -1], [4, -1], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [4, -1], [4, -1], [4, -1], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0018","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70d20a94c9907e9eceb457e0d9d52020d6d5b334fc207a0d1a767eb611018957","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254018,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[5,0],\"target\":[5,0],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[5, 0], target=[5, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [5, 0], [5, 0], [5, 0], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [5, 0], [5, 0], [5, 0], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0019","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d9c7bad0f7b896808a3ac21742a882a71a0746c3aca02d334b825cc75758468","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254019,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[6,1],\"target\":[6,1],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[6, 1], target=[6, 1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [6, 1], [6, 1], [6, 1], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [6, 1], [6, 1], [6, 1], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0020","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44aa3ac2bbeb181a7351da157086fd658d3f4174181ff29e5c3b81db5fef7828","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254020,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[7,2],\"target\":[7,2],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[7, 2], target=[7, 2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [7, 2], [7, 2], [7, 2], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [7, 2], [7, 2], [7, 2], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0021","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1636d0bf93a343e4fc2c7ae0e15fca5f4f5b12bc8661a928c323a81f4339df49","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254021,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[8,3],\"target\":[9,3],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[8, 3], target=[9, 3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [8, 3], [9, 3], [8, 3], No)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [8, 3], [9, 3], [8, 3], No)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0022","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9492e836f3255889279efb90118cc88efe808fd10a5d500ef0bde32952e9f74","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254022,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[9,-3],\"target\":[9,-3],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[9, -3], target=[9, -3]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [9, -3], [9, -3], [9, -3], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [9, -3], [9, -3], [9, -3], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0023","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b83a65d5f5809b5c51dd8b8d1b48fc67d3fb689ec252817c1153afa7b6e25b4a","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254023,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[10,-2],\"target\":[10,-2],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[10, -2], target=[10, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [10, -2], [10, -2], [10, -2], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [10, -2], [10, -2], [10, -2], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0024","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a14f2069a39fe900b1e94c3e767df3d46d0aac47bc943bdb84982a2df2c772ce","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254024,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[11,-1],\"target\":[11,-1],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[11, -1], target=[11, -1]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [11, -1], [11, -1], [11, -1], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [11, -1], [11, -1], [11, -1], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0025","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e198383391f0938335f66624199c718af5676a1861fa0818bdc7ab71c4083ee","family_id":"qf-mod25-basisdeep-span-witness-audit","seed":202610254025,"semantic_template":"qf-mod25-basisdeep-span-witness-audit/{\"coefficients\":[12,0],\"target\":[12,0],\"vectors\":[[1,0],[0,1]]}"},"fields":{"title":"Explicit Reconstruction without Conflating Span and Independence","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1]], coefficients=[12, 0], target=[12, 0]. Give (original vectors,coefficients,target,reconstructed sum,equal?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [12, 0], [12, 0], [12, 0], Yes)","explanation":"A matching coefficient list witnesses membership in the span; it does not by itself make the generators independent or a basis. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [12, 0], [12, 0], [12, 0], Yes)."},"skills":["Explicit Reconstruction without Conflating Span and Independence"],"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-20261013-advance-20261025-r04-linear-independence-basis-0026","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2005420fad790dc531fd8416f9e492127ec7ba65466a5c3c2a0b40cac8a62cb","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254026,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[-2,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [-2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 0], [-2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0027","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eed898ccacdc1ec3e1557531a5bd04e26fb9d99d7b7e90c8e6e7f50f0478514","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254027,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-11],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -11], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -11], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -11], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0028","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd92cc6a64b56eeefd3060e7d6e0274c83c5ac545b33e7b2eabe7b779cb71d8c","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254028,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-10],[0,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -10], [0, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -10], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -10], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0029","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3d6e48f45f1067f92d2bbc431b33f4f2e475bc2c0473292e680c388141f1df6","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254029,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[1,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[0, 0], [1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0030","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ef977ebbe4fa451938f81364de80f0d3290427bba010e0d4ecbf52cb2adda11","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254030,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-8],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -8], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -8], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -8], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0031","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d96e4675cc27de57187b82bc0fa3697fc64a0ab0e00cf36227362c09fed5acf","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254031,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-7],[-2,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -7], [-2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -7], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -7], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0032","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca26692818a8f874b1fe87f95ce5d04c0bdce93b1c1068cd308b2ec2925ab0f1","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254032,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[-1,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [-1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 0], [-1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0033","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc999f3d9b2b1d0b0356117b466b0aaaf04c7e3f98aaf34cc7e2ca4dd104f0e1","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254033,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-5],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -5], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -5], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -5], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0034","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ea5af565ff2f05048c97e07ccb550e0e4902cc8a1e22dba68630c0064d0f1fc","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254034,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-4],[1,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -4], [1, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -4], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -4], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0035","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b68c6b8c5ce257a2c347aa60432d854b1bf7350fb33035b3d6bddc82e66d6d8","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254035,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[2,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[0, 0], [2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0036","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e8cc11bb7676f0573ea911f764ddcddbc7cb3b99c03e135acdcd14f7978ad23","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254036,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-2],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -2], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -2], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0037","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73b020c01c355e05d4b9651df070021f852cbe4e87a45c23ee246b587f7a7a3e","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254037,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,-1],[-1,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, -1], [-1, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, -1], [-1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [-1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0038","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e472ca15163885c447a1094c1bd7402ebc58ad23fd36539d19666b58f7786fc","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254038,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[0,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [0, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0039","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9fac5baecd605a7e2c482e032dabf7a651714a56e62c6b09a2797086e12fba0","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254039,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,1],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 1], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 1], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0040","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b14ab217fa788bd021e847c2e00370806f16e2563b374687576c75ca2cd33be","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254040,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,2],[2,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 2], [2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 2], [2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0041","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bd77f4502f5884b20a8fce67511f69030ff6dad5814b7a21349d600a485ff6f","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254041,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[-2,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [-2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[0, 0], [-2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [-2, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0042","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b65142f3a6478a27cbe65acb7334ae24246ea53b2435895bdb6853582e3a23c","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254042,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,4],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 4], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 4], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 4], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0043","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"938f830550c0a5cba06478718ab8394b0587077884541d5f6f68d77aa9608faa","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254043,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,5],[0,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 5], [0, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 5], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 5], [0, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0044","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96004e6c93c32855b7a247cdcf935f05b9f84d68145aee813473427ab9546c57","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254044,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[1,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 0], [1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [1, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0045","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da60c03d0025c5be363f5db254f3ac52df00b2d238c479364c5c475ee1ea4b84","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254045,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,7],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 7], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 7], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 7], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0046","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88b3a3e7729610b61fb54083ce93d24edbdb2bc07ac097d52b745b3e83b22398","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254046,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,8],[-2,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 8], [-2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 8], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 8], [-2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0047","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47910fc89fdcc35546d678eb5c171bf1f4819123009c2430915d96b6ab58ac90","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254047,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[0,0],[-1,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[0, 0], [-1, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[0, 0], [-1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [-1, 1], [1, 1]], 0, [0, 0], Yes, [1, 0, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0048","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fba8c9e6e666501c807361021e593f7c03a5e4c00a70b9845ae13fe647d83d1a","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254048,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,10],[0,0],[1,1]],\"zero_index\":1}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 10], [0, 0], [1, 1]], selected zero-based index=1. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 10], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 10], [0, 0], [1, 1]], 1, [0, 0], Yes, [0, 1, 0], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0049","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a3994bd24b5701c779464c616a727b878ec2efe7bdcedb7be421d0848e74dc7","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254049,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,11],[1,1],[0,0]],\"zero_index\":2}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 11], [1, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 11], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 11], [1, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0050","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d342a42e7e2ac3030b795591b60e5ddfa5e8db9932fd7e1601d453066abaa560","family_id":"qf-mod25-basisdeep-zero-vector-dependence-certificate","seed":202610254050,"semantic_template":"qf-mod25-basisdeep-zero-vector-dependence-certificate/{\"vectors\":[[1,0],[2,1],[1,1]],\"zero_index\":0}"},"fields":{"title":"A Selected Zero Vector Gives a Nontrivial Zero Relation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 0], [2, 1], [1, 1]], selected zero-based index=0. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.","answer":{"kind":"expression","value":"([[1, 0], [2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)","explanation":"The proposed coefficients are nontrivial by construction. They give a zero sum exactly when the selected vector is zero. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [2, 1], [1, 1]], 0, [1, 0], No, [1, 0, 0], No)."},"skills":["A Selected Zero Vector Gives a Nontrivial Zero Relation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0051","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"038aceb2f001ce2ffa1ca86b28def7b6c6005e17d33024a03c77d1ff196fd05b","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254051,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[1,2],\"right\":[1,3],\"scalar\":1}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[1, 2], right=[1, 3], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([1, 2], [1, 3], 1, [1, 2], [0, -1], No)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2], [1, 3], 1, [1, 2], [0, -1], No)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0052","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4e89cf1399f482f970ad21cf276dbdfbab401ace240420f5e0bd5c6d99dd4de","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254052,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[2,3],\"right\":[4,6],\"scalar\":2}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[2, 3], right=[4, 6], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([2, 3], [4, 6], 2, [4, 6], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3], [4, 6], 2, [4, 6], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0053","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c83fa1873021930b728d5cb296accb6d800a779717a8a1d7a04d117969090a51","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254053,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[3,4],\"right\":[9,12],\"scalar\":3}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[3, 4], right=[9, 12], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([3, 4], [9, 12], 3, [9, 12], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 4], [9, 12], 3, [9, 12], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0054","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d92f2074fc678c7e63f94b3d662ba4561f940bac815af855513ca37ff9033216","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254054,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[4,5],\"right\":[16,20],\"scalar\":4}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[4, 5], right=[16, 20], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([4, 5], [16, 20], 4, [16, 20], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, 5], [16, 20], 4, [16, 20], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0055","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55eb498cf456549cbd84d336b7a6b20840c0b0ba9467b31b13b844f0a83d3163","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254055,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[5,6],\"right\":[25,30],\"scalar\":5}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[5, 6], right=[25, 30], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([5, 6], [25, 30], 5, [25, 30], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 6], [25, 30], 5, [25, 30], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0056","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5749e97b066b4e372d183642c03057593f80e9c0229d3e9d7a3edd6c7bfc4e83","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254056,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[6,7],\"right\":[6,7],\"scalar\":1}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[6, 7], right=[6, 7], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([6, 7], [6, 7], 1, [6, 7], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 7], [6, 7], 1, [6, 7], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0057","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07e75bb8a94d95b18391a0a21d646c1a159f8cf253ef371de6f8fd77657c44e7","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254057,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[7,8],\"right\":[14,17],\"scalar\":2}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[7, 8], right=[14, 17], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([7, 8], [14, 17], 2, [14, 16], [0, -1], No)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 8], [14, 17], 2, [14, 16], [0, -1], No)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0058","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31bdb7bfaa2945949e33e5a31466cb099aa68cf670e4cd0b13ea2b631baccf6b","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254058,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[8,9],\"right\":[24,27],\"scalar\":3}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[8, 9], right=[24, 27], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([8, 9], [24, 27], 3, [24, 27], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, 9], [24, 27], 3, [24, 27], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0059","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"634a0628748f7aba94225e93143aa6ea437b403d7215d77c193ee4b6e92208bc","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254059,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[9,10],\"right\":[36,40],\"scalar\":4}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[9, 10], right=[36, 40], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([9, 10], [36, 40], 4, [36, 40], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, 10], [36, 40], 4, [36, 40], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0060","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47bc8e9e698bc2ea7eabe2ee17e899e4a31c2d509109ecb237da109b4e315e52","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254060,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[10,11],\"right\":[50,55],\"scalar\":5}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[10, 11], right=[50, 55], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([10, 11], [50, 55], 5, [50, 55], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 11], [50, 55], 5, [50, 55], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0061","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4918a479f8a4cb79a49d2c10b182f1cf093e2984f039dbdee7f04d61e749353","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254061,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[11,12],\"right\":[11,12],\"scalar\":1}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[11, 12], right=[11, 12], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([11, 12], [11, 12], 1, [11, 12], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 12], [11, 12], 1, [11, 12], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0062","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b77593bf4d00d52469a2c6d284d67b8c6d7addce20e695578e709a851834940","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254062,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[12,13],\"right\":[24,26],\"scalar\":2}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[12, 13], right=[24, 26], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([12, 13], [24, 26], 2, [24, 26], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 13], [24, 26], 2, [24, 26], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0063","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"261052eb3c1e762e830464bccff5d4b44d7924113c3fa1c75c8c73b6c5882da0","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254063,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[13,14],\"right\":[39,43],\"scalar\":3}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[13, 14], right=[39, 43], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([13, 14], [39, 43], 3, [39, 42], [0, -1], No)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([13, 14], [39, 43], 3, [39, 42], [0, -1], No)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0064","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3266fcf208774d708aa2f4d39a926fe5e23ae4687e4b7bb1538d9d848b944229","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254064,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[14,15],\"right\":[56,60],\"scalar\":4}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[14, 15], right=[56, 60], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([14, 15], [56, 60], 4, [56, 60], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([14, 15], [56, 60], 4, [56, 60], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0065","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58b257291557fd3b34cfb88b7ff96e16aec15af01ba42f61de88de59caf75bab","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254065,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[15,16],\"right\":[75,80],\"scalar\":5}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[15, 16], right=[75, 80], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([15, 16], [75, 80], 5, [75, 80], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([15, 16], [75, 80], 5, [75, 80], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0066","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f386a5ddbbaaee4fd1fd209f0484e0bc28304c4c64baf94e3e043f9769215a6","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254066,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[16,17],\"right\":[16,17],\"scalar\":1}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[16, 17], right=[16, 17], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([16, 17], [16, 17], 1, [16, 17], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([16, 17], [16, 17], 1, [16, 17], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0067","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"261fc81b2b82dbca69ab3fb55af7ead1d782f4d1a8dab96b10c3f3a5fa2ad9be","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254067,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[17,18],\"right\":[34,36],\"scalar\":2}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[17, 18], right=[34, 36], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([17, 18], [34, 36], 2, [34, 36], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([17, 18], [34, 36], 2, [34, 36], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0068","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90797dd10bec12486e6a78366beac74986d3cb4b029905383da55c84e9172908","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254068,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[18,19],\"right\":[54,57],\"scalar\":3}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[18, 19], right=[54, 57], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([18, 19], [54, 57], 3, [54, 57], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([18, 19], [54, 57], 3, [54, 57], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0069","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b0763fbd68937ee977f2b13377c97df1d6de6e934f9a6992322c82046f3b797","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254069,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[19,20],\"right\":[76,81],\"scalar\":4}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[19, 20], right=[76, 81], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([19, 20], [76, 81], 4, [76, 80], [0, -1], No)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([19, 20], [76, 81], 4, [76, 80], [0, -1], No)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0070","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2031df307813cfca1092000ffec8b9e390d2271eed1186dcea5775da34e841f0","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254070,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[20,21],\"right\":[100,105],\"scalar\":5}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[20, 21], right=[100, 105], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([20, 21], [100, 105], 5, [100, 105], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([20, 21], [100, 105], 5, [100, 105], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0071","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28087569244da061b9c69f18e881076c1f81e72d39b6803c79e0a978b41a79b2","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254071,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[21,22],\"right\":[21,22],\"scalar\":1}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[21, 22], right=[21, 22], nonzero integer scalar=1. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([21, 22], [21, 22], 1, [21, 22], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([21, 22], [21, 22], 1, [21, 22], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0072","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a97dee217c0e3edbc6faa63b1e981a065aba96394197c3de5a4ac5286948a75","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254072,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[22,23],\"right\":[44,46],\"scalar\":2}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[22, 23], right=[44, 46], nonzero integer scalar=2. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([22, 23], [44, 46], 2, [44, 46], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([22, 23], [44, 46], 2, [44, 46], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0073","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd3d3c40ec20b2b1c1623d8e8e88fe1666d77b908b5cbea7069befcdc93ef73b","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254073,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[23,24],\"right\":[69,72],\"scalar\":3}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[23, 24], right=[69, 72], nonzero integer scalar=3. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([23, 24], [69, 72], 3, [69, 72], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([23, 24], [69, 72], 3, [69, 72], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0074","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"972c2b7784cfa0bd929ce84cdb7c779a94c5999aaf6dffc845aed77b0ce1c751","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254074,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[24,25],\"right\":[96,100],\"scalar\":4}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[24, 25], right=[96, 100], nonzero integer scalar=4. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([24, 25], [96, 100], 4, [96, 100], [0, 0], Yes)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([24, 25], [96, 100], 4, [96, 100], [0, 0], Yes)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0075","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a44192ea6f2d2bbb740f222dd4f6121e2fbe8ddc00384cd4c4145bf49a661134","family_id":"qf-mod25-basisdeep-proportional-pair-dependence-certificate","seed":202610254075,"semantic_template":"qf-mod25-basisdeep-proportional-pair-dependence-certificate/{\"left\":[25,26],\"right\":[125,131],\"scalar\":5}"},"fields":{"title":"A Nonzero Proportional-Pair Witness Must Actually Match","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. nonzero left=[25, 26], right=[125, 131], nonzero integer scalar=5. Give (left,right,scalar,scalar*left,scalar*left−right,valid proportional dependence certificate?).","answer":{"kind":"expression","value":"([25, 26], [125, 131], 5, [125, 130], [0, -1], No)","explanation":"Check every coordinate of the proposed relation. A nonzero scalar and nonzero first vector are premises, not substitutes for equality with the supplied second vector. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([25, 26], [125, 131], 5, [125, 130], [0, -1], No)."},"skills":["A Nonzero Proportional-Pair Witness Must Actually Match"],"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-20261013-advance-20261025-r04-linear-independence-basis-0076","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e3e619bd081579b81ae38d1322a0a3e01ed3b2b2666e466bbe2aaaee5e39c28","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254076,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[-12,-3],\"second_coordinates\":[-11,-3],\"target\":[-12,21]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-12, 21], first integer coordinates=[-12, -3], second=[-11, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-12, 21], 1, [-12, -3], [-12, 21], Yes, [-11, -3], [-11, 19], No, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-12, 21], 1, [-12, -3], [-12, 21], Yes, [-11, -3], [-11, 19], No, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0077","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8303368b9c55f66384449e5f1a388a4250a0ef55e1ebabb5a90d5b4e4f73b968","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254077,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[-11,-2],\"second_coordinates\":[-11,-2],\"target\":[-11,9]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[-11, 9], first integer coordinates=[-11, -2], second=[-11, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-11, 9], 1, [-11, -2], [-11, 9], Yes, [-11, -2], [-11, 9], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-11, 9], 1, [-11, -2], [-11, 9], Yes, [-11, -2], [-11, 9], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0078","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98eb34888cdebe58a1e10ac59bb9456583033576545a5452ab9a77d8c5deeb53","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254078,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[-10,-1],\"second_coordinates\":[-10,-1],\"target\":[-10,-1]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[-10, -1], first integer coordinates=[-10, -1], second=[-10, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-10, -1], 1, [-10, -1], [-10, -1], Yes, [-10, -1], [-10, -1], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-10, -1], 1, [-10, -1], [-10, -1], Yes, [-10, -1], [-10, -1], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0079","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef7674602764bce5b073e7dc2c1e2d7699cd310010de7f4e09ab63c2953d070c","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254079,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[-9,0],\"second_coordinates\":[-9,0],\"target\":[-9,-9]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[-9, -9], first integer coordinates=[-9, 0], second=[-9, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-9, -9], 1, [-9, 0], [-9, -9], Yes, [-9, 0], [-9, -9], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-9, -9], 1, [-9, 0], [-9, -9], Yes, [-9, 0], [-9, -9], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0080","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b7f9271c2276d22a1a23724bf5efa52fdc6b73c0b6be690fe651b93f318afba","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254080,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[-8,1],\"second_coordinates\":[-7,1],\"target\":[-8,-15]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[-8, -15], first integer coordinates=[-8, 1], second=[-7, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-8, -15], 1, [-8, 1], [-8, -15], Yes, [-7, 1], [-7, -13], No, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-8, -15], 1, [-8, 1], [-8, -15], Yes, [-7, 1], [-7, -13], No, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0081","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49fcf4bb99f6ba944911811a5cffceb294dfb42279a3860d5897b9b4cc22571b","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254081,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[-7,2],\"second_coordinates\":[-7,2],\"target\":[-7,16]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-7, 16], first integer coordinates=[-7, 2], second=[-7, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-7, 16], 1, [-7, 2], [-7, 16], Yes, [-7, 2], [-7, 16], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-7, 16], 1, [-7, 2], [-7, 16], Yes, [-7, 2], [-7, 16], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0082","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8db0ca54958b4cfa77537b359e4a0d2df14279e65e45ac7d4f72dcff367e4fd2","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254082,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[-6,3],\"second_coordinates\":[-6,3],\"target\":[-6,9]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[-6, 9], first integer coordinates=[-6, 3], second=[-6, 3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-6, 9], 1, [-6, 3], [-6, 9], Yes, [-6, 3], [-6, 9], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-6, 9], 1, [-6, 3], [-6, 9], Yes, [-6, 3], [-6, 9], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0083","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b96b7afe5f6f067750a20a114407db5cdbe1286c2da6946731db5ff9809dc778","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254083,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[-5,-3],\"second_coordinates\":[-5,-3],\"target\":[-5,-3]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[-5, -3], first integer coordinates=[-5, -3], second=[-5, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [-5, -3], 1, [-5, -3], [-5, -3], Yes, [-5, -3], [-5, -3], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-5, -3], 1, [-5, -3], [-5, -3], Yes, [-5, -3], [-5, -3], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0084","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84a365d278021b9bef36f9d5fe2a7eb6c5a549c59a816a047ab92a7f4954958f","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254084,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[-4,-2],\"second_coordinates\":[-3,-2],\"target\":[-4,-6]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[-4, -6], first integer coordinates=[-4, -2], second=[-3, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [-4, -6], 1, [-4, -2], [-4, -6], Yes, [-3, -2], [-3, -5], No, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-4, -6], 1, [-4, -2], [-4, -6], Yes, [-3, -2], [-3, -5], No, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0085","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b46f73cffe5843dac4c375dc98b735289dc3d86539093c3944edc0bfda7d074","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254085,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[-3,-1],\"second_coordinates\":[-3,-1],\"target\":[-3,-7]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[-3, -7], first integer coordinates=[-3, -1], second=[-3, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [-3, -7], 1, [-3, -1], [-3, -7], Yes, [-3, -1], [-3, -7], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-3, -7], 1, [-3, -1], [-3, -7], Yes, [-3, -1], [-3, -7], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0086","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"410b8ed05ee0afdd2f80d0d590e63b64e02a70e3afea2799ec762e5625f90638","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254086,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[-2,0],\"second_coordinates\":[-2,0],\"target\":[-2,4]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[-2, 4], first integer coordinates=[-2, 0], second=[-2, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [-2, 4], 1, [-2, 0], [-2, 4], Yes, [-2, 0], [-2, 4], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-2, 4], 1, [-2, 0], [-2, 4], Yes, [-2, 0], [-2, 4], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0087","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdf9a9ca57872370a58a5573adac0ede4c0629d0b4fc2088533bac547bb4f3dc","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254087,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[-1,1],\"second_coordinates\":[-1,1],\"target\":[-1,2]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[-1, 2], first integer coordinates=[-1, 1], second=[-1, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [-1, 2], 1, [-1, 1], [-1, 2], Yes, [-1, 1], [-1, 2], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-1, 2], 1, [-1, 1], [-1, 2], Yes, [-1, 1], [-1, 2], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0088","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb24867f440a9c9cb52e9a77f5ea29ef142b32dc2f1777bcbe261b78db876d22","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254088,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[0,2],\"second_coordinates\":[1,2],\"target\":[0,2]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[0, 2], first integer coordinates=[0, 2], second=[1, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [0, 2], 1, [0, 2], [0, 2], Yes, [1, 2], [1, 2], No, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [0, 2], 1, [0, 2], [0, 2], Yes, [1, 2], [1, 2], No, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0089","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6baee2c2c4e8489753604cea8242b7c792dab8096e4c045c4a5041eaf7da01e2","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254089,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[1,3],\"second_coordinates\":[1,3],\"target\":[1,4]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[1, 4], first integer coordinates=[1, 3], second=[1, 3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [1, 4], 1, [1, 3], [1, 4], Yes, [1, 3], [1, 4], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [1, 4], 1, [1, 3], [1, 4], Yes, [1, 3], [1, 4], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0090","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c8a1e2822146a9875f75f2b380de242211d7d9689eebd2a6662f313efaa1d4e","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254090,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[2,-3],\"second_coordinates\":[2,-3],\"target\":[2,1]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[2, 1], first integer coordinates=[2, -3], second=[2, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [2, 1], 1, [2, -3], [2, 1], Yes, [2, -3], [2, 1], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [2, 1], 1, [2, -3], [2, 1], Yes, [2, -3], [2, 1], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0091","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d14f7e7672a0b194cc236936425e7ced45a927afb56d6278796b6d8a7e5b88c","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254091,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[3,-2],\"second_coordinates\":[3,-2],\"target\":[3,-8]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[3, -8], first integer coordinates=[3, -2], second=[3, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [3, -8], 1, [3, -2], [3, -8], Yes, [3, -2], [3, -8], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [3, -8], 1, [3, -2], [3, -8], Yes, [3, -2], [3, -8], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0092","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69cc5fb3f787e93cde5e8781414836aeab97af4811f8381c4ed90c64e8930c65","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254092,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[4,-1],\"second_coordinates\":[5,-1],\"target\":[4,-5]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[4, -5], first integer coordinates=[4, -1], second=[5, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [4, -5], 1, [4, -1], [4, -5], Yes, [5, -1], [5, -6], No, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [4, -5], 1, [4, -1], [4, -5], Yes, [5, -1], [5, -6], No, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0093","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30f1703f102cc9c4bd4bb0917cd0be249206acb4084186c4a33b950645b4f6e0","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254093,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[5,0],\"second_coordinates\":[5,0],\"target\":[5,0]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[5, 0], first integer coordinates=[5, 0], second=[5, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [5, 0], 1, [5, 0], [5, 0], Yes, [5, 0], [5, 0], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [5, 0], 1, [5, 0], [5, 0], Yes, [5, 0], [5, 0], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0094","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eda98e9509f683bdf316b892144f633be2a4a0b8a01a08e9bbd373d7e273607b","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254094,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[6,1],\"second_coordinates\":[6,1],\"target\":[6,7]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[6, 7], first integer coordinates=[6, 1], second=[6, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [6, 7], 1, [6, 1], [6, 7], Yes, [6, 1], [6, 7], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [6, 7], 1, [6, 1], [6, 7], Yes, [6, 1], [6, 7], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0095","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daf48ea6e352f03f6873cf0c9bda4518cdf0c2731b461b7d4b13be2e3dba8cdb","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254095,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[7,2],\"second_coordinates\":[7,2],\"target\":[7,16]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[7, 16], first integer coordinates=[7, 2], second=[7, 2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], Yes, [7, 2], [7, 16], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [7, 16], 1, [7, 2], [7, 16], Yes, [7, 2], [7, 16], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0096","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7d5cb41a5654463b7257bce3cdc3c9ca82ad38597b24026fd40750a6f55724d","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254096,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-2],[0,1]],\"first_coordinates\":[8,3],\"second_coordinates\":[9,3],\"target\":[8,-13]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -2], [0, 1]], target=[8, -13], first integer coordinates=[8, 3], second=[9, 3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -2], [0, 1]], [8, -13], 1, [8, 3], [8, -13], Yes, [9, 3], [9, -15], No, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [8, -13], 1, [8, 3], [8, -13], Yes, [9, 3], [9, -15], No, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0097","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef152d88e2713613b3462686b8f829ec735c725ead43653f3c002708b4f5ffc9","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254097,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,-1],[0,1]],\"first_coordinates\":[9,-3],\"second_coordinates\":[9,-3],\"target\":[9,-12]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, -1], [0, 1]], target=[9, -12], first integer coordinates=[9, -3], second=[9, -3]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, -1], [0, 1]], [9, -12], 1, [9, -3], [9, -12], Yes, [9, -3], [9, -12], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [9, -12], 1, [9, -3], [9, -12], Yes, [9, -3], [9, -12], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0098","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0e03d564a10ff9bac33e12255be02005043d1b124e0ccf869e3429641014314","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254098,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,0],[0,1]],\"first_coordinates\":[10,-2],\"second_coordinates\":[10,-2],\"target\":[10,-2]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 0], [0, 1]], target=[10, -2], first integer coordinates=[10, -2], second=[10, -2]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], [10, -2], 1, [10, -2], [10, -2], Yes, [10, -2], [10, -2], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [10, -2], 1, [10, -2], [10, -2], Yes, [10, -2], [10, -2], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0099","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f13965e66a3224971b9d61cd9ec19fc7c9a8bc9394d3accf076ae399b0d05a4d","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254099,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,1],[0,1]],\"first_coordinates\":[11,-1],\"second_coordinates\":[11,-1],\"target\":[11,10]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 1], [0, 1]], target=[11, 10], first integer coordinates=[11, -1], second=[11, -1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 1], [0, 1]], [11, 10], 1, [11, -1], [11, 10], Yes, [11, -1], [11, 10], Yes, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [11, 10], 1, [11, -1], [11, 10], Yes, [11, -1], [11, 10], Yes, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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-20261013-advance-20261025-r04-linear-independence-basis-0100","batch_id":"batch-20261013-advance-20261025-r04-linear-independence-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3884c6898f82ca7ec36565a0b1c65657179548838fbd5b7bb9aa94b949f504b7","family_id":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit","seed":202610254100,"semantic_template":"qf-mod25-basisdeep-r2-coordinate-uniqueness-audit/{\"basis\":[[1,2],[0,1]],\"first_coordinates\":[12,0],\"second_coordinates\":[13,0],\"target\":[12,24]}"},"fields":{"title":"Candidate Coordinate Lists and the Conditional Uniqueness Obligation","stem":"Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. ordered basis=[[1, 2], [0, 1]], target=[12, 24], first integer coordinates=[12, 0], second=[13, 0]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).","answer":{"kind":"expression","value":"([[1, 2], [0, 1]], [12, 24], 1, [12, 0], [12, 24], Yes, [13, 0], [13, 26], No, Yes)","explanation":"Validate the basis first, then reconstruct both lists. The final implication can be true when one list is invalid; it does not certify both lists as representations. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [12, 24], 1, [12, 0], [12, 24], Yes, [13, 0], [13, 26], No, Yes)."},"skills":["Candidate Coordinate Lists and the Conditional Uniqueness Obligation"],"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."}}