{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0001","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45a64a343dd396e868d4ba05cb138464799770f6d3727df93427dea7ec4fd311","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147001,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3]] and unique candidate subset=[v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [v1, v2, v3], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v1, v2], [v2, v3]], [v1, v2, v3], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0002","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5da136ed88e6938569bcbb15f5fa26e2ddfa63f48afc58ddba9671aa3bbc03d","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147002,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]] and unique candidate subset=[v0, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [v0, v2, v3, v4], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v1, v2], [v2, v3], [v3, v4]], [v0, v2, v3, v4], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0003","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e45a71c1191ef0180270bea04bb899bb12da2d99fe92affaea9f0cf9f3f60df3","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147003,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]] and unique candidate subset=[v0, v1, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0004","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"103355a2d882df6cc56d016be35978d1f87d56d7e917cb3797888d2b1fa1542b","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147004,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]] and unique candidate subset=[v0, v1, v2, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v4, v5, v6], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v4, v5, v6], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0005","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d55c6e23b4621639173388cde98d0a205f8fec67584d55c557acfe76f3a68c7","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147005,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]] and unique candidate subset=[v0, v1, v2, v3, v5, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v5, v6, v7], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v5, v6, v7], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0006","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf8fd755f7860d464349cfa30338e52f75b5ecc72c0b273965165774766b6a10","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147006,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0007","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eeee17aae4975c099692b7ca250bee0460d391318afc4d93444205fdf967144","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147007,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v5, v7, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0008","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b84edcab1e97aadd48cfbd6cf0a5c878cd745e9a9329f37d699e0f74b3c19c9","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147008,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v5, v6, v8, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0009","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fc7f90dc45c32943ec21dec9ab89c166e2a6d1695f0796f80645308edbe9149","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147009,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v9, v10, v11]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v9], [[[v0, v1], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0010","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10a60045af063b7c5272c5e4f43e147b9eb0cacca8b70aabb43d8212db813ae9","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147010,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]] and unique candidate subset=[v0, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [v0, v2, v3], [[[v0, v1], Yes], [[v0, v3], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v0, v3], [v1, v2], [v2, v3]], [v0, v2, v3], [[[v0, v1], Yes], [[v0, v3], Yes], [[v1, v2], Yes], [[v2, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0011","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"684d749061f52323080e4af0c46ddbe1e127b3c906c1f3bfecea933cdea2af3e","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147011,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]] and unique candidate subset=[v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v4], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v0, v4], [v1, v2], [v2, v3], [v3, v4]], [v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v4], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0012","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"709d98a0a1edb1abb52bd4e7495bf20a8e525b2d5e6af3bc43078f160be06231","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147012,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v5], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v5], [v1, v2], [v2, v3], [v3, v4], [v4, v5]], [v0, v1, v2, v3, v4], [[[v0, v1], Yes], [[v0, v5], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0013","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ec0ecc8b9672e418bc836044614dac70737928228d1e39d19a17c4c873ce143","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147013,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v3, v4, v6], [[[v0, v1], Yes], [[v0, v6], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v6], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]], [v0, v1, v2, v3, v4, v6], [[[v0, v1], Yes], [[v0, v6], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0014","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9da5c06c0b2d8c454159e4306d001a0e63ba832c14d6ea5cb28882a68d1da875","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147014,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v4, v6, v7], [[[v0, v1], Yes], [[v0, v7], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v7], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]], [v0, v1, v2, v3, v4, v6, v7], [[[v0, v1], Yes], [[v0, v7], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0015","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fad2e4a674e878662ab5ad57b690266b131cc36bc407ee1b49fde8d8ddffe217","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147015,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v8], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v8], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v8], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0016","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2c8becdd54d28e5ba3851cd9ab9ce85af7b1853b9d4b95ebb2d176ab18ff265","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147016,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v9], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v9], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v9], Yes], [[v1, v2], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0017","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51a476033279e1848937a49e84ce9e0a61e69e5ddf2e77ccef21447fa080df00","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147017,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v10], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v10], [v1, v2], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v10], Yes], [[v1, v2], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0018","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0768d65a88170522543255d2a4e45249aba0e3dbfd779b10caeecdde990df1bb","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147018,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8, v9, v10, v11]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v11], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v11], [v1, v2], [v10, v11], [v10, v9], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]], [v0, v1, v10, v11, v2, v3, v4, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v11], Yes], [[v1, v2], Yes], [[v10, v11], Yes], [[v10, v9], Yes], [[v2, v3], Yes], [[v3, v4], Yes], [[v4, v5], Yes], [[v5, v6], Yes], [[v6, v7], Yes], [[v7, v8], Yes], [[v8, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0019","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f8e9dff2d8ba2d53b2219d22bec255365bd1f64cc54c558559b31a4b298b570","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147019,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3]] and unique candidate subset=[v0, v1, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [v0, v1, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3], [[v0, v1], [v0, v2], [v0, v3]], [v0, v1, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0020","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd9b1c3299e92ee9fe8f99775158dec9ee4da7dab408e93e90c4cf7f5e0e0e22","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147020,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [v0, v1, v2, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4], [[v0, v1], [v0, v2], [v0, v3], [v0, v4]], [v0, v1, v2, v3], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0021","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b2c455fcf642c4658df60acefc52f5fdf726abb317947ae0ac17edc35af56a2","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147021,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]] and unique candidate subset=[v0, v1, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]], [v0, v1, v3, v4, v5], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0022","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6662b4d46cad64eec78337e90daf7ae927babbe02244297df60a325d0212f28","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147022,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]] and unique candidate subset=[v1, v2, v3, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [v1, v2, v3, v4, v5, v6], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]], [v1, v2, v3, v4, v5, v6], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0023","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39701a9fc9c35cbf25d41b6ef84fdce321e1b89540689b125c5e2a28f823e415","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147023,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [v0, v1, v2, v3, v4, v5, v7], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]], [v0, v1, v2, v3, v4, v5, v7], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0024","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ea6cf4a418ce65fd836ce8cdbff714ee89268b846186eb17ec3fea0e1b0074","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147024,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]], [v0, v1, v2, v3, v4, v6, v7, v8], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0025","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56552e7f978041499fcafa6efd5cd61ceee023afabab6237a5fc53b574b5da9e","family_id":"qf-mod45-vcover-complete-edge-audit","seed":202611147025,"semantic_template":"qf-mod45-vcover-complete-edge-audit/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Every Undirected Edge Needs at Least One Selected Endpoint","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]] and unique candidate subset=[v0, v1, v2, v3, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (vertices, edges, candidate, edge_coverage_rows, cover, uncovered_edge_or_null, certificate).","answer":{"kind":"expression","value":"([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes], [[v0, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)","explanation":"List every [edge,covered] in canonical edge order. A single uncovered edge refutes cover status; report the first such witness or null. A cover is a vertex subset, not an edge cover or a directed orientation condition. The exact result, retaining the requested order, is ([v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], [[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]], [v0, v1, v2, v3, v5, v6, v7, v8, v9], [[[v0, v1], Yes], [[v0, v2], Yes], [[v0, v3], Yes], [[v0, v4], Yes], [[v0, v5], Yes], [[v0, v6], Yes], [[v0, v7], Yes], [[v0, v8], Yes], [[v0, v9], Yes]], Yes, null, vertex_cover_requires_every_edge_to_have_at_least_one_selected_endpoint)."},"skills":["Every Undirected Edge Needs at Least One Selected Endpoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0026","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a27daecc3bd6e9b2bdc8c963e717f6882159126555444bd00feb96a22b700ad","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147026,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0027","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c173a5c7ba4ca06142dd61c803dbc20c05ab8e32c9a1767f96ad46ea574b32a","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147027,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0028","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ab59af1c558068ab3922d3eaf5645061f469ca0169e6d7b13662b73a161d357","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147028,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0029","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf322cf98d046972eb35946278175ebc0ef8bde36d7a5ce8366777a19eefeb32","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147029,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0030","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cb3574bc115f83a969f558c670bc48d0c15cce9bb4b22a4bda45d4e04bdadc4","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147030,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0031","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adda45a4c5c1e246be03a83e37b1a97a584b3cbe4f8ed30921add5c75668f79b","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147031,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0032","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"259536aa01f69b653bff4cda88a239aed0e913240392b7b1226f1ceb20bdfb5d","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147032,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0033","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39faf09bacae05cba5c143753dfc413d7ebf38e0555a857322fcd900e663eeec","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147033,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0034","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65e9ed4921b3481fec2033a005339538ee594821feef29bcc6b491ff107e0147","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147034,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0035","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e8ecc2937329dcdb631fe6ea373580785da0b3378882981bb02b40d0f48aec8","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147035,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v3], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v3], No], [[v1, v2], No], [[v2, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0036","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a345c88e5e0fcb5b721b93bbf5afd1510176cddefcb1927b25407fbb10b0f90","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147036,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v4], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v4], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0037","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1955444a77d487d9592e772b749a1b5ed757114ceca314eb46b1d8b6220b7fed","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147037,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v5], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v5], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0038","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c73be8f600899e236ad10a94c8938ca075d81b97acd8a716656d2cdaa12db13","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147038,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v6\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v6], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v6], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0039","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd90a032d2e92bb4e8827067df1a0d42951f6b8d5fbdea0579615ab2c944109f","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147039,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v7\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v7], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v7], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0040","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d097a12b45447420cc406abb071342ee124af5a9b3570c67f4206b511b490ca1","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147040,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v8\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v8], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v8], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0041","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c4d4beab6af0635c35d985252fbf35691ee762693868c4b129cb110c8c68589","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147041,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v9\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v9], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v9], No], [[v1, v2], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0042","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae625bae21f38db9175f4b4cdc3abdf34838ae3a8886b316c12abded29466cbd","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147042,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v10], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v10], No], [[v1, v2], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0043","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dbc3739b606a409adb2fbdbb6d4f6c5dae8106eb86c5506116bc8dd95f8465b","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147043,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v11\"],[\"v1\",\"v2\"],[\"v10\",\"v11\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10], [v10, v11], [v11, v0]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v11], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v11], No], [[v1, v2], No], [[v10, v11], No], [[v10, v9], No], [[v2, v3], No], [[v3, v4], No], [[v4, v5], No], [[v5, v6], No], [[v6, v7], No], [[v7, v8], No], [[v8, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0044","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f258dd135a9f6a7e01ebd6c982e38352b280f2dac85d556d32bd0297aab3e3b","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147044,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0045","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75ba0fcf577ae48b5388fb7036dffde05485605c796aa07f02a8836d227655a3","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147045,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0046","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a41a40a58f97077ff6d355db9d2871cffadc74dd8e352a296c17fdc75c97bf69","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147046,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0047","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16b9e114d6270b570de3f2880f78930a520644e3e06553823282a81e6395c5c8","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147047,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0048","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f315c0ab87e88cb956ce0419ed82f7a23e26b0a477dfe7490eedac7dbed0b3a1","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147048,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0049","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c433abe0b070e2a5499cfcfd8db4ef25a2067c4229c9101bcf535da29252026","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147049,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0050","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e386cc0d2942d427330f112f30a5115a17d2c178e9f396eabf8771943e59f155","family_id":"qf-mod45-vcover-uncovered-edge-witness","seed":202611147050,"semantic_template":"qf-mod45-vcover-uncovered-edge-witness/{\"candidate\":[],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"One Edge Whose Both Endpoints Are Missing Refutes the Candidate","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, edge_coverage_rows, uncovered_edge, cover, boundary).","answer":{"kind":"expression","value":"([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No], [[v0, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)","explanation":"Require an actual noncover and retain the complete edge rows plus the first uncovered edge. An outside/duplicate candidate vertex is a type error, not this well-typed false-cover witness. The exact result, retaining the requested order, is ([], [[[v0, v1], No], [[v0, v2], No], [[v0, v3], No], [[v0, v4], No], [[v0, v5], No], [[v0, v6], No], [[v0, v7], No], [[v0, v8], No], [[v0, v9], No]], [v0, v1], No, one_edge_with_both_endpoints_outside_C_refutes_vertex_cover)."},"skills":["One Edge Whose Both Endpoints Are Missing Refutes the Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0051","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"304645a1772e468f5ea7ea7d6507f0d3e1545d4e80700c4a6a3c2730f3119327","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147051,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[],\"edges\":[],\"superset\":[\"v1\"],\"vertices\":[\"v0\",\"v1\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1], loop-free simple undirected edges=[] and unique candidate subset=[]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v1], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [v0, v1], [], Yes, [v1], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (Yes, Yes, [v0, v1], [], Yes, [v1], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0052","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e50d97140f21eeb5870b9fc2605f52ead69a4671cfa33136dd1c30a821dbd410","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147052,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2]] and unique candidate subset=[v0, v1, v2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0053","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e73eb9c174b27960bd0a3430d32946882998efd9932fd167af781b5adde1d5d","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147053,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0054","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16c883410ec9f04643cfdc228feb30dec82c0b7219dc7ed73d9c30ec7c25708d","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147054,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0055","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27002a4a5e9db88b7c3cb7dfa27961a952148949eaa0903cac9628f9fe70b841","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147055,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5]] and unique candidate subset=[v0, v1, v2, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0056","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a40da92262fad83743b7e7a7c2ee66143040190072b03cc4b603618f4c21e1b","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147056,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0057","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4e6c9de0453954f8c6c7db9b8b1ac740b3a9c2ef0fbc4d4df43374787ba0c67","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147057,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0058","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4adaa8f00fe1afb528d62b12965d488f7faf073463488dda81f93761c3598c13","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147058,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0059","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bb2c6f1654536bfddf3982a755b347b944df85426c8492cbf6076dc82cdcbdb","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147059,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0060","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb8651d855088319fe45f5d6c104446086b16b2a034cc3e1ac8f66e37ed41fa4","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147060,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v1\",\"v2\"],[\"v10\",\"v9\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"],[\"v5\",\"v6\"],[\"v6\",\"v7\"],[\"v7\",\"v8\"],[\"v8\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v6], [v6, v7], [v7, v8], [v8, v9], [v9, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0061","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76851cf5916f8c3884e3ad1ff3295755d58e08af3b47c39bb3de98001b2f9d9b","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147061,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\"],\"edges\":[[\"v0\",\"v1\"]],\"superset\":[\"v0\",\"v1\",\"v2\"],\"vertices\":[\"v0\",\"v1\",\"v2\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2], loop-free simple undirected edges=[[v0, v1]] and unique candidate subset=[v0, v1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0062","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b83896d7cdc304cfa863957d3ae0bbaf610ce3f475dbcd32762e8331bfdcc42","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147062,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v0, v2]] and unique candidate subset=[v0, v1, v2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0063","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"304fd505c9f299537ef84ce4f6131d73df44702d3a6221f63af30f440e1afa89","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147063,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0064","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efefcbff6ab83d331f3e7bea129fba597ace9e06f8b2603b174b3ad2a43c021a","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147064,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0065","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f71b26871580ae6e3f509b5f00fae524c4b9c898e64251453db87e2c5ffffb6e","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147065,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5]] and unique candidate subset=[v0, v1, v2, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0066","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d0b5f745157578430595801cc01561b89879fcaa83f1e2934dd8046368d5024","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147066,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v7], [v0, v1, v2, v3, v4, v5, v6], Yes, [v0, v1, v2, v3, v4, v5, v6, v7], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0067","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c946ef00b0da747f06faad31781b4992f461ae016f7df23adb1d04121f8c301a","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147067,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v8], [v0, v1, v2, v3, v4, v5, v6, v7], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0068","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f5ae428320d7c8a97858504aab0ede16fdae77bdc456ba6c966425c425e193b","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147068,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v9], [v0, v1, v2, v3, v4, v5, v6, v7, v8], Yes, [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0069","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb5a7a9f6652cd71d202a682f8439b1256e084ed853e122a0cd834734105ea40","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147069,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v10], [v0, v1, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0070","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3166a91849547ab017e9c66769c62bf2b28539faf47283b2b7c3bcc1845e2e25","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147070,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v10\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v10\"],[\"v0\",\"v2\"],[\"v0\",\"v3\"],[\"v0\",\"v4\"],[\"v0\",\"v5\"],[\"v0\",\"v6\"],[\"v0\",\"v7\"],[\"v0\",\"v8\"],[\"v0\",\"v9\"]],\"superset\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"],\"vertices\":[\"v0\",\"v1\",\"v10\",\"v11\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\",\"v7\",\"v8\",\"v9\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], loop-free simple undirected edges=[[v0, v1], [v0, v2], [v0, v3], [v0, v4], [v0, v5], [v0, v6], [v0, v7], [v0, v8], [v0, v9], [v0, v10]] and unique candidate subset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v11], [v0, v1, v10, v2, v3, v4, v5, v6, v7, v8, v9], Yes, [v0, v1, v10, v11, v2, v3, v4, v5, v6, v7, v8, v9], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0071","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"840555eac0890f3c58497faaf4559bbeee7be3906356fe2f52aa50cfec8a881f","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147071,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\"],\"edges\":[],\"superset\":[\"v0\",\"v1\",\"v2\"],\"vertices\":[\"v0\",\"v1\",\"v2\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2], loop-free simple undirected edges=[] and unique candidate subset=[v0, v1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [v0, v1, v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (Yes, Yes, [v0, v1, v2], [v0, v1], Yes, [v0, v1, v2], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0072","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15e0ec2dca72947fcca64c2c7b5bae513de07d2405923257c63274c17e482a50","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147072,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v2\"],[\"v1\",\"v2\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v0]] and unique candidate subset=[v0, v1, v2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v3], [v0, v1, v2], Yes, [v0, v1, v2, v3], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0073","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddefc1e82486a339e966feb95eb887ab6caa9b26822e57a323a263fdcd0fd685","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147073,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v3\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v0]] and unique candidate subset=[v0, v1, v2, v3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v4], [v0, v1, v2, v3], Yes, [v0, v1, v2, v3, v4], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0074","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88102acc7aa7eb364898d31764645c53f299ca266a9e787049ef25d963021c85","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147074,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v4\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v0]] and unique candidate subset=[v0, v1, v2, v3, v4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v5], [v0, v1, v2, v3, v4], Yes, [v0, v1, v2, v3, v4, v5], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0075","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce26cfa584c0ca31c89f10aad4f9689508db797068180dfc4d8673ab1029c6a4","family_id":"qf-mod45-vcover-empty-isolated-superset","seed":202611147075,"semantic_template":"qf-mod45-vcover-empty-isolated-superset/{\"candidate\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\"],\"edges\":[[\"v0\",\"v1\"],[\"v0\",\"v5\"],[\"v1\",\"v2\"],[\"v2\",\"v3\"],[\"v3\",\"v4\"],[\"v4\",\"v5\"]],\"superset\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"],\"vertices\":[\"v0\",\"v1\",\"v2\",\"v3\",\"v4\",\"v5\",\"v6\"]}"},"fields":{"title":"Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks","stem":"Use complete finite unique vertices=[v0, v1, v2, v3, v4, v5, v6], loop-free simple undirected edges=[[v0, v1], [v1, v2], [v2, v3], [v3, v4], [v4, v5], [v5, v0]] and unique candidate subset=[v0, v1, v2, v3, v4, v5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Given superset=[v0, v1, v2, v3, v4, v5, v6], possibly equal to the candidate. Give (edgeless_graph, empty_cover_valid_if_edgeless_or_null, isolated_vertices, candidate, candidate_cover, superset, superset_cover, certificate).","answer":{"kind":"expression","value":"(No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)","explanation":"Require the candidate to be a cover and the new set actually to contain it. An edgeless graph allows the empty cover; otherwise the empty-cover conditional slot is null rather than an invented certificate. Adding vertices preserves coverage, including optional isolated vertices. The exact result, retaining the requested order, is (No, null, [v6], [v0, v1, v2, v3, v4, v5], Yes, [v0, v1, v2, v3, v4, v5, v6], Yes, empty_edge_graph_empty_cover_is_valid_isolated_vertices_are_optional_and_cover_supersets_remain_covers)."},"skills":["Edgeless Graphs, Optional Isolated Vertices and Cover Supersets Have Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0076","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df4976552378d451d0b5cabedc510bd5e545d5e5445925afa27dc22c6ad905b8","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147076,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1]] and unique candidate subset=[l0_0, l0_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1], [[l0_0, [l0_1], No], [l0_1, [l0_0], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1], [[l0_0, [l0_1], No], [l0_1, [l0_0], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0077","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"672a101f5863f5db57e2e1f66f8ef841de77c99b012e7b8f95e7d32fe0c85d32","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147077,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2]] and unique candidate subset=[l0_0, l0_1, l0_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], No], [l0_1, [l0_0, l0_2], No], [l0_2, [l0_0, l0_1], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2], [[l0_0, [l0_1, l0_2], No], [l0_1, [l0_0, l0_2], No], [l0_2, [l0_0, l0_1], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0078","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"086d91f2054f8dff9ac43c2e8ee2d96d15c19b0dfe57154c3fccb358f597e9ed","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147078,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3], [[l0_0, [l0_1, l0_2, l0_3], No], [l0_1, [l0_0, l0_2, l0_3], No], [l0_2, [l0_0, l0_1, l0_3], No], [l0_3, [l0_0, l0_1, l0_2], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0079","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a59c0a957680d61cd888769ae781e0ef9085a3de0ff120f0da512437e33aef8","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147079,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4], [[l0_0, [l0_1, l0_2, l0_3, l0_4], No], [l0_1, [l0_0, l0_2, l0_3, l0_4], No], [l0_2, [l0_0, l0_1, l0_3, l0_4], No], [l0_3, [l0_0, l0_1, l0_2, l0_4], No], [l0_4, [l0_0, l0_1, l0_2, l0_3], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0080","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dab5fbed2d276e487c50416881f133e9bcc6ebded2cfd3aecb0079c7d3bc9331","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147080,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0081","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17a4bf10a16ba2300645a60a267964eb3099a3bc6943e3e9093f1cc939e0f3cc","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147081,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0082","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9de879dae0a1f7653ba9f7695626498c48467befaf5b4dd4941bea9a935eefe2","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147082,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0083","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca051a558495798b17fa4c3f6883e086e49858cdff6ffae9d1287a166c047b3","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147083,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0084","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5911393ae57d1d24599cc45d409b311d1571a711f7fd3f20c83849f0247ba089","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147084,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8], [c0, l0_9]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0085","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4dcac957562051cd2ba88a8de433ad3183c454148e96ef66cf50345bb5433c0","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147085,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_10\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c0\",\"l0_4\"],[\"c0\",\"l0_5\"],[\"c0\",\"l0_6\"],[\"c0\",\"l0_7\"],[\"c0\",\"l0_8\"],[\"c0\",\"l0_9\"]],\"vertices\":[\"c0\",\"l0_0\",\"l0_1\",\"l0_10\",\"l0_2\",\"l0_3\",\"l0_4\",\"l0_5\",\"l0_6\",\"l0_7\",\"l0_8\",\"l0_9\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9, l0_10], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c0, l0_4], [c0, l0_5], [c0, l0_6], [c0, l0_7], [c0, l0_8], [c0, l0_9], [c0, l0_10]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9, l0_10]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], [[l0_0, [l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_1, [l0_0, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_10, [l0_0, l0_1, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_2, [l0_0, l0_1, l0_10, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_3, [l0_0, l0_1, l0_10, l0_2, l0_4, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_4, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_5, l0_6, l0_7, l0_8, l0_9], No], [l0_5, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_6, l0_7, l0_8, l0_9], No], [l0_6, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_7, l0_8, l0_9], No], [l0_7, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_8, l0_9], No], [l0_8, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_9], No], [l0_9, [l0_0, l0_1, l0_10, l0_2, l0_3, l0_4, l0_5, l0_6, l0_7, l0_8], No]], Yes, 1, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0086","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6f7d2153756a3299187ea18f2a177c902c3bf862af30037f8b295327cfd0774","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147086,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], No], [l0_1, [l0_0, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l1_1], No], [l1_1, [l0_0, l0_1, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1], [[l0_0, [l0_1, l1_0, l1_1], No], [l0_1, [l0_0, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l1_1], No], [l1_1, [l0_0, l0_1, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0087","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fe5c6a739975c9a20e68a1f4196e848560ba6c3b6af99dd9377efd6c7ede5b5","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147087,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0088","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbbe6aff7611d623ceddba5d7838932acae021c92d113fae6e439ef1429c7010","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147088,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0089","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feee124a540f6e95b8fdd01ce26eee3e520414fd8756e399f3cc49ebfae76f61","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147089,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0090","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d778d8ee0ccb87eafc468aaa31549cee03e9ea6f77a1fa5ec6d0f40f2f19c4bf","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147090,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0091","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"252824392c2e12545c2430e29e89a55eba79d516e6629992272b2dbd7de5b477","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147091,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0092","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00883fc9a26346e3cb2c478c3ffa37d9ef326adf06594b6461e975d1c10ba4b6","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147092,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"],[\"c1\",\"l1_7\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\",\"l1_7\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6], [c1, l1_7]] and unique candidate subset=[l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], No], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], [[l0_0, [l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l0_1, [l0_0, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_0, [l0_0, l0_1, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_1, [l0_0, l0_1, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_2, [l0_0, l0_1, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6, l1_7], No], [l1_3, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6, l1_7], No], [l1_4, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6, l1_7], No], [l1_5, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6, l1_7], No], [l1_6, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_7], No], [l1_7, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0093","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4a84f1ae77b0df50c1af11bdd453e6fdbeac77fd325152d28bd50400aa62b7c","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147093,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0094","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b96ac3cd492947da4aba05d83ae19caad0d2b50249a1c3ddccf2e08b520890e","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147094,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0095","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b6e79bad8138591088e7e1d05341e75c14e6f3c01b8442f4d47c9cbd244e822","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147095,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0096","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e6b88146a7cf4aa94d002c97212693e15c553bbc6db527783e6b5aad36507af","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147096,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0097","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b3cd440721b76d24f469acd56096ec424add11fd9777ff9ec45a0cfa9e9d994","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147097,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0098","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96b521d0ac06abc89f601f32043baf60aacb6328460b36d41f8646fe141a0010","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147098,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"],[\"c1\",\"l1_3\"],[\"c1\",\"l1_4\"],[\"c1\",\"l1_5\"],[\"c1\",\"l1_6\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l1_0\",\"l1_1\",\"l1_2\",\"l1_3\",\"l1_4\",\"l1_5\",\"l1_6\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4], [c1, l1_5], [c1, l1_6]] and unique candidate subset=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4, l1_5, l1_6], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4, l1_5, l1_6], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4, l1_5, l1_6], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_5, l1_6], No], [l1_5, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_6], No], [l1_6, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4, l1_5], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0099","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6f111dadcf6cd3bd946ba926a609f462cf1423829092fd743e5b4ac5c3f7cf9","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147099,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r07-vertex-cover-0100","batch_id":"batch-20261015-advance-20261114-r07-vertex-cover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e74e40f53d9f973b9b14ab8444eab153d4097f39dd0d9a2852dec8a28c1eeb1d","family_id":"qf-mod45-vcover-minimal-vs-minimum-boundary","seed":202611147100,"semantic_template":"qf-mod45-vcover-minimal-vs-minimum-boundary/{\"candidate\":[\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"],\"edges\":[[\"c0\",\"l0_0\"],[\"c0\",\"l0_1\"],[\"c0\",\"l0_2\"],[\"c0\",\"l0_3\"],[\"c1\",\"l1_0\"],[\"c1\",\"l1_1\"],[\"c1\",\"l1_2\"]],\"vertices\":[\"c0\",\"c1\",\"l0_0\",\"l0_1\",\"l0_2\",\"l0_3\",\"l1_0\",\"l1_1\",\"l1_2\"]}"},"fields":{"title":"Inclusion-Minimal Covers Need Not Have Minimum Cardinality","stem":"Use complete finite unique vertices=[c0, l0_0, l0_1, l0_2, l0_3, c1, l1_0, l1_1, l1_2], loop-free simple undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c0, l0_3], [c1, l1_0], [c1, l1_1], [c1, l1_2]] and unique candidate subset=[l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2]. Edges normalize to sorted unordered pairs and are duplicate-free. A cover selects at least one endpoint of every edge; isolated vertices need not be selected. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, minimum, minimality_tests_agree, certificate).","answer":{"kind":"expression","value":"([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)","explanation":"Require a cover, then report [deleted vertex,reduced candidate,still covers] for every single deletion. Inclusion-minimal means no deletion works; minimum compares cardinality against all covers of this complete bounded graph. The final flag only compares these two candidate-specific Booleans, so both No can agree. No general minimal-iff-minimum theorem is asserted. The exact result, retaining the requested order, is ([l0_0, l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], [[l0_0, [l0_1, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_1, [l0_0, l0_2, l0_3, l1_0, l1_1, l1_2], No], [l0_2, [l0_0, l0_1, l0_3, l1_0, l1_1, l1_2], No], [l0_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], No], [l1_0, [l0_0, l0_1, l0_2, l0_3, l1_1, l1_2], No], [l1_1, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_2], No], [l1_2, [l0_0, l0_1, l0_2, l0_3, l1_0, l1_1], No]], Yes, 2, No, No, inclusion_minimal_means_no_single_deletion_while_minimum_compares_all_covers)."},"skills":["Inclusion-Minimal Covers Need Not Have Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}