{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0001","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21980f945b6cb332f9eb9198f1ca5b3ff82bf1eb1663ff4901d0b96eb70e7c12","family_id":"qf-mod27-walk-qualification","seed":202610277001,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,1,0,1,1]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 1, 0, 1, 1]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 1], [[3, 1, 1]], No)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 0, 1, 1], [[3, 1, 1]], No)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0002","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23f41f4cf46cb0a4468840ced5afe3f7cf9b61aaa92bcbbe4cb452735d5e49ac","family_id":"qf-mod27-walk-qualification","seed":202610277002,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,1,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([2, 1, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 1, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0003","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"678d31c05a4cb955312123bb3869c3a747220f9e7f5c80c46fc07d2cea1fa0c0","family_id":"qf-mod27-walk-qualification","seed":202610277003,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,1,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([3, 1, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 1, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0004","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45246dcaa74d2c953ca8d046cf4e84a55df5d4bd1090af4e1cfcb064ada71f6b","family_id":"qf-mod27-walk-qualification","seed":202610277004,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,1,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([4, 1, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 1, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0005","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"788cd1a85cfbaf6f0dfc2f42a53004f5c516932fb8829b1f3bdb73508960e8ad","family_id":"qf-mod27-walk-qualification","seed":202610277005,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,1,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([5, 1, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 1, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0006","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a320d399476b83ec6f98f9f85ce071de27ed612af0cce6c0279c35042ccb0b0","family_id":"qf-mod27-walk-qualification","seed":202610277006,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,2,0,1,1]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 2, 0, 1, 1]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([0, 2, 0, 1, 1], [[3, 1, 1]], No)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 2, 0, 1, 1], [[3, 1, 1]], No)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0007","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1482984eaca56c3844b876f4fd5065533533dc6741ebbf6c3a10394a61ba081","family_id":"qf-mod27-walk-qualification","seed":202610277007,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,2,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([1, 2, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0008","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c61361cc312c0e9c5e200c1c527d658da9e57bb6ba52e97795e8fd9e6ad77fc7","family_id":"qf-mod27-walk-qualification","seed":202610277008,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,2,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([3, 2, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 2, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0009","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c55148aaa8c687b6df14a29469a6f9270447c934f9ffea1a8e559fc39f3f56ac","family_id":"qf-mod27-walk-qualification","seed":202610277009,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,2,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([4, 2, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 2, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0010","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebd5e96b2c8a6b43bb9f272927a22717edb614754c1dde776c38592a67970bc3","family_id":"qf-mod27-walk-qualification","seed":202610277010,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,2,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([5, 2, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 2, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0011","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daa9188242e67b9a546d6f45b4a1288f1232b751a47b2e5fff12c5e2a2af4be5","family_id":"qf-mod27-walk-qualification","seed":202610277011,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,3,0,1,1]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 3, 0, 1, 1]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([0, 3, 0, 1, 1], [[3, 1, 1]], No)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 3, 0, 1, 1], [[3, 1, 1]], No)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0012","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ae67294de1a4f286ca39052ecbbef8c577388955112ba0bd7bf3fbd13df02b1","family_id":"qf-mod27-walk-qualification","seed":202610277012,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,3,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([1, 3, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 3, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0013","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4f4f5cf7379aec0b7bfa4707862eb2a38a1ce300490a75edfd62b05d997e8fd","family_id":"qf-mod27-walk-qualification","seed":202610277013,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,3,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([2, 3, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0014","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f99b5bdfd210b8691edf8f9226cd4a16452a617bbf3736748012c27c5d6ea7","family_id":"qf-mod27-walk-qualification","seed":202610277014,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,3,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([4, 3, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 3, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0015","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"155fbaae6b334be6f8b5463e0a3827783c64515872e413c673324bef5ca18c64","family_id":"qf-mod27-walk-qualification","seed":202610277015,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,3,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([5, 3, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 3, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0016","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99fdc1b2f493a67b64ddaa997fd28ebfe25240c1bf6730758b08b8c9cf31e86d","family_id":"qf-mod27-walk-qualification","seed":202610277016,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,4,0,1,1]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 4, 0, 1, 1]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([0, 4, 0, 1, 1], [[3, 1, 1]], No)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 4, 0, 1, 1], [[3, 1, 1]], No)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0017","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41a28e223c91ec97305a77bc0bfe5fd250f32129372d72c55e0b9634187b2d8f","family_id":"qf-mod27-walk-qualification","seed":202610277017,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,4,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([1, 4, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 4, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0018","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8b32f81b787dbd2333f6141cdc236335bcb73232f68bd4daae309271d13aea3","family_id":"qf-mod27-walk-qualification","seed":202610277018,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,4,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([2, 4, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 4, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0019","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf5408089a82951ff052c588b9eb2c10181fb85830e03c2c02137a39fcb24fd4","family_id":"qf-mod27-walk-qualification","seed":202610277019,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,4,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([3, 4, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 4, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0020","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"139bea9afc8560b7fa1155eed9d640b4cd67a37a11676f625f6f7aabedf2569d","family_id":"qf-mod27-walk-qualification","seed":202610277020,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,4,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([5, 4, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 4, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0021","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20ea633213dda474877c7a46b0201ef8a7a9871b069b9517de468ee0b2ff0394","family_id":"qf-mod27-walk-qualification","seed":202610277021,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,5,0,1,1]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 5, 0, 1, 1]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([0, 5, 0, 1, 1], [[3, 1, 1]], No)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 5, 0, 1, 1], [[3, 1, 1]], No)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0022","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71a029a9e6176cb30b37105e7f471549f2ca133a9024d54adc5c1440670b2bbf","family_id":"qf-mod27-walk-qualification","seed":202610277022,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,5,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([1, 5, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 5, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0023","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1363140ae02633e60fd8c901641992a8b2016fd48f09c99733cb3e76bcd6fb2b","family_id":"qf-mod27-walk-qualification","seed":202610277023,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,5,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([2, 5, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 5, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0024","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47afd6c3246a588830e87b13adbac2dd9ce3846373edfefc4ba58c40be9dccee","family_id":"qf-mod27-walk-qualification","seed":202610277024,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,5,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([3, 5, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 5, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0025","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccf55de7dbbc8ced4066ea37063ebf61e7a3555bd4f45e911294916b9591d43e","family_id":"qf-mod27-walk-qualification","seed":202610277025,"semantic_template":"qf-mod27-walk-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,5,0,1,0]}"},"fields":{"title":"Check Every Consecutive Step in a Vertex Sequence","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (vertices,all bad [zero-based step index,from,to] records,walk?).","answer":{"kind":"expression","value":"([4, 5, 0, 1, 0], [], Yes)","explanation":"Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 5, 0, 1, 0], [], Yes)."},"skills":["Check Every Consecutive Step in a Vertex Sequence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0026","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fdd06d4f2e7f1dc728c69159f1eddd88a212eb98269d27788079717d49cfee8","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277026,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,1,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0, 1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0027","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2d526c6670d169e984380d755cecf66ff2c2e6dbf6c00ecfaa21d33c63191f","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277027,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,1,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0, 1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0028","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5cd38a0d2088ed3ddc79e57dc5e3669dff59731e7bece6be663c8d38f56ee58","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277028,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,1,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0, 1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0029","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3344a93050bd225d0608bb177e9d88b206c779128aa554cb78c1f2ccc0b59699","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277029,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,1,0,1,4]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 1, 0, 1, 4]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [1, 4], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [1, 4], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0030","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"262711c9690ccbb43a62440a64390e61d4cddc1a26987f42fb0a389933a46ff3","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277030,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,1,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0, 1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0031","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3b8ed8ee8881c7334c51ef87126c7213f3e7848597766ac904bbdc0022ba0c5","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277031,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,2,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0032","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fb9eff392f4a2fa2d2d166cf02a84e434b9436923bca1e927f00fbe22156f9c","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277032,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,2,0,1,1]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 2, 0, 1, 1]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(No, [1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (No, [1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0033","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a879a975f66f64e7f91240ec42b73475f3f7695e49883f82286e73d584de06f","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277033,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,2,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0034","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57cf6c0eec94093d57a305d3c51f4b70102713e012a45febc3071ae35d9af0ca","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277034,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,2,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0035","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c018102654ae4fa392d95267dbe8df40ded9e77c659e21a529558b7997fe4f6","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277035,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,2,0,1,5]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 2, 0, 1, 5]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [5], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [5], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0036","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ae4bbaefb29d96968b880f82b065e7bfbb0994da929bd21dfe30b0a82aa2c0e","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277036,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,3,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0037","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d47309d243e1120c60ce10980913211d0ca07ec5da939cbfd18bdceacb3ff80","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277037,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,3,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0, 1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0038","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebe0e494295a5d994ee890f8cb3e2a46c51d94e2100fb2bd7503c20696225395","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277038,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,3,0,1,2]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 3, 0, 1, 2]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [2], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [2], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0039","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb34707dec15f4995cc39a92fb31492824378e268fbb4d0db924fc981a943b7","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277039,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,3,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0040","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a24f45563708b95a3592577120e56727f83b1e7c43fa9955e35a080bab0bede","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277040,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,3,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0041","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a2b0060fff4a0ba57e556112aa02e9f8633b74c5885ad17f0e87097e6a0bc09","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277041,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,4,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0042","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c39b7614bb6f15c07b82e1da35a7eac3b2e579c578c836e8bd5720b74f317b7","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277042,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,4,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0, 1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0043","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f306220dfca76f1a63a85c2945deaeed890fa4bb61f307e3505160500c50b6e","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277043,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,4,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0044","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb39a8789ad2965942d3e09cfd1f141852a1ee959a2aadb4f3da8e965c4f8d6e","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277044,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,4,0,1,3]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 3]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [3], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [3], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0045","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0c5f4a1c105d8873f89a2b43f25e3dc00b86d446d8f387c91738879a042dd27","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277045,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,4,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0046","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6491eaac1a3a333732162bf16c0e3815e39b0d8b0c81d615e706a719c4e6ad00","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277046,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,5,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0047","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6314cdfcb2795a361cdbd2a1e8b116adcaedf6c4fffa484f0afc2da63f22b5b","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277047,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,5,0,1,1]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 5, 0, 1, 1]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(No, [1], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (No, [1], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0048","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6269bdfe1eb1d51c0ec356fde7dd0d6b3440866c1879fedd37421b463f2d2c9c","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277048,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,5,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0049","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c0e0edf5f0aee929245731cea10174a93de0fcd459399f3c250fbae34b03968","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277049,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,5,0,1,0]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [0], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0050","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db8c76606918335af42e0da649529cb88728074f62aa6ff6335cc6b37c43f15b","family_id":"qf-mod27-walk-simple-path-qualification","seed":202610277050,"semantic_template":"qf-mod27-walk-simple-path-qualification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,5,0,1,4]}"},"fields":{"title":"A Walk Is Simple Only when Every Vertex Is Distinct","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 5, 0, 1, 4]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).","answer":{"kind":"expression","value":"(Yes, [4], No)","explanation":"First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [4], No)."},"skills":["A Walk Is Simple Only when Every Vertex Is Distinct"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0051","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"310c7193b1591d7a53584897fd119316a8f09f8227dbe19b71aad02262653e50","family_id":"qf-mod27-walk-length-endpoints","seed":202610277051,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,1,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0052","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d17c4c1a946eb84c163317b2a366ce6bda39f0c5a01314c51a9eb2b39a02ed46","family_id":"qf-mod27-walk-length-endpoints","seed":202610277052,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,1,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0053","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ee50ade449cea5116f26e4d11a2b9eae454ec5357fc5c3365ce458c07ed2db8","family_id":"qf-mod27-walk-length-endpoints","seed":202610277053,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,1,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0054","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0970161152212dd18c1082e23555fb7c37de7ca26db3e7aa1677fdf57f8a9bbd","family_id":"qf-mod27-walk-length-endpoints","seed":202610277054,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,1,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0055","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8e9a04222fb4aab4fcb8fc32d2c4c8194a2fd62d1092322df923e6d70b8ac52","family_id":"qf-mod27-walk-length-endpoints","seed":202610277055,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,1,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0056","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d7a632ebacb565639e0883cd1399319c694ab6f5fca72f5783abd79b9b097df","family_id":"qf-mod27-walk-length-endpoints","seed":202610277056,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,2,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0057","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62de18fe1427f267f68932ed37a22b78ef564fc1e3ecbaf08681faaee1b63fa3","family_id":"qf-mod27-walk-length-endpoints","seed":202610277057,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,2,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0058","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7b037b207c18197fbbbb5bd0612205b9d3d14264c95a2de5a4856b980ac5278","family_id":"qf-mod27-walk-length-endpoints","seed":202610277058,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,2,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0059","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a681acd9cac88b86c30e22a072fc3db9fdc0ddd67b7265ebb630d88916300c4f","family_id":"qf-mod27-walk-length-endpoints","seed":202610277059,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,2,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0060","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd16acb8a14de1bd308c7c99338a0849949e179589935f8fabf7fcbf1cd5fa8e","family_id":"qf-mod27-walk-length-endpoints","seed":202610277060,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,2,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0061","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80cc95fd9c88aa908454e32027baace9f3396442989e84b4294bd26138ca0e34","family_id":"qf-mod27-walk-length-endpoints","seed":202610277061,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,3,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0062","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12fb65035b76716fb88f51d7540edb47e91ad673f19bce04b7c6108922cc9c1a","family_id":"qf-mod27-walk-length-endpoints","seed":202610277062,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,3,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0063","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e2244870862be15d602677e8865fa4dac512847faa4b443e6d6f1abd70cb120","family_id":"qf-mod27-walk-length-endpoints","seed":202610277063,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,3,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0064","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"271b58c11a1aa904f4b0c696e5d2abd5b600410fb9b9d367fc3bcce6541dcd82","family_id":"qf-mod27-walk-length-endpoints","seed":202610277064,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,3,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0065","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3547d3a6381d6c71f67092445cd25aae0107050258a51130365a6dcd9aba61a","family_id":"qf-mod27-walk-length-endpoints","seed":202610277065,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,3,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0066","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f75c68a897d9a90c3ac3abac140dc96f8e4e2224661c08d9055d2e7f909d1aff","family_id":"qf-mod27-walk-length-endpoints","seed":202610277066,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,4,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0067","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14534d0e95f8464175e551afd8c0f7b30285e925e8172f1da6c1593a51af9ce7","family_id":"qf-mod27-walk-length-endpoints","seed":202610277067,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,4,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0068","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ea6ff354c83bfc566f170457aec29cf63c7fbc2cc1d184568f253deb9a352c","family_id":"qf-mod27-walk-length-endpoints","seed":202610277068,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,4,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0069","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea25e325d286190b6c814e03bdff710933cfa67faa5c1db9b4b84b24855f56b1","family_id":"qf-mod27-walk-length-endpoints","seed":202610277069,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,4,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0070","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72050b8e22d08930918cb0309ddd2e0cf696039784e15616c6e17b4be922b50a","family_id":"qf-mod27-walk-length-endpoints","seed":202610277070,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,4,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0071","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf12977fd0431bf4cf309a531a6c4151e68825a68f7a124c7e60f540edf79f63","family_id":"qf-mod27-walk-length-endpoints","seed":202610277071,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,5,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0072","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfcdc0147f745f4549707735e3d1a0a2aa430796d235721f657201398e3f7b10","family_id":"qf-mod27-walk-length-endpoints","seed":202610277072,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,5,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0073","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"368904d1f1812483c509938caafc50112f3ac53d83bcc3dc74cb17adab6b7aed","family_id":"qf-mod27-walk-length-endpoints","seed":202610277073,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,5,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0074","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e1c2e4fd840e86ea6b0cb5ae250d2c0610d4b9ba26a9d859827b98c6e5ad576","family_id":"qf-mod27-walk-length-endpoints","seed":202610277074,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,5,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0075","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec5bc5d7c3c50410b497ac758790a2e39fb4682ca16b2b8e26e2d45d6ce4ac7a","family_id":"qf-mod27-walk-length-endpoints","seed":202610277075,"semantic_template":"qf-mod27-walk-length-endpoints/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,5,0,1,0]}"},"fields":{"title":"Separate Candidate Length and Endpoints from Walk Qualification","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.","answer":{"kind":"expression","value":"(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)","explanation":"Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)."},"skills":["Separate Candidate Length and Endpoints from Walk 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-20261014-advance-20261027-r07-graph-walk-simple-path-0076","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5bdf3ae7e31ad98aa5054bd7620ce8ad1b13a8804f9fe2e31dc30ef1d7deec9","family_id":"qf-mod27-walk-repetition-classification","seed":202610277076,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,1,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, Yes)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, Yes)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0077","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abe431db855220d707d06f60c5d787441f8116d20cb0b11be52ec7d5018482cd","family_id":"qf-mod27-walk-repetition-classification","seed":202610277077,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,1,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0078","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d48cafa04d4fa4f3cb0c1e82134777b3165c2b4018fd33f63810350e2012d7c","family_id":"qf-mod27-walk-repetition-classification","seed":202610277078,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,1,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0079","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11cb2ad636ac547c38dfa9745c390df410a87ba7c6a6a0ea47871bf7abd2edc8","family_id":"qf-mod27-walk-repetition-classification","seed":202610277079,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,1,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0080","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"807e0c8e3df5f3c7b9aaedcd41582cb8dec0542b057b1b592c5c9e1a2862218f","family_id":"qf-mod27-walk-repetition-classification","seed":202610277080,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,1,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 1, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0081","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3f429abe2331b77f1b91aec56487e45b46aaea294972ed03461307f0b6bd2de","family_id":"qf-mod27-walk-repetition-classification","seed":202610277081,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,2,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 2], [0, 1]], No, No, Yes)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 2], [0, 1]], No, No, Yes)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0082","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b3305aed5bf4db691f8b409eb8488c2b180efbc883017e58d3ed07b4d8c4006","family_id":"qf-mod27-walk-repetition-classification","seed":202610277082,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,2,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0083","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04b863a17621f54f55843d61c0b30638017968f9cc0db6b59f1b0d344f00f4a7","family_id":"qf-mod27-walk-repetition-classification","seed":202610277083,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,2,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0084","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20cac3586f3c94479e7147aba00ed7fafbcb1e3d3110084802dac34867539f21","family_id":"qf-mod27-walk-repetition-classification","seed":202610277084,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,2,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0085","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e5d3516c43ff447858d1ae0c228461a4caf396c065c31b9c5255c3b7bc18ffd","family_id":"qf-mod27-walk-repetition-classification","seed":202610277085,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,2,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0086","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b15e9d9c657fe484398597323f91703e889202b6ca4f1d81ae4835aa009ff24c","family_id":"qf-mod27-walk-repetition-classification","seed":202610277086,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,3,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 3], [0, 1]], No, No, Yes)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 3], [0, 1]], No, No, Yes)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0087","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4c8c86b3b91147da53973b02af45a05b47ec6e297d4087a00c411dd24f47888","family_id":"qf-mod27-walk-repetition-classification","seed":202610277087,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,3,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0088","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9f4c0890c11c8ce740b5525fe4674dd42ae2cc575c729453b2c309348bbc2e3","family_id":"qf-mod27-walk-repetition-classification","seed":202610277088,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,3,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0089","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f53c501fc4f74bfcca9c173c286bcc16e5a8102ced448abc3675dc9f285ac3f","family_id":"qf-mod27-walk-repetition-classification","seed":202610277089,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,3,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0090","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19a3db51800ce84cc6d66cf7e523079aab0bc1e64c36f587a3d3ad04600e637e","family_id":"qf-mod27-walk-repetition-classification","seed":202610277090,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,3,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0091","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93c4b75e71a204db8397e71207ac15858dd22dcada537476438cc93ce915b59f","family_id":"qf-mod27-walk-repetition-classification","seed":202610277091,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,4,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 4], [0, 1]], No, No, Yes)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 4], [0, 1]], No, No, Yes)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0092","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bedd0f72d62a7f0ebf3d0f0515e62018236778b05804906a8ff66f67dd52e42c","family_id":"qf-mod27-walk-repetition-classification","seed":202610277092,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,4,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0093","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecdbccfcb96e5168fb2b3c78d65f54c3b41d54d1c3fa8978adb88a144da151df","family_id":"qf-mod27-walk-repetition-classification","seed":202610277093,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,4,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0094","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0e8597f16ef5b958a7ffa55ecbc3d019a90caf3e103343e384d018d872be45e","family_id":"qf-mod27-walk-repetition-classification","seed":202610277094,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,4,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0095","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d77c0c6c0cbef250d11c3d2d744de94d2d248a9f744d6786b0420de4b9fbb42b","family_id":"qf-mod27-walk-repetition-classification","seed":202610277095,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[5,4,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 4, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0096","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5757fa157bc25d53e2ec78f572afcdbbf56e8a6ec361d4fe7031307583b9ca4e","family_id":"qf-mod27-walk-repetition-classification","seed":202610277096,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[0,5,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[0, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 5], [0, 1]], No, No, Yes)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 5], [0, 1]], No, No, Yes)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0097","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"611ed28b136b2472659941d7093ca6479da526dde79f2463a16468a1c5771162","family_id":"qf-mod27-walk-repetition-classification","seed":202610277097,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[1,5,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[1, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0, 1], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0098","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c07aa56f3a7cccb26890ec6368d167dfe29a8eccf2ef823b56e3f4a67ea359c1","family_id":"qf-mod27-walk-repetition-classification","seed":202610277098,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[2,5,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[2, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0099","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c6873b3085a7ec9b8e1e31db04ffd0fdc40a4618bfc5a1ac804d7b0720466b6","family_id":"qf-mod27-walk-repetition-classification","seed":202610277099,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,5,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r07-graph-walk-simple-path-0100","batch_id":"batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d38c5459e7cd29d6fac641175dcb57c3180c89be03e40d08e141ae70af96e7f","family_id":"qf-mod27-walk-repetition-classification","seed":202610277100,"semantic_template":"qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[4,5,0,1,0]}"},"fields":{"title":"Distinguish Repeated Vertices, Repeated Edges and Closed Walks","stem":"Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[4, 5, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.","answer":{"kind":"expression","value":"(Yes, 4, [0], [[0, 1]], No, No, No)","explanation":"Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)."},"skills":["Distinguish Repeated Vertices, Repeated Edges and Closed Walks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}