{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0001","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f6d206831683b23b6b3e232f200fb4c45e5f3656452f2117582c5f5abac6c2d","family_id":"qf-mod31-countable-finite-certificate","seed":202610311001,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-2,1,40]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[1, -2, 40] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([1, -2, 40], [[1, 0], [-2, 1], [40, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([1, -2, 40], [[1, 0], [-2, 1], [40, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0002","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17843d623bcb5173625f787b35e598c6435b4694e754c44677fd721a8e6c39c7","family_id":"qf-mod31-countable-finite-certificate","seed":202610311002,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-3,2,42]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[2, -3, 42] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([2, -3, 42], [[2, 0], [-3, 1], [42, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([2, -3, 42], [[2, 0], [-3, 1], [42, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0003","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ba4a1beffdad69678bae09512c2df3a94d040d1c998891d7d1855ec4fc844db","family_id":"qf-mod31-countable-finite-certificate","seed":202610311003,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-4,3,44]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[3, -4, 44] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([3, -4, 44], [[3, 0], [-4, 1], [44, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([3, -4, 44], [[3, 0], [-4, 1], [44, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0004","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"968627c867d3cfdae9c606f5ef386c3124960a0f8af134d239b19b8e25a821e1","family_id":"qf-mod31-countable-finite-certificate","seed":202610311004,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-5,4,46]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[4, -5, 46] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([4, -5, 46], [[4, 0], [-5, 1], [46, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([4, -5, 46], [[4, 0], [-5, 1], [46, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0005","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62fbb65f594d04aea330a374d17afedcb20dd5eb620ee84a82167a033de802b2","family_id":"qf-mod31-countable-finite-certificate","seed":202610311005,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-6,5,48]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[5, -6, 48] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([5, -6, 48], [[5, 0], [-6, 1], [48, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([5, -6, 48], [[5, 0], [-6, 1], [48, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0006","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eb3ed0a09a194089f90835594d12e9abe579d64a3edaf20f55f97e7faf18a67","family_id":"qf-mod31-countable-finite-certificate","seed":202610311006,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-7,6,50]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[6, -7, 50] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([6, -7, 50], [[6, 0], [-7, 1], [50, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([6, -7, 50], [[6, 0], [-7, 1], [50, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0007","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21226e190605f136bbceadb0184bd2ea16764d26245f4fcb73d3298caa09ddec","family_id":"qf-mod31-countable-finite-certificate","seed":202610311007,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-8,7,52]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[7, -8, 52] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0008","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe60d607175e477ec83fc8400805f574339679494832f6df04b248cec22d9e5","family_id":"qf-mod31-countable-finite-certificate","seed":202610311008,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-9,8,54]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[8, -9, 54] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([8, -9, 54], [[8, 0], [-9, 1], [54, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([8, -9, 54], [[8, 0], [-9, 1], [54, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0009","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f60404a35ef542618b0ecb2119a8efae724c41c5b9b842ed203117c4c634013","family_id":"qf-mod31-countable-finite-certificate","seed":202610311009,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-10,9,56]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[9, -10, 56] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([9, -10, 56], [[9, 0], [-10, 1], [56, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([9, -10, 56], [[9, 0], [-10, 1], [56, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0010","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0415d08d424b89af588ed07b577f497656b2b40aa766742c02bd454330471a47","family_id":"qf-mod31-countable-finite-certificate","seed":202610311010,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-11,10,58]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[10, -11, 58] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([10, -11, 58], [[10, 0], [-11, 1], [58, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([10, -11, 58], [[10, 0], [-11, 1], [58, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0011","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9209634931526cac942b96752df6090c3cb1b4ecfbb84708e6abb0178ce0a4ca","family_id":"qf-mod31-countable-finite-certificate","seed":202610311011,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-12,11,60]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[11, -12, 60] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([11, -12, 60], [[11, 0], [-12, 1], [60, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([11, -12, 60], [[11, 0], [-12, 1], [60, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0012","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f4b8b0cb222b64dd720393d17e3b764825c72ead356db3517435d4facaacccb","family_id":"qf-mod31-countable-finite-certificate","seed":202610311012,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-13,12,62]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[12, -13, 62] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([12, -13, 62], [[12, 0], [-13, 1], [62, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([12, -13, 62], [[12, 0], [-13, 1], [62, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0013","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4c4d16b7a743103d049691106ac8fffa04e74778dc7998bb741b74d3152ed96","family_id":"qf-mod31-countable-finite-certificate","seed":202610311013,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-14,13,64]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[13, -14, 64] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([13, -14, 64], [[13, 0], [-14, 1], [64, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([13, -14, 64], [[13, 0], [-14, 1], [64, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0014","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ddcea23a96cc273d45bbf0f5543526777d5819822473eb5e362195cb9053db4","family_id":"qf-mod31-countable-finite-certificate","seed":202610311014,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-15,14,66]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[14, -15, 66] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([14, -15, 66], [[14, 0], [-15, 1], [66, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([14, -15, 66], [[14, 0], [-15, 1], [66, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0015","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8c7e93c891f16680761414d719ac741085d87df59ae51eed7eb734f72fd5f95","family_id":"qf-mod31-countable-finite-certificate","seed":202610311015,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-16,15,68]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[15, -16, 68] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([15, -16, 68], [[15, 0], [-16, 1], [68, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([15, -16, 68], [[15, 0], [-16, 1], [68, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0016","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8371ca895643c860fa1a4e615a98bb58ac7abdff624311359e99176e8e4ca49a","family_id":"qf-mod31-countable-finite-certificate","seed":202610311016,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-17,16,70]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[16, -17, 70] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([16, -17, 70], [[16, 0], [-17, 1], [70, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([16, -17, 70], [[16, 0], [-17, 1], [70, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0017","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aed48aac62ae7c518d8817df2a7dcdb8506227fbc22c0b0cd210aaf64b3caf5","family_id":"qf-mod31-countable-finite-certificate","seed":202610311017,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-18,17,72]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[17, -18, 72] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([17, -18, 72], [[17, 0], [-18, 1], [72, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([17, -18, 72], [[17, 0], [-18, 1], [72, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0018","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f2e52343847770c264845df5887b1657bb2fc3de14f4fa3a8d7129058a882f9","family_id":"qf-mod31-countable-finite-certificate","seed":202610311018,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-19,18,74]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[18, -19, 74] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([18, -19, 74], [[18, 0], [-19, 1], [74, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([18, -19, 74], [[18, 0], [-19, 1], [74, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0019","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10d6d69991266da88328fd9103f81014ee13b7dcfceae18f9dd21d2b24efd354","family_id":"qf-mod31-countable-finite-certificate","seed":202610311019,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-20,19,76]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[19, -20, 76] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([19, -20, 76], [[19, 0], [-20, 1], [76, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([19, -20, 76], [[19, 0], [-20, 1], [76, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0020","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6f167732a472291f008942690a54c3f1baec047adcf764d3de447309d121027","family_id":"qf-mod31-countable-finite-certificate","seed":202610311020,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-21,20,78]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[20, -21, 78] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([20, -21, 78], [[20, 0], [-21, 1], [78, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([20, -21, 78], [[20, 0], [-21, 1], [78, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0021","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4305d6de9c992a36e72567e57a8094e9e4f7230f948e7166e64501dbada107ef","family_id":"qf-mod31-countable-finite-certificate","seed":202610311021,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-22,21,80]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[21, -22, 80] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([21, -22, 80], [[21, 0], [-22, 1], [80, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([21, -22, 80], [[21, 0], [-22, 1], [80, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0022","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2186a80316719e78079933a8c5f38370b0ecc55973f222fa39a684e5a4de6bc3","family_id":"qf-mod31-countable-finite-certificate","seed":202610311022,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-23,22,82]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[22, -23, 82] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([22, -23, 82], [[22, 0], [-23, 1], [82, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([22, -23, 82], [[22, 0], [-23, 1], [82, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0023","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c79f848f5a611cfb566ec8d7cb5297dfe1353ea0ecc9d02a5f9d837c376e8b3d","family_id":"qf-mod31-countable-finite-certificate","seed":202610311023,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-24,23,84]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[23, -24, 84] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([23, -24, 84], [[23, 0], [-24, 1], [84, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([23, -24, 84], [[23, 0], [-24, 1], [84, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0024","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c530bae85879f194f27b55045ddf85dd7ee44654be62cb07b063d9ad4805cd0","family_id":"qf-mod31-countable-finite-certificate","seed":202610311024,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-25,24,86]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[24, -25, 86] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([24, -25, 86], [[24, 0], [-25, 1], [86, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([24, -25, 86], [[24, 0], [-25, 1], [86, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0025","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1877475d5eeac0afac0ef69c3a94dd915f8fe47d4187cc628a66f59ef119e9a2","family_id":"qf-mod31-countable-finite-certificate","seed":202610311025,"semantic_template":"qf-mod31-countable-finite-certificate/{\"elements\":[-26,25,88]}"},"fields":{"title":"A Complete Finite List Gives an At-Most-Countable Certificate","stem":"Complete nonempty duplicate-free finite integer list elements=[25, -26, 88] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].","answer":{"kind":"expression","value":"([25, -26, 88], [[25, 0], [-26, 1], [88, 2]], Yes, finite_implies_at_most_countable)","explanation":"Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([25, -26, 88], [[25, 0], [-26, 1], [88, 2]], Yes, finite_implies_at_most_countable)."},"skills":["A Complete Finite List Gives an At-Most-Countable Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0026","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"105a463719600001173347abffd36570ecfbd805e2a9160e5389f3fc4412def5","family_id":"qf-mod31-countable-affine-injection","seed":202610311026,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-12,\"left\":-1,\"right\":2,\"slope\":1}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=-12, selected left=-1, right=2. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(1, -12, -1, 2, -13, -10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, -12, -1, 2, -13, -10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0027","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f61a09e3f0ed511c167c6b4c9c2fbb8b510e2f1efb477745011f3002c95f955","family_id":"qf-mod31-countable-affine-injection","seed":202610311027,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-11,\"left\":-2,\"right\":3,\"slope\":2}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=-11, selected left=-2, right=3. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(2, -11, -2, 3, -15, -5, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, -11, -2, 3, -15, -5, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0028","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35d27a63304a9f600029198cf81c193f6538a46024d0ed8395310a8700be7036","family_id":"qf-mod31-countable-affine-injection","seed":202610311028,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-10,\"left\":-3,\"right\":4,\"slope\":3}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=-10, selected left=-3, right=4. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(3, -10, -3, 4, -19, 2, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, -10, -3, 4, -19, 2, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0029","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0de1ebaa702cb6c2fb0c15c3379a3dcb8c8b3928a4ea9b38dc26fd4aacaed06b","family_id":"qf-mod31-countable-affine-injection","seed":202610311029,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-9,\"left\":-4,\"right\":5,\"slope\":4}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=-9, selected left=-4, right=5. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(4, -9, -4, 5, -25, 11, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, -9, -4, 5, -25, 11, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0030","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15905e2abd95c3662281b6dbdf4918154beae1c822bfb5e6127c912004dd478f","family_id":"qf-mod31-countable-affine-injection","seed":202610311030,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-8,\"left\":-5,\"right\":6,\"slope\":5}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=-8, selected left=-5, right=6. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(5, -8, -5, 6, -33, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, -8, -5, 6, -33, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0031","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ad2c7a495e4e84cf30133a53f6ea3b42b467c80c11304074f13b6a962bcc8f2","family_id":"qf-mod31-countable-affine-injection","seed":202610311031,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-7,\"left\":-6,\"right\":7,\"slope\":1}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=-7, selected left=-6, right=7. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(1, -7, -6, 7, -13, 0, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, -7, -6, 7, -13, 0, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0032","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35680117fd37c935140785380e28185339d14f6f7717b44a60b35dd4c06fb092","family_id":"qf-mod31-countable-affine-injection","seed":202610311032,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-6,\"left\":-7,\"right\":8,\"slope\":2}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=-6, selected left=-7, right=8. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(2, -6, -7, 8, -20, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, -6, -7, 8, -20, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0033","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5fa3ac57c5b4245e65ce1c33fd737d4be7babde2cfa3243b170a2650104f599","family_id":"qf-mod31-countable-affine-injection","seed":202610311033,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-5,\"left\":-8,\"right\":9,\"slope\":3}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=-5, selected left=-8, right=9. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(3, -5, -8, 9, -29, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, -5, -8, 9, -29, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0034","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6ca3d2a17de5aae5cf5b760d9e3221cc3f3d893afe1e4ceed03d6315bf9a859","family_id":"qf-mod31-countable-affine-injection","seed":202610311034,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-4,\"left\":-9,\"right\":10,\"slope\":4}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=-4, selected left=-9, right=10. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(4, -4, -9, 10, -40, 36, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, -4, -9, 10, -40, 36, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0035","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"915d2f2d52d3f19dab8d794975f77021db0b6ab00b55f6b490a622b25d3e7f19","family_id":"qf-mod31-countable-affine-injection","seed":202610311035,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-3,\"left\":-10,\"right\":11,\"slope\":5}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=-3, selected left=-10, right=11. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(5, -3, -10, 11, -53, 52, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, -3, -10, 11, -53, 52, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0036","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62ccb61b98c4ece760baa64f8423e583e7a3e1405f2f8329008b11a1de75fefe","family_id":"qf-mod31-countable-affine-injection","seed":202610311036,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-2,\"left\":-11,\"right\":12,\"slope\":1}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=-2, selected left=-11, right=12. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(1, -2, -11, 12, -13, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, -2, -11, 12, -13, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0037","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7234b1aca57ae688247e5363f8141f04ddc9e48d7ccb0cede12195c0fc34666","family_id":"qf-mod31-countable-affine-injection","seed":202610311037,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":-1,\"left\":-12,\"right\":13,\"slope\":2}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=-1, selected left=-12, right=13. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(2, -1, -12, 13, -25, 25, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, -1, -12, 13, -25, 25, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0038","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50d8a3fdf77ec8b9aabf9bd0395958b9bc05438eb1cdcf863268ab20037ef03a","family_id":"qf-mod31-countable-affine-injection","seed":202610311038,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":0,\"left\":-13,\"right\":14,\"slope\":3}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=0, selected left=-13, right=14. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(3, 0, -13, 14, -39, 42, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, 0, -13, 14, -39, 42, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0039","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7de8e66273b14eb57eb98b498abd11d6e8860e8bf7b0814fb688657ee9de4dd","family_id":"qf-mod31-countable-affine-injection","seed":202610311039,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":1,\"left\":-14,\"right\":15,\"slope\":4}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=1, selected left=-14, right=15. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(4, 1, -14, 15, -55, 61, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, 1, -14, 15, -55, 61, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0040","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7e73de0fef331fd8fc67075b46b7e3f104917a8cc31227472360189adc410b9","family_id":"qf-mod31-countable-affine-injection","seed":202610311040,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":2,\"left\":-15,\"right\":16,\"slope\":5}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=2, selected left=-15, right=16. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(5, 2, -15, 16, -73, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, 2, -15, 16, -73, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0041","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"878fe082faa28f0378b98eed90d29ec35331830aebb3fa4b37b9b14b437d0b2d","family_id":"qf-mod31-countable-affine-injection","seed":202610311041,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":3,\"left\":-16,\"right\":17,\"slope\":1}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=3, selected left=-16, right=17. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(1, 3, -16, 17, -13, 20, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, 3, -16, 17, -13, 20, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0042","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a83a3adfb917a433008d0175c0343cac09a9e7e43b05ac8045f1fc21a330d0f1","family_id":"qf-mod31-countable-affine-injection","seed":202610311042,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":4,\"left\":-17,\"right\":18,\"slope\":2}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=4, selected left=-17, right=18. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(2, 4, -17, 18, -30, 40, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, 4, -17, 18, -30, 40, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0043","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b8bd185b5c11ec5973ddd05d1c7d334a78b5f452ac97195364a465b88461e55","family_id":"qf-mod31-countable-affine-injection","seed":202610311043,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":5,\"left\":-18,\"right\":19,\"slope\":3}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=5, selected left=-18, right=19. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(3, 5, -18, 19, -49, 62, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, 5, -18, 19, -49, 62, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0044","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3ce7de7cfe0ad912d1d12e4807e522d4c0967176045a34ad6859beb9c864455","family_id":"qf-mod31-countable-affine-injection","seed":202610311044,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":6,\"left\":-19,\"right\":20,\"slope\":4}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=6, selected left=-19, right=20. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(4, 6, -19, 20, -70, 86, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, 6, -19, 20, -70, 86, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0045","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d6a4561bf66a1eacbbdad6ad61bf9968c3ed992ebcab7a1105d0ea8c954c1f0","family_id":"qf-mod31-countable-affine-injection","seed":202610311045,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":7,\"left\":-20,\"right\":21,\"slope\":5}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=7, selected left=-20, right=21. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(5, 7, -20, 21, -93, 112, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, 7, -20, 21, -93, 112, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0046","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b84a3df326a7b0ab13a1ec0ff19d4f6438958d27e9d98fcd82db9e34160e870","family_id":"qf-mod31-countable-affine-injection","seed":202610311046,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":8,\"left\":-21,\"right\":22,\"slope\":1}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=8, selected left=-21, right=22. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(1, 8, -21, 22, -13, 30, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, 8, -21, 22, -13, 30, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0047","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04ec10569b370ea0a1866289ca19441df05500813ca3c5cbbcad4e82459c46ff","family_id":"qf-mod31-countable-affine-injection","seed":202610311047,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":9,\"left\":-22,\"right\":23,\"slope\":2}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=9, selected left=-22, right=23. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(2, 9, -22, 23, -35, 55, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, 9, -22, 23, -35, 55, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0048","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c0ee4681afa9c43351bfeb771a84d39e1f5d2b43ec25def360930e2eb345c1c","family_id":"qf-mod31-countable-affine-injection","seed":202610311048,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":10,\"left\":-23,\"right\":24,\"slope\":3}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=10, selected left=-23, right=24. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(3, 10, -23, 24, -59, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, 10, -23, 24, -59, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0049","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b0c8c95e43124d94dc15f38dbf5c9b88e374caae5191d8ea4c3dd93e0253bdb","family_id":"qf-mod31-countable-affine-injection","seed":202610311049,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":11,\"left\":-24,\"right\":25,\"slope\":4}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=11, selected left=-24, right=25. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(4, 11, -24, 25, -85, 111, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, 11, -24, 25, -85, 111, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0050","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2401276bf7082bd478e8cfaca1fc1e76a6b9d2909c8ef8d30d59b077ec7ae1ca","family_id":"qf-mod31-countable-affine-injection","seed":202610311050,"semantic_template":"qf-mod31-countable-affine-injection/{\"intercept\":12,\"left\":-25,\"right\":26,\"slope\":5}"},"fields":{"title":"A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess","stem":"On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=12, selected left=-25, right=26. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).","answer":{"kind":"expression","value":"(5, 12, -25, 26, -113, 142, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)","explanation":"A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, 12, -25, 26, -113, 142, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)."},"skills":["A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0051","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830148a3391c0cb88e543b8d1699cb9361384d1418856724e2b188229415e4a9","family_id":"qf-mod31-countable-enumeration","seed":202610311051,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-1,0,50],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[0, 50, -1], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 0, [[0, 0], [1, 50], [2, -1]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 0, [[0, 0], [1, 50], [2, -1]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0052","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff8ef4032c0a61d5e2d8f66512de8b5081f0d2eadaf9073e11f113c713f92c93","family_id":"qf-mod31-countable-enumeration","seed":202610311052,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-2,1,51],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[1, 51, -2], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 51, [[0, 1], [1, 51], [2, -2]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 51, [[0, 1], [1, 51], [2, -2]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0053","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77f44634f211c9f2867f89d715aacf8f0d1e0f731952d77f85eeb9756226cdce","family_id":"qf-mod31-countable-enumeration","seed":202610311053,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-3,2,52],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[2, 52, -3], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -3, [[0, 2], [1, 52], [2, -3]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -3, [[0, 2], [1, 52], [2, -3]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0054","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c50b7ae2704d3a69ecd07df020be6c7993f37c481e100d9351b9ab9d904109d","family_id":"qf-mod31-countable-enumeration","seed":202610311054,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-4,3,53],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[3, 53, -4], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 3, [[0, 3], [1, 53], [2, -4]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 3, [[0, 3], [1, 53], [2, -4]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0055","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c4d144ff7e713d316617bfb4b6e7ac1fdd583fd2d7161fc4629e528a31ec121","family_id":"qf-mod31-countable-enumeration","seed":202610311055,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-5,4,54],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[4, 54, -5], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 54, [[0, 4], [1, 54], [2, -5]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 54, [[0, 4], [1, 54], [2, -5]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0056","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f675257082345986a211cf2cb0805105a79d5bc057bc6ab7eebe293ed817a034","family_id":"qf-mod31-countable-enumeration","seed":202610311056,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-6,5,55],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[5, 55, -6], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -6, [[0, 5], [1, 55], [2, -6]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -6, [[0, 5], [1, 55], [2, -6]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0057","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44138996d58cc3295e850af6d3bce5c4febe49514e873991436b0dd9a2bf79b7","family_id":"qf-mod31-countable-enumeration","seed":202610311057,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-7,6,56],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[6, 56, -7], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 6, [[0, 6], [1, 56], [2, -7]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 6, [[0, 6], [1, 56], [2, -7]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0058","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2145aa6bc9116439404d2692ea440c038cd5ec440048f7602d791f87e4d9688b","family_id":"qf-mod31-countable-enumeration","seed":202610311058,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-8,7,57],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[7, 57, -8], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 57, [[0, 7], [1, 57], [2, -8]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 57, [[0, 7], [1, 57], [2, -8]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0059","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f83ec66a23f1a0cbd50919497e096be3db356784bcfdd2805869297d6f9d4e0a","family_id":"qf-mod31-countable-enumeration","seed":202610311059,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-9,8,58],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[8, 58, -9], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -9, [[0, 8], [1, 58], [2, -9]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -9, [[0, 8], [1, 58], [2, -9]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0060","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7789d30f52426abaa4d951373005d0811f9e1c659a2894df3abbe5f1bcb30ee6","family_id":"qf-mod31-countable-enumeration","seed":202610311060,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-10,9,59],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[9, 59, -10], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 9, [[0, 9], [1, 59], [2, -10]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 9, [[0, 9], [1, 59], [2, -10]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0061","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fb92d78d216f2be2fdf0236452f8ea96289adb64ceb9aa60a9bb42b25f70948","family_id":"qf-mod31-countable-enumeration","seed":202610311061,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-11,10,60],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[10, 60, -11], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 60, [[0, 10], [1, 60], [2, -11]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 60, [[0, 10], [1, 60], [2, -11]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0062","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40043e927036b732fce05be6223c69c31055fc9989ed1184eaa75e3fe067f3da","family_id":"qf-mod31-countable-enumeration","seed":202610311062,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-12,11,61],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[11, 61, -12], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -12, [[0, 11], [1, 61], [2, -12]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -12, [[0, 11], [1, 61], [2, -12]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0063","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afd43a1c2bf8357b1f02a17ae570d84abc144b1ce894e7fcea12f86c8ea71fa8","family_id":"qf-mod31-countable-enumeration","seed":202610311063,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-13,12,62],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[12, 62, -13], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 12, [[0, 12], [1, 62], [2, -13]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 12, [[0, 12], [1, 62], [2, -13]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0064","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02418bc3ba6e3134feeb81c3c3059d2224656146cc268213b8db1b9f2da7b1ac","family_id":"qf-mod31-countable-enumeration","seed":202610311064,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-14,13,63],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[13, 63, -14], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 63, [[0, 13], [1, 63], [2, -14]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 63, [[0, 13], [1, 63], [2, -14]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0065","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60279f18b44546651b6c8dc0af99dd002d4d7a58335688d4744a32467441a8f5","family_id":"qf-mod31-countable-enumeration","seed":202610311065,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-15,14,64],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[14, 64, -15], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -15, [[0, 14], [1, 64], [2, -15]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -15, [[0, 14], [1, 64], [2, -15]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0066","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb6e9fed0af34bfe1115974c76aca445447b33ea5eba641cce6c4690a8d04035","family_id":"qf-mod31-countable-enumeration","seed":202610311066,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-16,15,65],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[15, 65, -16], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 15, [[0, 15], [1, 65], [2, -16]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 15, [[0, 15], [1, 65], [2, -16]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0067","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbfceb5efa4e3cea0e07b1fc9461f645a31fb4af7ef7e8a796788aab2411cb1a","family_id":"qf-mod31-countable-enumeration","seed":202610311067,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-17,16,66],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[16, 66, -17], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 66, [[0, 16], [1, 66], [2, -17]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 66, [[0, 16], [1, 66], [2, -17]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0068","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b5fcffd1c67529be8977bac7f38f64cf70c549039b1175df2b3b6eb55167abf","family_id":"qf-mod31-countable-enumeration","seed":202610311068,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-18,17,67],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[17, 67, -18], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -18, [[0, 17], [1, 67], [2, -18]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -18, [[0, 17], [1, 67], [2, -18]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0069","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc8a453b349bd3e6b4bc6328e3744dcd6a0e4b424ffa962dbe881eb3604968c6","family_id":"qf-mod31-countable-enumeration","seed":202610311069,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-19,18,68],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[18, 68, -19], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0070","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"697f62ef7545d4f0422e33ddbe06f9e9f7608a3465a1489172b1ecee2b40e658","family_id":"qf-mod31-countable-enumeration","seed":202610311070,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-20,19,69],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[19, 69, -20], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 69, [[0, 19], [1, 69], [2, -20]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 69, [[0, 19], [1, 69], [2, -20]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0071","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b318a5f76d59b8aebfd94ed07baa4774db0729beb7669b321dd75f6fdd224361","family_id":"qf-mod31-countable-enumeration","seed":202610311071,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-21,20,70],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[20, 70, -21], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -21, [[0, 20], [1, 70], [2, -21]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -21, [[0, 20], [1, 70], [2, -21]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0072","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2f5b89b1b7ce0530ba0e0a59622b0d93cdcc330e03ecca3d59665258cfb846f","family_id":"qf-mod31-countable-enumeration","seed":202610311072,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-22,21,71],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[21, 71, -22], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 21, [[0, 21], [1, 71], [2, -22]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 21, [[0, 21], [1, 71], [2, -22]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0073","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae689f1403b6242214d45166900378abd892e02995d1c12de1962b9115a584cc","family_id":"qf-mod31-countable-enumeration","seed":202610311073,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-23,22,72],\"index\":1}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[22, 72, -23], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(1, 72, [[0, 22], [1, 72], [2, -23]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 72, [[0, 22], [1, 72], [2, -23]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0074","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f0bc15e8be46c0c2a02ad89a9b6917cb7c17d4d7c07f6cf3ecab1b8d19da24a","family_id":"qf-mod31-countable-enumeration","seed":202610311074,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-24,23,73],\"index\":2}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[23, 73, -24], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(2, -24, [[0, 23], [1, 73], [2, -24]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -24, [[0, 23], [1, 73], [2, -24]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0075","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83881670026c5bc6582f97abbaf5882bfc77cb6dff5d92aa604535710767f009","family_id":"qf-mod31-countable-enumeration","seed":202610311075,"semantic_template":"qf-mod31-countable-enumeration/{\"elements\":[-25,24,74],\"index\":0}"},"fields":{"title":"Keep Enumeration Direction and Original List Order Explicit","stem":"Complete nonempty duplicate-free finite integer enumeration elements=[24, 74, -25], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.","answer":{"kind":"expression","value":"(0, 24, [[0, 24], [1, 74], [2, -25]], finite_enumeration_complete)","explanation":"Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 24, [[0, 24], [1, 74], [2, -25]], finite_enumeration_complete)."},"skills":["Keep Enumeration Direction and Original List Order Explicit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0076","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"774e32691c1b969ce0c570c47cb658547376413e0f0b2b8da59e416af5e3700f","family_id":"qf-mod31-countable-classification","seed":202610311076,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":1,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=1. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 1, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 1, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0077","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c8968f59397e62dea52139d5663c6a265002b33d4d1b9a8cf5877a265b310cf","family_id":"qf-mod31-countable-classification","seed":202610311077,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":2,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=2. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 2, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 2, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0078","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13ee74541fd0c0892815fc8e8dec08d5b8942e1a533826bcc2eb43dbd5111638","family_id":"qf-mod31-countable-classification","seed":202610311078,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":3,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=3. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 3, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 3, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0079","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d21db799297067816290ca370344bb50d77504fcf947a0ecd6574d17b82afb20","family_id":"qf-mod31-countable-classification","seed":202610311079,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":4,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=4. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 4, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 4, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0080","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0158635b9f93276647038be0bf9d176887295b4ae86e9a419ec091d435a9d760","family_id":"qf-mod31-countable-classification","seed":202610311080,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":5,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=5. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 5, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 5, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0081","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c24c64ab76259b9e78ae7ddd8cfef4613d67f82ee2bbda588c402ac0591af371","family_id":"qf-mod31-countable-classification","seed":202610311081,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":6,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=6. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 6, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 6, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0082","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c7eca3f6690e90c535ccef107de95c0011d08062beefaa5024b3521468da6b1","family_id":"qf-mod31-countable-classification","seed":202610311082,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":7,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=7. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 7, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 7, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0083","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd7e0ac79a336bb1591311294444ec32ac9231b2453b6e413a55388a2cee3318","family_id":"qf-mod31-countable-classification","seed":202610311083,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":8,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=8. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 8, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 8, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0084","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce6cf276656b1a101ae5af241472c7d235a664b13b31f80bfbd04e1e01b837ec","family_id":"qf-mod31-countable-classification","seed":202610311084,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":9,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=9. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 9, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 9, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0085","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a66b0c73ce1aee1ff5c408f2f9f6c04ee91bba790f41381ec0d506d53eecf16a","family_id":"qf-mod31-countable-classification","seed":202610311085,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":10,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=10. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 10, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 10, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0086","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce6ad807fc80c64246e9a9febdb9573ed0f932dadbfe7478a7834c186c4f5d3a","family_id":"qf-mod31-countable-classification","seed":202610311086,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":11,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=11. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 11, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 11, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0087","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67c3bcc8ffc3b528530f19e98f1f35963ab57f35be84c0a7981a65a9a4871023","family_id":"qf-mod31-countable-classification","seed":202610311087,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":12,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=12. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 12, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 12, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0088","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec708f568017e86e6d094d91ce0cccec0d7aedc3f41b3d40d98acd44dcc0e123","family_id":"qf-mod31-countable-classification","seed":202610311088,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":13,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=13. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 13, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 13, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0089","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe0f35d1dacb58d2988a12bc94ab616c6bed0d0e7031898bfa6004ea1a4eb733","family_id":"qf-mod31-countable-classification","seed":202610311089,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":14,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=14. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 14, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 14, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0090","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0d993f984a61919fea942042726dccb5425a78a8014da4df277ef0d40913111","family_id":"qf-mod31-countable-classification","seed":202610311090,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":15,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=15. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 15, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 15, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0091","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb7f403085fbd6786a8adfaaf516c41a7d1f75cae23da3fda4fccc37d392c8e3","family_id":"qf-mod31-countable-classification","seed":202610311091,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":16,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=16. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 16, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 16, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0092","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"961a3337b59ee28fd06bc4fb5dfdb2eda4f0901ff722fc93572ea3d3192f5c41","family_id":"qf-mod31-countable-classification","seed":202610311092,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":17,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=17. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 17, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 17, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0093","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d12a0bd21820a807fd6f026dccba5ccdfe9f24245cae18b542dcb29748d178b7","family_id":"qf-mod31-countable-classification","seed":202610311093,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":18,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=18. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 18, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 18, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0094","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"262957c61b0295ff84aa45f27289a063cbc9ab79fc9fd5d64ede855fe484351b","family_id":"qf-mod31-countable-classification","seed":202610311094,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":19,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=19. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 19, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 19, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0095","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b5903656114ed8bf5f5560e392a63af074c3b631f08c25e5ae57f382c408f54","family_id":"qf-mod31-countable-classification","seed":202610311095,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":20,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=20. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 20, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 20, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0096","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02bb04e3ca4afa4d778e49e25409a6f4b7a0ae486dcac293f9a5335e55e9a9de","family_id":"qf-mod31-countable-classification","seed":202610311096,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":21,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=21. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 21, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 21, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0097","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee9462e987c2a632aac74fb7ff32da084216b4a0319c9800e4a3dd9e8abfe19e","family_id":"qf-mod31-countable-classification","seed":202610311097,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":22,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=22. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 22, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 22, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0098","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9425d219871e7e61ea2b75f8e2b67e9d55df43cea3c909ae79a6ca1c2afb0fd8","family_id":"qf-mod31-countable-classification","seed":202610311098,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":23,\"evidence_mode\":\"universal_affine_injection\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=universal_affine_injection, certificate_parameter=23. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(universal_affine_injection, 23, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 23, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0099","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43cf7bf4e31feb54c294a1b922a951ca97a0c38c7d4d087b1bb6684a20c18416","family_id":"qf-mod31-countable-classification","seed":202610311099,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":24,\"evidence_mode\":\"failed_candidate_encoding\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=failed_candidate_encoding, certificate_parameter=24. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(failed_candidate_encoding, 24, No, no_classification_from_failed_witness)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 24, No, no_classification_from_failed_witness)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r01-at-most-countable-set-0100","batch_id":"batch-20261014-advance-20261031-r01-at-most-countable-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed11df476b2e4bc7058bcd841127802275f988ad3a4e94a0492d1ba6d815c6a2","family_id":"qf-mod31-countable-classification","seed":202610311100,"semantic_template":"qf-mod31-countable-classification/{\"certificate_parameter\":25,\"evidence_mode\":\"complete_finite_list\"}"},"fields":{"title":"A Failed Encoding Does Not Prove Uncountability","stem":"Declared evidence_mode=complete_finite_list, certificate_parameter=25. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).","answer":{"kind":"expression","value":"(complete_finite_list, 25, Yes, at_most_countable)","explanation":"This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 25, Yes, at_most_countable)."},"skills":["A Failed Encoding Does Not Prove Uncountability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}