{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0001","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"814ca3989f5c9ba871a98f2c6d1855c6e2bdd8b5decb73af716b2c4bd4b3b5cb","family_id":"qf-density17-tree-component-qualification","seed":202610176001,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[],\"n\":0}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(0, [], [], Yes, No, No, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], [], Yes, No, No, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0002","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830d8c3f6d4bd9de6f89c8bbdc4b00aeb28ecc5a3f6c46051b2bf120a9be8239","family_id":"qf-density17-tree-component-qualification","seed":202610176002,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[],\"n\":1}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(1, [], [[0]], Yes, No, Yes, Yes)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], [[0]], Yes, No, Yes, Yes)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0003","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3019272ebadc2ddd2c8c5c3ba9fc0435ed8ffe1fe7cf163affb348e97814d5a5","family_id":"qf-density17-tree-component-qualification","seed":202610176003,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], [[0], [1, 2, 3]], No, Yes, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], [[0], [1, 2, 3]], No, Yes, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0004","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33656c9984d8294d76fea25956192c12c52b0586819f1531f153c652695019c2","family_id":"qf-density17-tree-component-qualification","seed":202610176004,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[2,3]],\"n\":4}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [2, 3]], [[0, 1], [2, 3]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0005","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d96912d49febdca0f59aff6ea06ae673465d337156cfaea72a766214ca3f188","family_id":"qf-density17-tree-component-qualification","seed":202610176005,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[1,2],[2,3]],\"n\":4}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(4, [[0, 1], [1, 2], [2, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [1, 2], [2, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0006","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91843fd0023d9a902d7d83b06f978d45d41288232eedf74bdb6d66f1b76d19f8","family_id":"qf-density17-tree-component-qualification","seed":202610176006,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3]], Yes, No, Yes, Yes)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0007","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"630de81abde93062b1ac06d05d0d1c366bc8f3ffd4c5358ef9e3bd82eef099a2","family_id":"qf-density17-tree-component-qualification","seed":202610176007,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1]], [[0, 1], [2], [3], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1]], [[0, 1], [2], [3], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0008","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"598fca67e37438d83ec68144e2286f495755d8df061137ddfb86bb3ca1c587d6","family_id":"qf-density17-tree-component-qualification","seed":202610176008,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 2]], [[0, 2], [1], [3], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2]], [[0, 2], [1], [3], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0009","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d41a62e421b74bdba16441e29a1c4334e4a5255a0bd5234e709b4a31c9085ed","family_id":"qf-density17-tree-component-qualification","seed":202610176009,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0010","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e5e597232a2fab0b5e4b18f211aca137cc15e549a07de4f65f7edff4d0e593e","family_id":"qf-density17-tree-component-qualification","seed":202610176010,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,3]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 3]], [[0, 3], [1], [2], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3]], [[0, 3], [1], [2], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0011","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67021cd0cfa3dbfd2af2a856c5f63bc61ebac715d3956fd8d0f291e90869ea4f","family_id":"qf-density17-tree-component-qualification","seed":202610176011,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,3]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3]], [[0, 1, 3], [2], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0012","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b5f2305083b6334b6f80afa7dccc5cbabb6f2f8a274b87ca2ed88305afaae0e","family_id":"qf-density17-tree-component-qualification","seed":202610176012,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3]], [[0, 2, 3], [1], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0013","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92e456d7e17cde6360c0136fee86dc4c3c5f6da6e69496bc56300faefb81028c","family_id":"qf-density17-tree-component-qualification","seed":202610176013,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,3]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3]], [[0, 1, 2, 3], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0014","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4eff6a6afe4be4b628a86d50550c2deb5b81af3042c7f877d603bc7d24a82aa","family_id":"qf-density17-tree-component-qualification","seed":202610176014,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 4]], [[0, 4], [1], [2], [3]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 4]], [[0, 4], [1], [2], [3]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0015","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b77150fd0b12ddb59e9232c64ca14355a24d323fe9b27cb5a57529928cfad2b7","family_id":"qf-density17-tree-component-qualification","seed":202610176015,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 4]], [[0, 1, 4], [2], [3]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0016","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5903485a0056be7eade4fc7077b67968cf7ed8efa9d012e895574421947c3960","family_id":"qf-density17-tree-component-qualification","seed":202610176016,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 4]], [[0, 2, 4], [1], [3]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0017","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e75ed6e534f167fb36ffb89de595176b96dfbc77ac457304d915593d42437adc","family_id":"qf-density17-tree-component-qualification","seed":202610176017,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 4]], [[0, 1, 2, 4], [3]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 4]], [[0, 1, 2, 4], [3]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0018","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a78ea27d635c6e6385840b3d83976687694e34fb0190ed50bc88a24802a10b56","family_id":"qf-density17-tree-component-qualification","seed":202610176018,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 3], [0, 4]], [[0, 3, 4], [1], [2]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0019","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37b8d2835af907566f43b0d04e6a621618eaa4b05f1b1ec50566c1089c6f7523","family_id":"qf-density17-tree-component-qualification","seed":202610176019,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 3], [0, 4]], [[0, 1, 3, 4], [2]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 3], [0, 4]], [[0, 1, 3, 4], [2]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0020","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23ea6f07fe94c66d8c844eb2a0854bcd3f4e1edd342da8d5c98395901e5b3334","family_id":"qf-density17-tree-component-qualification","seed":202610176020,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 2], [0, 3], [0, 4]], [[0, 2, 3, 4], [1]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [0, 3], [0, 4]], [[0, 2, 3, 4], [1]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0021","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d4e9aa0e8225bf0238dc1bf57ef9cabceb2d1620c6038e20b040a4d2ed0c0f2","family_id":"qf-density17-tree-component-qualification","seed":202610176021,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[0, 1, 2, 3, 4]], Yes, No, Yes, Yes)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [0, 3], [0, 4]], [[0, 1, 2, 3, 4]], Yes, No, Yes, Yes)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0022","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"865487c3e3dbcdfc8002222cd01a1d16a06f24eb55d433b0d89b0265924fa167","family_id":"qf-density17-tree-component-qualification","seed":202610176022,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[1,2]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[1, 2]], [[0], [1, 2], [3], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[1, 2]], [[0], [1, 2], [3], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0023","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94b7f5545789ce1072031d5cffa1218bd88172c002e58c1e1cf5468b46873e31","family_id":"qf-density17-tree-component-qualification","seed":202610176023,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[1,2]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0024","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2094225c14cdbaef99a8ce99685909f9771ceef8d5f6af3ce23a797909573232","family_id":"qf-density17-tree-component-qualification","seed":202610176024,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, No, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0025","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24edb9fc3977ff2fa77c38803b665721c10c88a5adb1fee2674af93c809c6788","family_id":"qf-density17-tree-component-qualification","seed":202610176025,"semantic_template":"qf-density17-tree-component-qualification/{\"edges\":[[0,1],[0,2],[1,2]],\"n\":5}"},"fields":{"title":"Components, Cycles, and Nonempty Tree Qualification","stem":"Use a complete simple undirected graph on Fin(5)={0,…,4} with raw edges=[[0, 1], [0, 2], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Give (n, normalized edges, sorted components, are all vertex pairs mutually reachable?, any cycle?, nonempty?, tree?). The pairwise condition is vacuously true on the empty vertex set.","answer":{"kind":"expression","value":"(5, [[0, 1], [0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, Yes, Yes, No)","explanation":"Check the full component and cycle structure, not just a numerical edge condition. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2]], [[0, 1, 2], [3], [4]], No, Yes, Yes, No)."},"skills":["Components, Cycles, and Nonempty Tree Qualification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0026","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f937b255d7a74989b090d3d468e629a0d25026f2f389d16900012d41d750942","family_id":"qf-density17-tree-unique-path-gate","seed":202610176026,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[],\"end\":\"untyped\",\"n\":0,\"start\":\"untyped\"}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=-1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(0, [], No, [No, No], not-a-tree, not-evaluated)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, [No, No], not-a-tree, not-evaluated)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0027","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92691befeacb81e949f22907524ae9264a56387edc24f652fcadc3b143f7eb7f","family_id":"qf-density17-tree-unique-path-gate","seed":202610176027,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[],\"end\":0,\"n\":1,\"start\":0}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(1, [], Yes, [Yes, Yes], [0], 0)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, [Yes, Yes], [0], 0)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0028","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2871308eb444e7d8aac14bce26802bb777e455d827d2369c17dc33e5c2d9fa2c","family_id":"qf-density17-tree-unique-path-gate","seed":202610176028,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[1,2],[1,3],[2,3]],\"end\":2,\"n\":4,\"start\":0}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], not-a-tree, not-evaluated)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], not-a-tree, not-evaluated)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0029","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5707e4f2a0e70f9494149f86e3b9fab67e718e1d5cd2adf3fd47eca3f6792027","family_id":"qf-density17-tree-unique-path-gate","seed":202610176029,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[1,2]],\"end\":2,\"n\":3,\"start\":\"untyped\"}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=-1, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], Yes, [No, Yes], outside-vertex-set, not-evaluated)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [No, Yes], outside-vertex-set, not-evaluated)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0030","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fe488581f4a40ea3583b1699072b2a8831440473d0426459e0328e39cd0b7a1","family_id":"qf-density17-tree-unique-path-gate","seed":202610176030,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[1,2]],\"end\":1,\"n\":3,\"start\":1}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], [1], 0)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], [1], 0)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0031","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"157244836c3ac653d825eebf204b11517474965a433a540e75836704039843a1","family_id":"qf-density17-tree-unique-path-gate","seed":202610176031,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,3]],\"end\":3,\"n\":4,\"start\":1}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], Yes, [Yes, Yes], [1, 0, 3], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], Yes, [Yes, Yes], [1, 0, 3], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0032","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45e930f41d76de82381182f2a8acf37a5cd9aaa9516e3eca28d98afd02caf411","family_id":"qf-density17-tree-unique-path-gate","seed":202610176032,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"end\":3,\"n\":6,\"start\":0}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], [0, 3], 1)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], [0, 3], 1)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0033","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cede19dc68ad6835979b95eb5918b6a9a3d719a5c6797fe7365d3f6faac0f30","family_id":"qf-density17-tree-unique-path-gate","seed":202610176033,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"end\":4,\"n\":6,\"start\":1}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [1, 0, 4], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [1, 0, 4], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0034","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5cb2ed0cfa87f5aacd63fb47dc3324cfe329c6ed37244653dc0047c5cb8ecc3","family_id":"qf-density17-tree-unique-path-gate","seed":202610176034,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2]],\"end\":5,\"n\":6,\"start\":2}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [2, 0, 5], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], [2, 0, 5], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0035","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2709f0d0f57ff049f1de00c0fe60d44f0dc20fff79956ec9a5ba979747c72a5","family_id":"qf-density17-tree-unique-path-gate","seed":202610176035,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,3]],\"end\":0,\"n\":6,\"start\":3}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [3, 0], 1)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [3, 0], 1)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0036","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beeb8a674e91516941670bcc756e3c28d17467fabc02c192a298aa65ef2fd998","family_id":"qf-density17-tree-unique-path-gate","seed":202610176036,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,4]],\"end\":1,\"n\":6,\"start\":4}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], [4, 1], 1)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], [4, 1], 1)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0037","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61b9a1464b5506ee49ff1bb266e6c0b3f9b95c21c1ce5c345e7d5f2cf4ad7a4e","family_id":"qf-density17-tree-unique-path-gate","seed":202610176037,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,5]],\"end\":2,\"n\":6,\"start\":5}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], [5, 0, 2], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], [5, 0, 2], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0038","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e2da37846ecd9c87e2865b8344e165c2e07ff0af93afc1ddee885f478856f1","family_id":"qf-density17-tree-unique-path-gate","seed":202610176038,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3]],\"end\":3,\"n\":6,\"start\":0}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [0, 1, 3], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], [0, 1, 3], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0039","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d7286d2337057d5f1b2d5a4fd8026c19b66acb584e59cec81ece7251e37c7b5","family_id":"qf-density17-tree-unique-path-gate","seed":202610176039,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"end\":4,\"n\":6,\"start\":1}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 0, 4], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 0, 4], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0040","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0988a59d246472dc50b7307882281aa907640a91eb74cd79d2ea7fcc03b92de7","family_id":"qf-density17-tree-unique-path-gate","seed":202610176040,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"end\":5,\"n\":6,\"start\":2}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 1, 0, 5], 3)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 1, 0, 5], 3)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0041","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"588c112dafda67479efa56c515591be1d5a5ef984c175c33dd9653409e3a7d48","family_id":"qf-density17-tree-unique-path-gate","seed":202610176041,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,3]],\"end\":0,\"n\":6,\"start\":3}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 1, 0], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 1, 0], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0042","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4de556c220d9baad9335a86474c93ca7369761d7179f62ec4ad5b719211b1bdc","family_id":"qf-density17-tree-unique-path-gate","seed":202610176042,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"end\":1,\"n\":6,\"start\":4}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], [4, 0, 1], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], [4, 0, 1], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0043","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46caae0c9e7a9ac12fa3fbb0ea12d33bb805a41f09bab977ea328385740c4f2a","family_id":"qf-density17-tree-unique-path-gate","seed":202610176043,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,5]],\"end\":2,\"n\":6,\"start\":5}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], [5, 2], 1)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], [5, 2], 1)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0044","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f4576269c0e9820a9578bad4d3507119b5460f3b21fdae1ca6859b504198355","family_id":"qf-density17-tree-unique-path-gate","seed":202610176044,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[2,3]],\"end\":3,\"n\":6,\"start\":0}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 2, 3], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 2, 3], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0045","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ff1d1f6d59482009c2a36f09b4c772a6d47c21fe2fa268e8d7cb6caf4a994fd","family_id":"qf-density17-tree-unique-path-gate","seed":202610176045,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,2],[1,3]],\"end\":4,\"n\":6,\"start\":1}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=1, end=4. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 2, 0, 4], 3)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], [1, 2, 0, 4], 3)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0046","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0dbf7e1150a393fbd6f2b9f6575734b7a4f56d17c881ebd5c672cb139803c36","family_id":"qf-density17-tree-unique-path-gate","seed":202610176046,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,2],[2,3]],\"end\":5,\"n\":6,\"start\":2}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=2, end=5. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 0, 5], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], [2, 0, 5], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0047","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"048bc620b07d24e5069c065cbe6edffad4f28f871516dbf5873594a9b7c4c807","family_id":"qf-density17-tree-unique-path-gate","seed":202610176047,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[2,3]],\"end\":0,\"n\":6,\"start\":3}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=3, end=0. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 2, 0], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], [3, 2, 0], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0048","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e4d4ecf437a375dd9d90d8baf060a28a336f6ed87583120872152b0e505c566","family_id":"qf-density17-tree-unique-path-gate","seed":202610176048,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,4],[2,3]],\"end\":1,\"n\":6,\"start\":4}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=4, end=1. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], [4, 1], 1)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], [4, 1], 1)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0049","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce538a88eb464cc469cd7b0a16f8b6365d912310c8c8030b667938e713f10d4a","family_id":"qf-density17-tree-unique-path-gate","seed":202610176049,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,2],[0,4],[0,5],[1,5],[2,3]],\"end\":2,\"n\":6,\"start\":5}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=5, end=2. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], [5, 0, 2], 2)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], [5, 0, 2], 2)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0050","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6a1d9e57c8bdccab72602208ced03443c597abd6c67bf49cd641aea146c584d","family_id":"qf-density17-tree-unique-path-gate","seed":202610176050,"semantic_template":"qf-density17-tree-unique-path-gate/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[2,3]],\"end\":3,\"n\":6,\"start\":0}"},"fields":{"title":"Original-Tree and Endpoint Guards before a Unique Path","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. start=0, end=3. Give (n, normalized edges, tree?, two endpoint type flags, unique path or not-a-tree/outside-vertex-set, path edge count or not-evaluated). A failed original-tree guard takes precedence over the path lookup; equal valid endpoints give a one-vertex path.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 3], 1)","explanation":"Establish the original tree and both endpoint types before finding the path; its edge count is one less than its vertex count. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], [0, 3], 1)."},"skills":["Original-Tree and Endpoint Guards before a Unique Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0051","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"844492762340fe8c8aeff766a2313785daf0cdf966722092463cb48876772e8e","family_id":"qf-density17-tree-leaf-deletion","seed":202610176051,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":\"untyped\",\"edges\":[],\"n\":0}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(0, [], No, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0052","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4675a085af68911d21d43a6a10bf4d300da923b0ffc41f48ff9961fa596db89c","family_id":"qf-density17-tree-leaf-deletion","seed":202610176052,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":0,\"edges\":[],\"n\":1}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(1, [], Yes, Yes, 0, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, Yes, 0, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0053","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3b538bcf30e7eec2fe9ce119f40a8afc361746b22ec72d4bbb344c9b7011fa5","family_id":"qf-density17-tree-leaf-deletion","seed":202610176053,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":1,\"edges\":[[0,1]],\"n\":2}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=1. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(2, [[0, 1]], Yes, Yes, 1, Yes, [0], [], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[0, 1]], Yes, Yes, 1, Yes, [0], [], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0054","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47c4e8b2b01e308ce4a25ff196015d3ffc2e8588d34280fd0ebfc99a5196dc3e","family_id":"qf-density17-tree-leaf-deletion","seed":202610176054,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":0,\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=0. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], No, Yes, 0, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], No, Yes, 0, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0055","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f0fc99810d4c16bb364bfccc761487849e50f7b7bb12fee43bb2cf0e1a9354d","family_id":"qf-density17-tree-leaf-deletion","seed":202610176055,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":1,\"edges\":[[0,1],[1,2]],\"n\":3}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=1. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0056","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f19160ae061296e6596b98ebfd151db606dce16289d94565e6375fdf1c301a","family_id":"qf-density17-tree-leaf-deletion","seed":202610176056,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":\"untyped\",\"edges\":[[0,1],[1,2]],\"n\":3}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=3. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], Yes, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, No, outside-vertex-set, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0057","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9551807e5d6825557fc74ae959bb686c3607e6c9ed23bf1ab014d9ef0c93d84","family_id":"qf-density17-tree-leaf-deletion","seed":202610176057,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":2,\"edges\":[[0,1],[0,2],[0,3]],\"n\":4}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[0, 1], [0, 2], [0, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=2. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(4, [[0, 1], [0, 2], [0, 3]], Yes, Yes, 1, Yes, [0, 1, 3], [[0, 1], [0, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[0, 1], [0, 2], [0, 3]], Yes, Yes, 1, Yes, [0, 1, 3], [[0, 1], [0, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0058","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee9089f2285f69f4910e4302c2c2205f5df731db833d3f579632de8b8368d36","family_id":"qf-density17-tree-leaf-deletion","seed":202610176058,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [0, 4]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0059","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a046c3dc3314311525756dc7d7c393a18ad2c28862278c54d4bb6a03a83cc8d","family_id":"qf-density17-tree-leaf-deletion","seed":202610176059,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 3], [0, 4], [1, 2]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0060","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aafe58d35b1a4c7b58afad7ea7a0fabd875dd8ac5c27efabb37867f49a14d50f","family_id":"qf-density17-tree-leaf-deletion","seed":202610176060,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 2]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0061","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17fec66b2b59d483b7d7cc1e711bafb611f954c4d4d2185be931618d2b5448a6","family_id":"qf-density17-tree-leaf-deletion","seed":202610176061,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0062","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dd78c6013f99eb13a184e9026e0203993f4d09a08340ea3e3e5b6b6897767d7","family_id":"qf-density17-tree-leaf-deletion","seed":202610176062,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,4]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 3], [0, 4], [1, 4]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0063","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc9e6d03169f21f7697c1677d2689359ba4736b772831a432e8acd93c40d7cda","family_id":"qf-density17-tree-leaf-deletion","seed":202610176063,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,5]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0064","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c81cc9bfe397d64216f4f20c82f034550afbc22bdbb7f7d99b070397f855b885","family_id":"qf-density17-tree-leaf-deletion","seed":202610176064,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0065","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bd5ff4fb8e1e80f8d79f8f1d157d8fd6e99bd58d7da341ea929f211c09d1be0","family_id":"qf-density17-tree-leaf-deletion","seed":202610176065,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [1, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0066","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e77f026ccfa1dc0f21681ae7640f896400473f7f02f1b7143548e62e6100570","family_id":"qf-density17-tree-leaf-deletion","seed":202610176066,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 2], [2, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0067","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd95b5ae914080e0c962251a159ac0e63b7be62c8820f25e6a1617e715277a47","family_id":"qf-density17-tree-leaf-deletion","seed":202610176067,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0068","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f691e3f48389010a0912750c7e9bcc63be5e2aaec6308712304df89501dc80","family_id":"qf-density17-tree-leaf-deletion","seed":202610176068,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 3], [2, 4]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0069","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"776f05b52f8fb4309e4683943b97bebee2d7e50c7f2eaaa64c2401ef73b0a1bd","family_id":"qf-density17-tree-leaf-deletion","seed":202610176069,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,5]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0070","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2458cd0b1a78a000414db1cf01505b37424b7b7c175a9f510d95deffe7eaac40","family_id":"qf-density17-tree-leaf-deletion","seed":202610176070,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,1],[0,2],[0,4],[0,5],[2,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [2, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0071","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b86f35dc9a14b49210630539a8ebfdc4a212b603759a447bff03e894023f263","family_id":"qf-density17-tree-leaf-deletion","seed":202610176071,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [1, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0072","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc73b40c02c35988b94ea78cd78658bcb083fcdbee11f904d71866510910a818","family_id":"qf-density17-tree-leaf-deletion","seed":202610176072,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 2], [2, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0073","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5a192add042022f67eb245ed79ba55ed0282a94cdbe2ff6651597e027e0e4d6","family_id":"qf-density17-tree-leaf-deletion","seed":202610176073,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0074","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97d4a260d5e5c6113d86d619ded78d05a0584f0de07cdec2638939f2dc99491d","family_id":"qf-density17-tree-leaf-deletion","seed":202610176074,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,4],[2,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], Yes)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, Yes, 1, Yes, [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 4], [2, 3]], Yes)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0075","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dcc287ef9e73fe1e462221cd5175b75b2eef218677f43b0b12ee9179107492a","family_id":"qf-density17-tree-leaf-deletion","seed":202610176075,"semantic_template":"qf-density17-tree-leaf-deletion/{\"at\":5,\"edges\":[[0,2],[0,4],[0,5],[1,5],[2,3]],\"n\":6}"},"fields":{"title":"Degree-One Leaf Deletion without Renumbering","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate deleted vertex at=5. Give (n, normalized edges, original tree?, vertex typed?, degree or outside-vertex-set, qualified leaf?, retained labels or not-applicable, retained edges or not-applicable, later tree? or not-evaluated).","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)","explanation":"A deletion is qualified only for a typed degree-one vertex of the original tree; preserve all remaining original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, Yes, 2, No, not-applicable, not-applicable, not-evaluated)."},"skills":["Degree-One Leaf Deletion without Renumbering"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r06-tree-0076","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"960034a044d7489e8043521e115f96232d06c0781a5946ab7fdf6cbcfadc80da","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176076,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,1],\"edges\":[],\"n\":0}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(0)=∅ with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(0, [], No, [No, No], Yes, No, No, not-applicable, not-applicable)","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0, [], No, [No, No], Yes, No, No, not-applicable, not-applicable)."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0077","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6f015d477319eac981854b11ed5bd6a6d6ba118a05889b88666207500da57d0","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176077,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,0],\"edges\":[],\"n\":1}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(1)={0,…,0} with raw edges=[]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 0]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(1, [], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1, [], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0078","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e87cd4416d72f2b3211fee15b1264cf9605eaafb85322d42a04f471d975de07e","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176078,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,1],\"edges\":[[0,1]],\"n\":2}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(2)={0,…,1} with raw edges=[[0, 1]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(2, [[0, 1]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2, [[0, 1]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0079","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e66f6ba6f492e36f2669898eaf2950d1c945ad62c9acfff5639adbde0f9781a2","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176079,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,2],\"edges\":[[0,1],[1,2]],\"n\":3}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[2, 0]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [0, 1, 2], [0, 1, 2, 0])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [0, 1, 2], [0, 1, 2, 0])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0080","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f9c8ab8dd2b224360fa17cdeed6539c2693d6fcfa086baf8da57b8b077e999c","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176080,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[-1,2],\"edges\":[[0,1],[1,2]],\"n\":3}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[-1, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], Yes, [No, Yes], Yes, No, No, not-applicable, not-applicable)","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [No, Yes], Yes, No, No, not-applicable, not-applicable)."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0081","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e20d4db69f0622233c8f4bad465eb00316019cb218122199a853025d080e8c6","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176081,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[1,1],\"edges\":[[0,1],[1,2]],\"n\":3}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(3)={0,…,2} with raw edges=[[0, 1], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[1, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(3, [[0, 1], [1, 2]], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[0, 1], [1, 2]], Yes, [Yes, Yes], No, No, No, not-applicable, not-applicable)."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0082","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1df92246a478d439713820fc93a2e1a94fecd4e7e5d0d8c69150e0b243c6a70","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176082,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[0,1],\"edges\":[[1,2],[1,3],[2,3]],\"n\":4}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(4)={0,…,3} with raw edges=[[1, 2], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[0, 1]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], Yes, No, No, not-applicable, not-applicable)","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[1, 2], [1, 3], [2, 3]], No, [Yes, Yes], Yes, No, No, not-applicable, not-applicable)."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0083","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27fdaf1e7b93165c3d841295c2de051a5f7697e32818ae0d698b74ffa7ebae2b","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176083,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0084","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8bf0a0f07a1f4ccda85e0693cc2ba5ca4d97410a6419635a0e9568819bc32b5","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176084,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0085","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1775c895c9f3fe86ac61142aae81f4d2e690eb25b31f94524caa47f56d8bab5","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176085,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 2]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0086","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f770a843886d45b7869922fa39dd22b4db812d92dc8838e549a8e9047536adf5","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176086,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0087","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb071d94f0bc51e290870ec6f35a5dba551c841cb335185970b574b8431c56a","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176087,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,4]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0088","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a08eb85a18028ab149290375373d98dd8861178ed48ddd0790697d7bcb44ec43","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176088,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,5]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3], [0, 4], [0, 5], [1, 5]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0089","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e4fe93f69728c25189d17263df5246634df2313dc75d3df1e38d7d72211f14a","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176089,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0090","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e55668db2b06b6ae563d5b928fb804de153e384c4e3cc1da2ff434b8d7800df0","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176090,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0091","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"397c934debb095d6ea82bf5f429eb69d633c7188e07308555805e723fe8c810e","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176091,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 1, 0, 5], [2, 1, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0092","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fd0129e45126db8fab64cee2db8fd7a5b4535a962c507163e9fabc8abeb1d24","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176092,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 3, 1, 0, 5], [2, 3, 1, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0093","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"484665ab7039ef9ddef70698eb737f9a503f33c1010eafc557662bee4218b9e2","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176093,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,4]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 4, 0, 5], [2, 4, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 4]], Yes, [Yes, Yes], Yes, No, Yes, [2, 4, 0, 5], [2, 4, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0094","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c896d18716601b67adf49dfa5303b75fd6bf2b823d64cb8db33cfd1c020ed196","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176094,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,4],[0,5],[1,3],[2,5]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 4], [0, 5], [1, 3], [2, 5]], Yes, [Yes, Yes], Yes, Yes, No, not-applicable, not-applicable)."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0095","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c304f5a2184e4f15dd02bf4cf3d104e7c7a61f415450f5ad640f8b27baa0ba56","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176095,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,1],[0,2],[0,4],[0,5],[2,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 4], [0, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0096","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a0103b3bfac05ab9b14f6c5faa2c061aecfe08837ac64ec950eaa59344254d4","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176096,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,2],[1,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [1, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0097","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"addd90893ebc391ee2b34212f052694c030be4383851ab829313b0b4e903e327","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176097,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,2],[2,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 2], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0098","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b34fd54e220e3498a9a75a068c944cb23653ea2f3029aa115890cdd8d1cee65","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176098,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,3],[2,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 3], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0099","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbbd4bcd529d9ddee420e70887dcc90df6027054938cc41114f255083809dfb3","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176099,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,4],[2,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 4], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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-20261012-advance-20261017-r06-tree-0100","batch_id":"batch-20261012-advance-20261017-r06-tree-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1635b9d94a10a1a4da1ff92282659d22a83e5c078ba017d7e5c2eb42282593d","family_id":"qf-density17-tree-add-nonedge-cycle","seed":202610176100,"semantic_template":"qf-density17-tree-add-nonedge-cycle/{\"edge\":[2,5],\"edges\":[[0,2],[0,4],[0,5],[1,5],[2,3]],\"n\":6}"},"fields":{"title":"Qualified Tree Nonedge and Its Unique Cycle Witness","stem":"Use a complete simple undirected graph on Fin(6)={0,…,5} with raw edges=[[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]]. Endpoints are typed and loops are prohibited. Reversed or repeated edges count once; normalize endpoints and edge order. Do not infer adjacency from a drawing. A tree is nonempty, connected and acyclic. Edge count alone does not decide this. The empty graph is not a tree; a singleton is a tree, but its degree-0 vertex is not a leaf. Preserve original vertex labels after deletion. Candidate added edge=[5, 2]. Give (n, normalized edges, original tree?, endpoint type flags, distinct endpoints?, already an edge?, qualification?, original path or not-applicable, closed cycle or not-applicable). A witness starts from the smaller normalized endpoint.","answer":{"kind":"expression","value":"(6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])","explanation":"Only a new nonloop edge between valid vertices of an existing tree closes its unique original path into a cycle. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [[0, 2], [0, 4], [0, 5], [1, 5], [2, 3]], Yes, [Yes, Yes], Yes, No, Yes, [2, 0, 5], [2, 0, 5, 2])."},"skills":["Qualified Tree Nonedge and Its Unique Cycle 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."}}