{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0001","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c6399f2ba0677e8372e671aa489c545876a6d5f43a967773fffe5eabc8faea","family_id":"qf-mod48-product-rectangle-generation","seed":202611174001,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], 2, 2, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], []], 2, 2, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0002","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be09c860b9161af9ad946930dcfd54336ef1fde2fa2fd2969a91bf6db0841b9d","family_id":"qf-mod48-product-rectangle-generation","seed":202611174002,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0003","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ba9599f18f15e913655b296067857c5dde37393fad5d4803b3c2fc360208055","family_id":"qf-mod48-product-rectangle-generation","seed":202611174003,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0004","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb920400fcf3c8c99a894bc0520b3bcb7941163e1f6e7ea80efe278cda11454e","family_id":"qf-mod48-product-rectangle-generation","seed":202611174004,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0005","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40496733dcd2bb6fcaee010f49903bddefa3a1e7fdd6b00d2830f1a928ea885b","family_id":"qf-mod48-product-rectangle-generation","seed":202611174005,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0006","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9d54fe7b71d25ee1a1bc4bbcb66344d023ab8da66da4d8a39b1b27e0095a37d","family_id":"qf-mod48-product-rectangle-generation","seed":202611174006,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0007","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2896d1aca6f1d218f4d32d1e3ae73471edb49f7bca53e5988307e47164491d1","family_id":"qf-mod48-product-rectangle-generation","seed":202611174007,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [0], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0008","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e65e42b2cda9975604137f07a2fceea5d0e11aaa356a0cf244781be024965d39","family_id":"qf-mod48-product-rectangle-generation","seed":202611174008,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0009","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"230d0d81d27ac9c31c17316fcadb284c2c1172083a1ca12eefdb92baa9143138","family_id":"qf-mod48-product-rectangle-generation","seed":202611174009,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0010","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feb3c44862094a5bac76df6f026540c54c4477f3a2ba85e08784a0140381faf5","family_id":"qf-mod48-product-rectangle-generation","seed":202611174010,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0011","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3519fa2db45e48c88d2da6371ae94c03b7ba12f3570dffc330015fae95c53b35","family_id":"qf-mod48-product-rectangle-generation","seed":202611174011,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0012","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b827d2bd3affc5209b4302e754aae5860808fcbe4d55b41acfcfc46d8e65d7ef","family_id":"qf-mod48-product-rectangle-generation","seed":202611174012,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [0], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], [[2, 100], [2, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0013","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9cb879e732c0815cf0434aca9210b841837d6befc9f715c59b4e89ba8264b5a","family_id":"qf-mod48-product-rectangle-generation","seed":202611174013,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0014","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c17e0cba5ef464c48bfe82ae249a69f3c6aef17c38b15937c06319113d4da75","family_id":"qf-mod48-product-rectangle-generation","seed":202611174014,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0015","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1d98682d9fb05af06b25c35efaaa0764315ea7771610fcbd1c4274f566dceb0","family_id":"qf-mod48-product-rectangle-generation","seed":202611174015,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0016","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42c143962f5bebcf1933f818650495f168a190c53ec9ff58d65c9e304d64d41c","family_id":"qf-mod48-product-rectangle-generation","seed":202611174016,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [1], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0017","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01a24b7ae9671a39a86929f9c233480ca49abcf364f621b8f0e9ccc8f6cff146","family_id":"qf-mod48-product-rectangle-generation","seed":202611174017,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], []], 3, 3, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0018","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78eb3fd145bb0155839d8a00baaa39fe041938dae3d75066ac5a6e642191efb5","family_id":"qf-mod48-product-rectangle-generation","seed":202611174018,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0019","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"955b536a27069fd734a1bb7e82a724cd8bc906c4ac50d930c76defe71b464bb1","family_id":"qf-mod48-product-rectangle-generation","seed":202611174019,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0020","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edc59af7fae8d93ccf9d0369d7c026c231c362c05922b5a0e709a18eab1021b5","family_id":"qf-mod48-product-rectangle-generation","seed":202611174020,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0021","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14dec42c8cfc8f060f481d2a45b324bf5e2d12b99b5dcfa35b92092793509249","family_id":"qf-mod48-product-rectangle-generation","seed":202611174021,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0022","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87057cfc2d39f9ae4a0de132a2b58cfd72b4c4f52ff3637378834fe099ce208e","family_id":"qf-mod48-product-rectangle-generation","seed":202611174022,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[1, 100], [1, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0023","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e887db2df9db60611b1f9f6374147550b58d284989c31ab1c41305a91ae8f8e","family_id":"qf-mod48-product-rectangle-generation","seed":202611174023,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [0], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 6, 6, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[1, 100], [1, 101]], []], 6, 6, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0024","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c66b97cb3f1235dfee5338c7516df38f653e1bb2d048a8271545a6d76ffe44fc","family_id":"qf-mod48-product-rectangle-generation","seed":202611174024,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[2, 100], [2, 101]], []], 4, 4, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0025","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5102dcb7735186003dcf33ec14c1e5b249a558a1d3f6bfac4215c333208f4f3","family_id":"qf-mod48-product-rectangle-generation","seed":202611174025,"semantic_template":"product-rectangle-generation|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"The Entire Product Topology Contains Every Union of Basic Rectangles","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (product_points, basic_rectangles, product_topology, rectangle_count, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)","explanation":"Retain the complete typed Cartesian carrier, all distinct rectangles and the complete union-generated topology in source canonical order. Rectangles alone need not be closed under unions or exhaust all opens. This exact family uses two nonempty factors and no zero-factor convention. The exact result, retaining the requested order, is ([[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], [[[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], [[0, 100], [0, 101], [1, 100], [1, 101]], [[2, 100], [2, 101]], []], 5, 5, Yes, rectangles_generate_opens_by_arbitrary_unions)."},"skills":["The Entire Product Topology Contains Every Union of Basic Rectangles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0026","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889a1183c59e9c6b61aec548195e5b127f95266397fe7c9178e544074c428585","family_id":"qf-mod48-product-projection-continuity","seed":202611174026,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0027","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d0efabbb8380820152985b3f1f0f4eb0dbac7f24221561727fed78b06918fca","family_id":"qf-mod48-product-projection-continuity","seed":202611174027,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0028","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff127bdf5d41632207cee3d0513b2def3dfc9e323e5681dfef649b564059e781","family_id":"qf-mod48-product-projection-continuity","seed":202611174028,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0029","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30e64025cfdbd9fb71c082e6b092511c9d8efc2a3cbc82df9546754cb127bcfa","family_id":"qf-mod48-product-projection-continuity","seed":202611174029,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0030","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b466e65bf43be78fe817b238ed80a825ac1c2c586e31283a7a75d41cad0577fc","family_id":"qf-mod48-product-projection-continuity","seed":202611174030,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0031","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34fc8f6142e06f8000a79d594d6c191b28415c3b64c163f4de8d333f403ef4eb","family_id":"qf-mod48-product-projection-continuity","seed":202611174031,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0032","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cc45fc038ba57976d7deba6f619a65163a2ff65b5ab9004029b4e31c6eaf542","family_id":"qf-mod48-product-projection-continuity","seed":202611174032,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [0], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0033","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"543e0db816fff8fa19383063dc813998c0c235ff31afea7f4503d845285ea8e6","family_id":"qf-mod48-product-projection-continuity","seed":202611174033,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0034","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"724e19fdabace2b0c543a75deb14ab6af262fc4b022bf260bb1e9e74b0be03b3","family_id":"qf-mod48-product-projection-continuity","seed":202611174034,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0035","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e6d891f46693a0d945965111648919f46255e27ec1ca09a7c656b8130e3ad36","family_id":"qf-mod48-product-projection-continuity","seed":202611174035,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0036","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51aefe767874930fce2796ca0d8b99e0c58c8102fb3f21a8f81724e8c1762dbf","family_id":"qf-mod48-product-projection-continuity","seed":202611174036,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0037","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a34b7dba5e2d0ad7d2e0f8b4115f98407ea94a5b2ad50a2bbf1d17b2051c1b94","family_id":"qf-mod48-product-projection-continuity","seed":202611174037,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [0], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 2], [[0, 100], [0, 101], [2, 100], [2, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0038","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcdbd1ec6cb3b1f5ad5c8e88601f85e5d5a6d8dd31ccd5452cbd743fa52cd972","family_id":"qf-mod48-product-projection-continuity","seed":202611174038,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0039","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d676859b3874da58fdcd8de4e818f1f82d6e704dcb0352ffce9337d97bdb99d","family_id":"qf-mod48-product-projection-continuity","seed":202611174039,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0040","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b2b9d3a304a01473b28629310fc8e999aa3c21d154bc318eb9f68efe083a2e0","family_id":"qf-mod48-product-projection-continuity","seed":202611174040,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0041","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ab145f68efe5018dbfa8e75898dde18f2186c128297902769cd2d4f50da6804","family_id":"qf-mod48-product-projection-continuity","seed":202611174041,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [1], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[1, 2], [[1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0042","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ab0502d5d8e7358dcc7dd7f219b2c0f4b881509974cb3573967948777b85df2","family_id":"qf-mod48-product-projection-continuity","seed":202611174042,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0043","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"026356c1fb4a10f4ad13a16ce0cb70a78a2d897f20c1c277738cce15ea71e8e4","family_id":"qf-mod48-product-projection-continuity","seed":202611174043,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0044","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3284db4c871546e36f84343bf63507bee69aa0ed3078938e62f0413d3ec5e8d4","family_id":"qf-mod48-product-projection-continuity","seed":202611174044,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0045","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b722db5a2468c21653cea8ae83aca18128c5d12d7c00b02ec1ee9833584f513","family_id":"qf-mod48-product-projection-continuity","seed":202611174045,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0046","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"505d4a0b98b523b0e4a33550b108f1ec1434acb9ce603b522ccb7d70b3ea6e50","family_id":"qf-mod48-product-projection-continuity","seed":202611174046,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0047","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"457c0f1b7de7ace340f015d7abd5edf51848d288eb17adc5f4a6d98ead695a11","family_id":"qf-mod48-product-projection-continuity","seed":202611174047,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0048","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ced6e225ccb15ef2dc481913a3b7872eae90346026ae962d8387a730f8057c79","family_id":"qf-mod48-product-projection-continuity","seed":202611174048,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [0], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[0], [[0, 100], [0, 101]], Yes], [[1], [[1, 100], [1, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0049","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7a02481d660af4a9b2d5de36364f5ac2bb82f123fa201429694ce616320a772","family_id":"qf-mod48-product-projection-continuity","seed":202611174049,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0050","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2421b1f5189627e71e4d1c70df0f802d98fc58e01f7867e1799cd4ee024f108c","family_id":"qf-mod48-product-projection-continuity","seed":202611174050,"semantic_template":"product-projection-continuity|{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}"},"fields":{"title":"Each Coordinate Projection Has Open Preimages for Every Factor Open","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Give (first_projection_preimages, second_projection_preimages, first_projection_continuous, second_projection_continuous, projection_certificate, certificate).","answer":{"kind":"expression","value":"([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)","explanation":"For every open of each factor retain its entire cylinder preimage and open flag in the full generated product. Check both projections, not just their values on a sample of points. No general universal-property proof is supplied by finite enumeration. The exact result, retaining the requested order, is ([[[], [], Yes], [[0, 1, 2, 3], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes], [[0, 1, 2], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101]], Yes], [[0, 1], [[0, 100], [0, 101], [1, 100], [1, 101]], Yes], [[2], [[2, 100], [2, 101]], Yes]], [[[], [], Yes], [[100, 101], [[0, 100], [0, 101], [1, 100], [1, 101], [2, 100], [2, 101], [3, 100], [3, 101]], Yes]], Yes, Yes, Yes, all_projection_preimages_are_open)."},"skills":["Each Coordinate Projection Has Open Preimages for Every Factor Open"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0051","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56f4d98c8c41e4d03dc3a8b31d1829375d7b08c51bcad2ba58d13a38af99b498","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174051,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0052","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de9863c9968e05fd5fbc9a3b6f4ce4da9ef5c7d95a4fd90f0d1f56e5166b79a1","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174052,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0053","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7bdd095fd0edef11f42e6559974ddc816e2d4a432079e110c2e3588382165c6","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174053,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0054","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52cdf7d64a50f8346bd0c310ac5a36a3d3e81a8501fa2b0e6309d0bf269e96f7","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174054,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0055","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12c6e33bb75683d95822dce65d94ae0f24b64a759640d060a5a981b46c245fdc","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174055,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0056","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2765bc997616191221bedacf55a344e3ac2ababc37e5c443e51408f7143fd99","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174056,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0057","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdbf4de2abad1d2c31e17a67b82035d4281a73470b234321b355b6dcc308168b","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174057,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1], [0], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0058","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eb0fce32cb127f459a89a2ed94b0f243bc8b55151aac8b798de17fe6962d0df","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174058,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0059","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89d37ebb833da6dc81b74670f0156d09a1837d291e95304f3adedc5488843fe8","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174059,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0060","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bfdf2ed6be4f25a94cdd6e672816fc576c4f74fa79e67848953fef7fb9847f1","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174060,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0061","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7ca4312635ecf8b34305e03b77de8588ccd5b148aaa3035a56a02e6aca98cab","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174061,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0062","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40cdbc36ac52133d8fa208f039c6d2233254896c8a7f7777f75513ec73a293de","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174062,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 2], [0], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0063","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fe3d10659d1716ddea4eb4225e5dc3d060ccf5695d5d03fe9820787d076c2b4","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174063,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0064","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d95bda779f4c837b50c824eefc82064005a09d97aa8858fa6d6183fbdcdc3754","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174064,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0065","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e2a323214bf2c836b5d2a72aef9074cb7561db925d7726f12ef0186a785201c","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174065,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0066","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f15377ba2f0dd774f6fa3f233aee7433efedb1208d7aa6a5efb48539263016d","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174066,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [1, 2], [1], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0067","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dda581dd73cc20641f815766425442bf881465590d317384fcc823b18d02b875","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174067,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0068","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ece73dcd69ef4fee04f3d99d2c1f30afd4f0a4d35080c50dde5fefb9bb9253e","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174068,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0069","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04634ff3a41f3e1726e5d988e1d55564e580a3692806bd845ee75fb26c79a39e","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174069,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0070","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b60da6728c1e2b2e382e79c3c3db499054d6b94948c28dea47dea3befa8d59a7","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174070,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0071","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cfd3a0e5c77bc6c52219234288b76bb9e0de91002c2c0fa77e45ff5c49eaa81","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174071,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [0]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0072","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b74e4d5bc8730073c77e4c976010c39f389ebcaa8d674e1deebe933547913bb","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174072,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0073","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e743c53b4d4d74d3e644a5eef6b8096f8445575f498c6a99ae2c2c5f82b2b01a","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174073,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [0], [1]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (No, Yes, No, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0074","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"634a5ea36233d5080d00600f76344acff05f4fcbe6dde4baea91a21e9244530d","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174074,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1],[0],[1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0], [1], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0075","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ff263fcbb194fa683a61f04fe74865901323e209d42cf01408a25833979f952","family_id":"qf-mod48-product-coordinate-criterion","seed":202611174075,"semantic_template":"product-coordinate-criterion|{map_x:[[0,0],[1,1]],map_y:[[0,100],[1,101]],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[2]],y_points:[100,101],y_topology:[[],[100,101]],z_points:[0,1],z_topology:[[],[0,1]]}"},"fields":{"title":"Continuity into a Product Is Equivalent to Continuity of Both Coordinates","stem":"Use complete first-factor points=[0, 1, 2, 3], topology=[[], [0, 1, 2, 3], [0, 1, 2], [0, 1], [2]], second-factor points=[100, 101], topology=[[], [100, 101]]. Ordered product pairs retain coordinate types; qualify both entire finite topologies before generating every rectangle union. Source Z=[0, 1], topology=[[], [0, 1]], aligned coordinate maps=[0, 1] and [100, 101]. Give (first_coordinate_continuous, second_coordinate_continuous, paired_map_continuous, criterion_equivalence, paired_map_rows, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)","explanation":"Audit each coordinate map and the paired map on the entire finite source. Compare the paired-map flag with the conjunction of coordinate flags; the equivalence can be Yes even when some continuity flags are No. Preserve map associations when sorting source labels. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, [[0, [0, 100]], [1, [1, 101]]], map_into_finite_product_continuous_iff_coordinates_continuous)."},"skills":["Continuity into a Product Is Equivalent to Continuity of Both Coordinates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0076","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"557ff82e3ae5d584bd87d6136bf80df312b2a4db2d117ac6bf3046a75565bc29","family_id":"qf-mod48-product-topology-boundary","seed":202611174076,"semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[0,100],[1,101]],x_points:[0,1],x_topology:[[],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101],[100],[101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=nonrectangle_open with the complete declared model={\"x_points\":[0,1],\"x_topology\":[[],[0],[1],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100],[101],[100,101]],\"candidate_open\":[[0,100],[1,101]]}. Give (mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate).","answer":{"kind":"expression","value":"(nonrectangle_open, [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [1, 101]], Yes, No, [[[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[0, 100], [1, 100]], [[0, 100]], [[0, 101], [1, 101]], [[0, 101]], [[1, 100], [1, 101]], [[1, 100]], [[1, 101]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (nonrectangle_open, [[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [1, 101]], Yes, No, [[[0, 100], [0, 101], [1, 100], [1, 101]], [[0, 100], [0, 101]], [[0, 100], [1, 100]], [[0, 100]], [[0, 101], [1, 101]], [[0, 101]], [[1, 100], [1, 101]], [[1, 100]], [[1, 101]], []], open_union_of_rectangles_need_not_be_one_rectangle)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0077","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce019104fefcc8c8fcdf4a4553fed4a7a30f034a5307f13470b5ee5ac689b087","family_id":"qf-mod48-product-topology-boundary","seed":202611174077,"semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=invalid_factor with the complete declared model={\"x_points\":[0,1],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).","answer":{"kind":"expression","value":"(invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0078","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd9aabf7ef138f6a7eb5acd53a8fe4a3f2d43b472d4d4273dffb76561f3389e5","family_id":"qf-mod48-product-topology-boundary","seed":202611174078,"semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=finite_box_no_extrapolation with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}. Give (mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary).","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 3, 3, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (finite_box_no_extrapolation, 8, 3, 3, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0079","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da1c416708c21cfb048230a0c98e4a4b85f96fb80b6cbc546f9017482d3ddae2","family_id":"qf-mod48-product-topology-boundary","seed":202611174079,"semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=ordered_pair_direction with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}. Give (mode, pair, reversed_pair, pair_identity_claim, boundary).","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0080","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e2969afa39e665bbb0bcc8f1ba7d93873c216475a984e30e87c9b7fd43607ff","family_id":"qf-mod48-product-topology-boundary","seed":202611174080,"semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[40,140],[41,141]],x_points:[40,41],x_topology:[[],[40,41],[40],[41]],y_points:[140,141],y_topology:[[],[140,141],[140],[141]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=nonrectangle_open with the complete declared model={\"x_points\":[40,41],\"x_topology\":[[],[40],[41],[40,41]],\"y_points\":[140,141],\"y_topology\":[[],[140],[141],[140,141]],\"candidate_open\":[[40,140],[41,141]]}. Give (mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate).","answer":{"kind":"expression","value":"(nonrectangle_open, [[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [41, 141]], Yes, No, [[[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [40, 141]], [[40, 140], [41, 140]], [[40, 140]], [[40, 141], [41, 141]], [[40, 141]], [[41, 140], [41, 141]], [[41, 140]], [[41, 141]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (nonrectangle_open, [[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [41, 141]], Yes, No, [[[40, 140], [40, 141], [41, 140], [41, 141]], [[40, 140], [40, 141]], [[40, 140], [41, 140]], [[40, 140]], [[40, 141], [41, 141]], [[40, 141]], [[41, 140], [41, 141]], [[41, 140]], [[41, 141]], []], open_union_of_rectangles_need_not_be_one_rectangle)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0081","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3777316d21062fbcb886ca8eff06d18fa679ff6e519ccab3e3928de6950011b","family_id":"qf-mod48-product-topology-boundary","seed":202611174081,"semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=invalid_factor with the complete declared model={\"x_points\":[0,1,2],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).","answer":{"kind":"expression","value":"(invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0082","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e011ab6856e1730a1f2c3275b490a40fec3ea1ff9e0466a267c9ba057582f54","family_id":"qf-mod48-product-topology-boundary","seed":202611174082,"semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=finite_box_no_extrapolation with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}. Give (mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary).","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 5, 5, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (finite_box_no_extrapolation, 8, 5, 5, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0083","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c74714523830543240c5842d8744be636bd9dc6e68699069ed40b3522d3bb8d1","family_id":"qf-mod48-product-topology-boundary","seed":202611174083,"semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=ordered_pair_direction with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}. Give (mode, pair, reversed_pair, pair_identity_claim, boundary).","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0084","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8cb929f02d3c82f5f222251983e8a9b3a7665c214e6a4425578a4e154f267ed","family_id":"qf-mod48-product-topology-boundary","seed":202611174084,"semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[80,180],[81,181]],x_points:[80,81],x_topology:[[],[80,81],[80],[81]],y_points:[180,181],y_topology:[[],[180,181],[180],[181]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=nonrectangle_open with the complete declared model={\"x_points\":[80,81],\"x_topology\":[[],[80],[81],[80,81]],\"y_points\":[180,181],\"y_topology\":[[],[180],[181],[180,181]],\"candidate_open\":[[80,180],[81,181]]}. Give (mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate).","answer":{"kind":"expression","value":"(nonrectangle_open, [[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [81, 181]], Yes, No, [[[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [80, 181]], [[80, 180], [81, 180]], [[80, 180]], [[80, 181], [81, 181]], [[80, 181]], [[81, 180], [81, 181]], [[81, 180]], [[81, 181]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (nonrectangle_open, [[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [81, 181]], Yes, No, [[[80, 180], [80, 181], [81, 180], [81, 181]], [[80, 180], [80, 181]], [[80, 180], [81, 180]], [[80, 180]], [[80, 181], [81, 181]], [[80, 181]], [[81, 180], [81, 181]], [[81, 180]], [[81, 181]], []], open_union_of_rectangles_need_not_be_one_rectangle)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0085","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df634be90a17c5d47539e5cd95a0d81df99c1ef6528c55ea3a4a513eac9dd9bf","family_id":"qf-mod48-product-topology-boundary","seed":202611174085,"semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=invalid_factor with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).","answer":{"kind":"expression","value":"(invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0086","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ccd6b787db58df463e29c274a460980bdfa43c72204a1d25e1c9e8d3ad631c7","family_id":"qf-mod48-product-topology-boundary","seed":202611174086,"semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=finite_box_no_extrapolation with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}. Give (mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary).","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 4, 4, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (finite_box_no_extrapolation, 8, 4, 4, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0087","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9043440ef260e6345367333e8739cc0332889d7176dd3f057190ba740c87b1eb","family_id":"qf-mod48-product-topology-boundary","seed":202611174087,"semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,2],[0],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=ordered_pair_direction with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,2],[0],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}. Give (mode, pair, reversed_pair, pair_identity_claim, boundary).","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0088","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4c81215f6e052167167836c6d6667737a08056cccc916d840b8494af0b5e1de","family_id":"qf-mod48-product-topology-boundary","seed":202611174088,"semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[120,220],[121,221]],x_points:[120,121],x_topology:[[],[120,121],[120],[121]],y_points:[220,221],y_topology:[[],[220,221],[220],[221]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=nonrectangle_open with the complete declared model={\"x_points\":[120,121],\"x_topology\":[[],[120],[121],[120,121]],\"y_points\":[220,221],\"y_topology\":[[],[220],[221],[220,221]],\"candidate_open\":[[120,220],[121,221]]}. Give (mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate).","answer":{"kind":"expression","value":"(nonrectangle_open, [[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [121, 221]], Yes, No, [[[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [120, 221]], [[120, 220], [121, 220]], [[120, 220]], [[120, 221], [121, 221]], [[120, 221]], [[121, 220], [121, 221]], [[121, 220]], [[121, 221]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (nonrectangle_open, [[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [121, 221]], Yes, No, [[[120, 220], [120, 221], [121, 220], [121, 221]], [[120, 220], [120, 221]], [[120, 220], [121, 220]], [[120, 220]], [[120, 221], [121, 221]], [[120, 221]], [[121, 220], [121, 221]], [[121, 220]], [[121, 221]], []], open_union_of_rectangles_need_not_be_one_rectangle)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0089","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3454ac84fb709fc6a0885afab12963d286548234a345a9ef3b55a61e50ad1eab","family_id":"qf-mod48-product-topology-boundary","seed":202611174089,"semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3,4],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=invalid_factor with the complete declared model={\"x_points\":[0,1,2,3,4],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).","answer":{"kind":"expression","value":"(invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0090","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"389f0540a9e0f91bf0b2dd486463702bd439a940c5a3f8d66676af0b2a9f1021","family_id":"qf-mod48-product-topology-boundary","seed":202611174090,"semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=finite_box_no_extrapolation with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}. Give (mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary).","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 4, 4, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (finite_box_no_extrapolation, 8, 4, 4, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0091","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e6ef80f43e993376053d9621af4956428103d67d70ee36774e423682aa685ff","family_id":"qf-mod48-product-topology-boundary","seed":202611174091,"semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[1,2],[1],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=ordered_pair_direction with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[1,2],[1],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}. Give (mode, pair, reversed_pair, pair_identity_claim, boundary).","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0092","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5342b9368e892f00052d5ada06a73fec532db523f0564793d69abe78f4715cfb","family_id":"qf-mod48-product-topology-boundary","seed":202611174092,"semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[160,260],[161,261]],x_points:[160,161],x_topology:[[],[160,161],[160],[161]],y_points:[260,261],y_topology:[[],[260,261],[260],[261]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=nonrectangle_open with the complete declared model={\"x_points\":[160,161],\"x_topology\":[[],[160],[161],[160,161]],\"y_points\":[260,261],\"y_topology\":[[],[260],[261],[260,261]],\"candidate_open\":[[160,260],[161,261]]}. Give (mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate).","answer":{"kind":"expression","value":"(nonrectangle_open, [[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [161, 261]], Yes, No, [[[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [160, 261]], [[160, 260], [161, 260]], [[160, 260]], [[160, 261], [161, 261]], [[160, 261]], [[161, 260], [161, 261]], [[161, 260]], [[161, 261]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (nonrectangle_open, [[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [161, 261]], Yes, No, [[[160, 260], [160, 261], [161, 260], [161, 261]], [[160, 260], [160, 261]], [[160, 260], [161, 260]], [[160, 260]], [[160, 261], [161, 261]], [[160, 261]], [[161, 260], [161, 261]], [[161, 260]], [[161, 261]], []], open_union_of_rectangles_need_not_be_one_rectangle)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0093","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a41b31637addf5d5395e8ad3efd57ae6c8d8bb0972d3cec53e4200a36389f4e","family_id":"qf-mod48-product-topology-boundary","seed":202611174093,"semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3,4,5],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=invalid_factor with the complete declared model={\"x_points\":[0,1,2,3,4,5],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).","answer":{"kind":"expression","value":"(invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0094","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9b4183a536a4c359fe5efe0ca3d21a9ea4139197b257dc0a2a75c2ecb6ca1ee","family_id":"qf-mod48-product-topology-boundary","seed":202611174094,"semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=finite_box_no_extrapolation with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}. Give (mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary).","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 4, 4, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (finite_box_no_extrapolation, 8, 4, 4, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0095","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21f3b129e50ea3e6f24cf9fedfa5e4e4e2cee020e9b8a311ae14750164a9c62f","family_id":"qf-mod48-product-topology-boundary","seed":202611174095,"semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=ordered_pair_direction with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}. Give (mode, pair, reversed_pair, pair_identity_claim, boundary).","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0096","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"426ae4f1af54b98841870f480ab5748d37564fe500b37c039da3df66afa5e54a","family_id":"qf-mod48-product-topology-boundary","seed":202611174096,"semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[200,300],[201,301]],x_points:[200,201],x_topology:[[],[200,201],[200],[201]],y_points:[300,301],y_topology:[[],[300,301],[300],[301]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=nonrectangle_open with the complete declared model={\"x_points\":[200,201],\"x_topology\":[[],[200],[201],[200,201]],\"y_points\":[300,301],\"y_topology\":[[],[300],[301],[300,301]],\"candidate_open\":[[200,300],[201,301]]}. Give (mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate).","answer":{"kind":"expression","value":"(nonrectangle_open, [[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [201, 301]], Yes, No, [[[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [200, 301]], [[200, 300], [201, 300]], [[200, 300]], [[200, 301], [201, 301]], [[200, 301]], [[201, 300], [201, 301]], [[201, 300]], [[201, 301]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (nonrectangle_open, [[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [201, 301]], Yes, No, [[[200, 300], [200, 301], [201, 300], [201, 301]], [[200, 300], [200, 301]], [[200, 300], [201, 300]], [[200, 300]], [[200, 301], [201, 301]], [[200, 301]], [[201, 300], [201, 301]], [[201, 300]], [[201, 301]], []], open_union_of_rectangles_need_not_be_one_rectangle)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0097","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24b7c2dd8fd79fb0a2cf6d1b91e53b699f6f06ccf153fb7a56e0d7383c6f1874","family_id":"qf-mod48-product-topology-boundary","seed":202611174097,"semantic_template":"product-topology-boundary|{mode:\"invalid_factor\",model:{invalid_side:\"x\",x_points:[0,1,2,3,4,5,6],x_topology:[[]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=invalid_factor with the complete declared model={\"x_points\":[0,1,2,3,4,5,6],\"x_topology\":[[]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"invalid_side\":\"x\"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).","answer":{"kind":"expression","value":"(invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0098","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00f4e290ea014ff7bd499714b4536f3d37772f8febce9e2909a31f71ca3b5169","family_id":"qf-mod48-product-topology-boundary","seed":202611174098,"semantic_template":"product-topology-boundary|{mode:\"finite_box_no_extrapolation\",model:{x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=finite_box_no_extrapolation with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[0,1],[0],[1]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]]}. Give (mode, product_point_count, rectangle_count, open_count, finite_model_certificate, finite_box_statement, infinite_extrapolation_claim, boundary).","answer":{"kind":"expression","value":"(finite_box_no_extrapolation, 8, 6, 6, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (finite_box_no_extrapolation, 8, 6, 6, Yes, finite_product_and_box_topologies_coincide, No, no_inference_to_infinite_products)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0099","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3477d228bd9ef4e3e6b192b67fe95c878a35cd214b31c1792312ca308f2cd1b8","family_id":"qf-mod48-product-topology-boundary","seed":202611174099,"semantic_template":"product-topology-boundary|{mode:\"ordered_pair_direction\",model:{pair:[0,101],x_points:[0,1,2,3],x_topology:[[],[0,1,2,3],[0,1,2],[2]],y_points:[100,101],y_topology:[[],[100,101]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=ordered_pair_direction with the complete declared model={\"x_points\":[0,1,2,3],\"x_topology\":[[],[0,1,2,3],[0,1,2],[2]],\"y_points\":[100,101],\"y_topology\":[[],[100,101]],\"pair\":[0,101]}. Give (mode, pair, reversed_pair, pair_identity_claim, boundary).","answer":{"kind":"expression","value":"(ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (ordered_pair_direction, [0, 101], [101, 0], No, ordered_pair_coordinates_are_typed_and_directional)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r04-finite-product-topology-0100","batch_id":"batch-20261015-advance-20261117-r04-finite-product-topology-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c563269788b5660efc21aff5faa3cf896d1a1523aa9e2a30f01f7e91fd89db28","family_id":"qf-mod48-product-topology-boundary","seed":202611174100,"semantic_template":"product-topology-boundary|{mode:\"nonrectangle_open\",model:{candidate_open:[[240,340],[241,341]],x_points:[240,241],x_topology:[[],[240,241],[240],[241]],y_points:[340,341],y_topology:[[],[340,341],[340],[341]]}}"},"fields":{"title":"Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs","stem":"Use mode=nonrectangle_open with the complete declared model={\"x_points\":[240,241],\"x_topology\":[[],[240],[241],[240,241]],\"y_points\":[340,341],\"y_topology\":[[],[340],[341],[340,341]],\"candidate_open\":[[240,340],[241,341]]}. Give (mode, product_points, candidate_open, open, basic_rectangle, basic_rectangles, certificate).","answer":{"kind":"expression","value":"(nonrectangle_open, [[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [241, 341]], Yes, No, [[[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [240, 341]], [[240, 340], [241, 340]], [[240, 340]], [[240, 341], [241, 341]], [[240, 341]], [[241, 340], [241, 341]], [[241, 340]], [[241, 341]], []], open_union_of_rectangles_need_not_be_one_rectangle)","explanation":"A nonrectangle witness must really be open but not a single rectangle. Invalid-factor mode retains each factor validity/error separately. Finite product and box topologies coincide here, without extrapolation to infinite products. A typed ordered pair is not identified with its reversal, which may not lie in the same product carrier. The exact result, retaining the requested order, is (nonrectangle_open, [[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [241, 341]], Yes, No, [[[240, 340], [240, 341], [241, 340], [241, 341]], [[240, 340], [240, 341]], [[240, 340], [241, 340]], [[240, 340]], [[240, 341], [241, 341]], [[240, 341]], [[241, 340], [241, 341]], [[241, 340]], [[241, 341]], []], open_union_of_rectangles_need_not_be_one_rectangle)."},"skills":["Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}