{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0001","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d57f0cfcf6a3ad07eaad097c3cc1dc651bda1de7da09fb90cb80b1ab4c53bf65","family_id":"qf-mod26-cauchy-finite-tail-audit","seed":202610263001,"semantic_template":"qf-mod26-cauchy-finite-tail-audit/{\"N\":0,\"epsilon\":[1,3],\"sequence\":[[1,2],[1,3],[1,4],[1,5],[1,6]]}"},"fields":{"title":"A Finite Tail Audit Cannot Establish General Cauchy Behavior","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, 1/2, -3], -5, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0027","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66ebf3b0b961bdeec51e730d42f1f727494207211d27f6e4e4fe72a5c23cf349","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263027,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-11,1],[2,2],[-2,2]],\"tail\":[-4,2]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, 1, -1], -2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0028","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c98cc6adad942a18e02e80e382885c68102e861d9df712b1babb48f40729a090","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263028,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-10,1],[3,2],[-1,3]],\"tail\":[-3,3]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, 3/2, -1/3], -1, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0029","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf00e7c0466d96ced43d22360faa1ddffc9bb90bf9b9ab2200d79a81353302d5","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263029,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-9,1],[4,2],[0,4]],\"tail\":[-2,4]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, 2, 0], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0030","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9cb0e9e3795b54e4fd006b7d136d2f2cba6db2d35fd4ae439515d3cdabd3023","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263030,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-8,1],[5,2],[1,5]],\"tail\":[-1,5]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 5/2, 1/5], -1/5, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0031","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5eb39160313a1b8a51f06ca2f407837847fcd550f50612b77b714fbf4a4dc52","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263031,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-7,1],[6,2],[2,1]],\"tail\":[0,6]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, 3, 2], 0, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0032","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8ff7a64b4175a025d8e7e9f5008ce4d0080708f55995a018bcc7cccc54bba0d","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263032,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-6,1],[7,2],[3,2]],\"tail\":[1,7]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, 7/2, 3/2], 1/7, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0033","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bf10d49f9844b27b36525417d0fbd28c51eff26dcf4e812f974a3385141cdf7","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263033,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-5,1],[8,2],[-3,3]],\"tail\":[2,1]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, 4, -1], 2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0034","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa5e02496eee3a38ca13ba3a395ad44472f82afeba11612f130bdf0d3dc93be2","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263034,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-4,1],[9,2],[-2,4]],\"tail\":[3,2]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 9/2, -1/2], 3/2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0035","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"934182d2765ff4b076e8ed7020587b73edbe36771d6f8aa766a10dda1cdfedf9","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263035,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-3,1],[10,2],[-1,5]],\"tail\":[4,3]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, 5, -1/5], 4/3, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0036","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"206b3c2ed0ce5fe82ad92e71778c680e908b4127979c58440f1a2cbf186829c8","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263036,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-2,1],[11,2],[0,1]],\"tail\":[5,4]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 11/2, 0], 5/4, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0037","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5b82455361c535633b62176dd992a4da867c573cf21ffdd9f3fac5d3b22e351","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263037,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[-1,1],[12,2],[1,2]],\"tail\":[-5,5]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, 6, 1/2], -1, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0038","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20f7dcc2cd24e049d36956e392fbc307681e867596180d621e3b598d92a90f56","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263038,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[0,1],[13,2],[2,3]],\"tail\":[-4,6]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 13/2, 2/3], -2/3, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0039","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cc4fe073e7871f2104ede0e6179a899e5899434884a0f2bf84c90bbaef13474","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263039,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[1,1],[14,2],[3,4]],\"tail\":[-3,7]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 7, 3/4], -3/7, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0040","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8e9b45815345025c67d7a261ac425f00e2a3355887f63ab1a751595972219c8","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263040,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[2,1],[15,2],[-3,5]],\"tail\":[-2,1]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 15/2, -3/5], -2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0041","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"318b99d80f3b645775fa37dc2ca18e76481e1b1f3dea840640c48b01345f1be9","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263041,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[3,1],[16,2],[-2,1]],\"tail\":[-1,2]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 8, -2], -1/2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0042","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"232dcb60a382a3f644d781b4fbbea88788394e8175a7b1df6deb6323bb4ff568","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263042,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[4,1],[17,2],[-1,2]],\"tail\":[0,3]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 17/2, -1/2], 0, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0043","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"458d9c2f34abe34083b46d6dfc35388ee9fda3f621748a72d42a00a094263170","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263043,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[5,1],[18,2],[0,3]],\"tail\":[1,4]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 9, 0], 1/4, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0044","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271fbf49dbd51972e9e8fb4642969606c27dacddec6c937c772c5e4e4178332d","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263044,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[6,1],[19,2],[1,4]],\"tail\":[2,5]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, 19/2, 1/4], 2/5, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0045","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1519935e822e643de76f072f6d69a18b099fdba72db1c9d2f3a665d54e2f4b8a","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263045,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[7,1],[20,2],[2,5]],\"tail\":[3,6]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, 10, 2/5], 1/2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0046","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bc54970f901b4874a70fe8c5343b7afee5ff2f3052c4086d7575dd899070f09","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263046,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[8,1],[21,2],[3,1]],\"tail\":[4,7]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 21/2, 3], 4/7, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0047","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4246ef20cf62aa5dd2eff03d4829bd70502244dceb4b184aa7723d4a704922d","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263047,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[9,1],[22,2],[-3,2]],\"tail\":[5,1]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 11, -3/2], 5, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0048","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"251533ed0d0ffaacb23e6bc94134234185f1e2024b242d4421d36fdd704068ed","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263048,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[10,1],[23,2],[-2,3]],\"tail\":[-5,2]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 23/2, -2/3], -5/2, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0049","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcf368a4732317ef9b844857ef57c282529de2d1533f4da303d61d1f00fff7fb","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263049,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[11,1],[24,2],[-1,4]],\"tail\":[-4,3]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, 12, -1/4], -4/3, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0050","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07da4da0b4586a2515214b56b7e16e201adc74ae1e9bdbd1cfa161ba97441a48","family_id":"qf-mod26-cauchy-ultimately-constant-certificate","seed":202610263050,"semantic_template":"qf-mod26-cauchy-ultimately-constant-certificate/{\"prefix\":[[12,1],[25,2],[0,5]],\"tail\":[-3,4]}"},"fields":{"title":"An Explicit Infinite Constant Tail Supplies a Cauchy Witness","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)0 choose N=prefix length, Yes)","explanation":"For any positive epsilon choose N equal to the prefix length. Every pair of terms at or beyond N is identical, so its distance is zero. This uses the declared infinite model rather than a finite sample. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 25/2, 0], -3/4, 3, for any epsilon>0 choose N=prefix length, Yes)."},"skills":["An Explicit Infinite Constant Tail Supplies a Cauchy Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261026-r03-cauchy-sequence-0051","batch_id":"batch-20261013-advance-20261026-r03-cauchy-sequence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7047641e2beb3509ffc632a488dab68ab1817034bf4ee75e5765281fea2ab5a","family_id":"qf-mod26-cauchy-reciprocal-epsilon-witness","seed":202610263051,"semantic_template":"qf-mod26-cauchy-reciprocal-epsilon-witness/{\"epsilon\":[1,3],\"shift\":1}"},"fields":{"title":"A Strict Reciprocal Bound for the Selected Epsilon","stem":"Keep the quantifier order: for every epsilon>0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)=N implies distance<=1/(N+shift)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)0 there exists N such that for all m,n≥N, d(x_m,x_n)