{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0001","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bd9b560f5658bad4f8204d4cee8e6fa162c453bbef7aba1c3ba82f9c0d9a742","family_id":"qf-mod37-clique-clique-audit","seed":202611062001,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0002","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe2fbd975e294a2ea04a1e751da2f31c5090684337c1fadb634acf4f3a74455d","family_id":"qf-mod37-clique-clique-audit","seed":202611062002,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0003","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e259edab812defb5b05ffd99cb0feab89774afbc76fb54b26c36a262d2ce5bb8","family_id":"qf-mod37-clique-clique-audit","seed":202611062003,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0004","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"273166648b6d936bf77d287b354b7eaa5c021496f12d401ef5e906bbef2b7fe1","family_id":"qf-mod37-clique-clique-audit","seed":202611062004,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0005","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"485ab628ef07803ca96af4a8964bed8af591317f116ef931a09e05716358b71d","family_id":"qf-mod37-clique-clique-audit","seed":202611062005,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0006","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc65980239f4a84cc4113e97968ea8cb7089375acae812aae75df1236b85b364","family_id":"qf-mod37-clique-clique-audit","seed":202611062006,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0007","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c76b2b9c478ec5d4d2aa85860122531e786fb63abecc0ce9097db9ad8257258","family_id":"qf-mod37-clique-clique-audit","seed":202611062007,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0008","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f04255669ab1cfc0617c5314db01eccfd28ea49bda26a875b2cabad3175407f2","family_id":"qf-mod37-clique-clique-audit","seed":202611062008,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0009","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ef12d72f6bff8b35647809b23bad172a9ab8d1cdb46da7d02a90618e33cb4e6","family_id":"qf-mod37-clique-clique-audit","seed":202611062009,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0010","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08bfba9f4fd38aa622f3b3e2a58f7b2eee88179bbce19f64f2cac96d8623eab5","family_id":"qf-mod37-clique-clique-audit","seed":202611062010,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0011","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9c97f1f509e5a558200c16a1b214a8d94d9e47c380ebdc98a23617e3b504194","family_id":"qf-mod37-clique-clique-audit","seed":202611062011,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0012","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b8646311631305bd41a0c6ab3fa934ccacb05812d2bb327ebd97193bd6a7d0e","family_id":"qf-mod37-clique-clique-audit","seed":202611062012,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0013","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"415cef16848f62550c5029ed8d7305aed34ef24a6d0d0afd0b5ef57af51c9db0","family_id":"qf-mod37-clique-clique-audit","seed":202611062013,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0014","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9193829050e6c106159f7742286fd8ce2a7d498285930daf3ec9dd0a6901d7e1","family_id":"qf-mod37-clique-clique-audit","seed":202611062014,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0015","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5101052c282a81953e37cdf08cf9f836610252675b8434331c0833a88b63fbd","family_id":"qf-mod37-clique-clique-audit","seed":202611062015,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0016","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dfb571756013b0f07eaa2fd3355088e2d5949e1225133c39267abe10f6e4e95","family_id":"qf-mod37-clique-clique-audit","seed":202611062016,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0017","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f522a179dd194170a4e119c0ef134082e68e5ba3c371ca3b04f9450643278fb4","family_id":"qf-mod37-clique-clique-audit","seed":202611062017,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0018","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ec37e04e46ab85f7743f734f96afe6f9ea7953bd3690004abc68194fb4434ea","family_id":"qf-mod37-clique-clique-audit","seed":202611062018,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0019","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83adf62650c756c651081c4ee3d8e6da39432090c96052f37405d7c1f6738e7b","family_id":"qf-mod37-clique-clique-audit","seed":202611062019,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0020","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fbc526bb9e5d4dd320c9bd94cc043d650a2054d626afb883fef90bb65c37c67","family_id":"qf-mod37-clique-clique-audit","seed":202611062020,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0021","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"024ffeb1952e830dd23de43f3bf297e6bb8626444582aaebb1dd0cf9236ac8f5","family_id":"qf-mod37-clique-clique-audit","seed":202611062021,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0022","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99306d6fcd40af5e3cf1752f786859ccbcc527418c4502aa88c2baad61db3368","family_id":"qf-mod37-clique-clique-audit","seed":202611062022,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0023","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2477a9f6b25f0b82a5dc2e2367bf1200c17e9d3240e25ada837f870fef6377a","family_id":"qf-mod37-clique-clique-audit","seed":202611062023,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0024","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c828c81d76b06b272220dfabba7be9b704f16eacb41b1411f1cec4749786c2e","family_id":"qf-mod37-clique-clique-audit","seed":202611062024,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0025","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c20dd4e1abfac5ca0461464e3b864d89ec5d38eb245d5a2c36f9ee147c817e3f","family_id":"qf-mod37-clique-clique-audit","seed":202611062025,"semantic_template":"qf-mod37-clique-clique-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[0,1,2]}"},"fields":{"title":"Only a Valid Clique Size Certifies a Clique-Number Lower Bound","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[0, 1, 2], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_clique, candidate_size, boundary).","answer":{"kind":"expression","value":"([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)","explanation":"Check adjacency of every distinct selected pair. The reported candidate size gives a lower bound on omega only if the validity flag is Yes; it is not by itself a maximum certificate or an exact chromatic number. The exact result, retaining the requested order, is ([0, 1, 2], Yes, 3, candidate_clique_only_lower_bound_on_omega)."},"skills":["Only a Valid Clique Size Certifies a Clique-Number Lower Bound"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0026","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6feec0b73c3aea3fce1e93ee1aee83dc214f6de2178e4cb737d5897ddd628b5c","family_id":"qf-mod37-clique-independent-audit","seed":202611062026,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0027","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9be55d8c4ea55ade37272faf56956dc3be171b87bdc103db9aa23824333a195","family_id":"qf-mod37-clique-independent-audit","seed":202611062027,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0028","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f99464cb6aa3e9983a9b68dd6bc84d3a866dd46a292e511fd2b2df1f015f1c8a","family_id":"qf-mod37-clique-independent-audit","seed":202611062028,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0029","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d7a1cd9a63cebdf5ea3fc35b6c206b68ce383fbc29e30f567f4255a1073433e","family_id":"qf-mod37-clique-independent-audit","seed":202611062029,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0030","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e92c7ea3d903f99b9d613271d925ef96029239ffac7b999d7c501077fecf4613","family_id":"qf-mod37-clique-independent-audit","seed":202611062030,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0031","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897da6d86b1f60c8342a08e3d062dd1e4aeeca045e3c7b5d34ca4275f3030e9a","family_id":"qf-mod37-clique-independent-audit","seed":202611062031,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0032","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4a9456c688362278cebbb14051af29d412c0752f022a497894c3cf93bc0c27c","family_id":"qf-mod37-clique-independent-audit","seed":202611062032,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0033","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9669819a998323bbd2bdcc821fa0b8925161fc0be65167307d60ab29be2177a4","family_id":"qf-mod37-clique-independent-audit","seed":202611062033,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0034","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0df26fa3594a66783b9d42934ea346281dd3a21756dd66b1289d4cdb8ee1d84f","family_id":"qf-mod37-clique-independent-audit","seed":202611062034,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0035","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05565e0163b114351fba9d131b2c3e72cb5b1616d6d6435204003529e95d82a0","family_id":"qf-mod37-clique-independent-audit","seed":202611062035,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0036","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"853a7fae51d3a92d3dcb723b80a79245bee724925545768a765d03dc599114f7","family_id":"qf-mod37-clique-independent-audit","seed":202611062036,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0037","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e2f82b9a335fddb72432dd3e89ec9655beda7e9a7f9931008828bf6f3b99871","family_id":"qf-mod37-clique-independent-audit","seed":202611062037,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0038","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d76605a319294c2321823f683d12607b383c247b2e0c3df7ad5a19928bb24b5","family_id":"qf-mod37-clique-independent-audit","seed":202611062038,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0039","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"857c991d5ab892dc4701afdf39464231e8f7d2bd3ba155325ab8cc331d65ef4e","family_id":"qf-mod37-clique-independent-audit","seed":202611062039,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0040","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d02a409e8d703f20e209ec4b777e9380296e0642ee81ab7b77f31d1f403665c6","family_id":"qf-mod37-clique-independent-audit","seed":202611062040,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0041","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb71ad4b1f2f748c4afc9af4cfb815dfda3c3d5f2f63b11ad2498c5a0207e949","family_id":"qf-mod37-clique-independent-audit","seed":202611062041,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0042","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a020c3a11dac8e4a9452b9c1931bebd22f391007c7550eddef79f442e517d209","family_id":"qf-mod37-clique-independent-audit","seed":202611062042,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0043","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6153eeef328f15eb47422c38466ccb200c3a9590964a7568ceedd473288a24f8","family_id":"qf-mod37-clique-independent-audit","seed":202611062043,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0044","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52984d700f1a16abb4cbb7a7ff5ec3257f9730f906432e711d14da8cfc7f5330","family_id":"qf-mod37-clique-independent-audit","seed":202611062044,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0045","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c89d948954330cf4d67f96749d3765f90234de91143fcbc037ba83f557f94529","family_id":"qf-mod37-clique-independent-audit","seed":202611062045,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0046","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d3c2f98fa54ef1f4fa9c6899727902c24bd17f50db970d965a00482f6145566","family_id":"qf-mod37-clique-independent-audit","seed":202611062046,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0047","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73c98b8b4b38863cb776b31e7e8e83dc3cd158bbd5d32e9f00fe662b3391d54c","family_id":"qf-mod37-clique-independent-audit","seed":202611062047,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0048","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9f57f0c54411a55475149df739315be4dcfd95e399acca63755ab95acddfa3a","family_id":"qf-mod37-clique-independent-audit","seed":202611062048,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0049","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea9e28cb46268d7680175091eec1287049dccfc99da50712ae88e10461a0b2fa","family_id":"qf-mod37-clique-independent-audit","seed":202611062049,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0050","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"971aa9a909885b9a33e79656ac177f43f925a25f003a92179fd601d9fa34d9da","family_id":"qf-mod37-clique-independent-audit","seed":202611062050,"semantic_template":"qf-mod37-clique-independent-audit/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":6,\"vertices\":[3,5]}"},"fields":{"title":"An Independent Set Is a Clique in the Complement","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 5], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Give (vertices, is_independent, clique_in_complement, candidate_size, boundary).","answer":{"kind":"expression","value":"([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)","explanation":"Check nonadjacency of every distinct selected pair and the corresponding complement-clique condition. A valid candidate size bounds alpha below, not above, and maximality/maximum need their own checks. The exact result, retaining the requested order, is ([3, 5], Yes, Yes, 2, candidate_independent_set_only_lower_bound_on_alpha)."},"skills":["An Independent Set Is a Clique in the Complement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0051","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f64518ccd9471229592c0edd4d8142967a6f3ed7f30f0bff147ce8ef4e4bd881","family_id":"qf-mod37-clique-exact-extrema","seed":202611062051,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0052","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dd803d78e772a921fc3d13e5cc07e21e161a09d56dd658bcb597e08d44aa4d4","family_id":"qf-mod37-clique-exact-extrema","seed":202611062052,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0053","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f95ea603402491637d1f53f3bbd72f7dbc64b7ce3c1c62f25f7d178e06e6ec6","family_id":"qf-mod37-clique-exact-extrema","seed":202611062053,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0054","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a28abcfe2ed5657d354a130addd78480372de6b0913b370af19a5f514e1db2c","family_id":"qf-mod37-clique-exact-extrema","seed":202611062054,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0055","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4336edd751e6a6fa364630580a245800aa2047233991acf64b4f3f91e3ae5a4d","family_id":"qf-mod37-clique-exact-extrema","seed":202611062055,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0056","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c947d3a81d4e5147ce24c3d71218e226d32a82f33160c972bb74ad02729d8f23","family_id":"qf-mod37-clique-exact-extrema","seed":202611062056,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0057","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70a49e1f2d7a6d3cfaacce0588267d094edc2364ae18a1ffa44b553fda81a1c9","family_id":"qf-mod37-clique-exact-extrema","seed":202611062057,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0058","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b3c2924f466563b1f911c158d548058dcee2f5b524190b3c5e5348ef0b4966b","family_id":"qf-mod37-clique-exact-extrema","seed":202611062058,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0059","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcabd4767dd9bfe1dbf857e72dab85fb5ff5d628c55c07d3c213da7d4220c371","family_id":"qf-mod37-clique-exact-extrema","seed":202611062059,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0060","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41b4d6c33fa92755f4ff3734d6e3b7f48915df9ee0a4f1ff8ec1d091f6278f43","family_id":"qf-mod37-clique-exact-extrema","seed":202611062060,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0061","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b29dafb8338640c06aab2fd225e2d752d9ab34ec5d77aacaef74649ddff13dc","family_id":"qf-mod37-clique-exact-extrema","seed":202611062061,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0062","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"197274c51d2aa87f3bdea2ae4432e4cb3b17303431b92b1d5141771221ba8b4b","family_id":"qf-mod37-clique-exact-extrema","seed":202611062062,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0063","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"818d45b9627fe1aea81dcc4da02d8e2feea840685954d8fe34aa1ddd5aceab73","family_id":"qf-mod37-clique-exact-extrema","seed":202611062063,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0064","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d455f51862578e72887f5077e90f78d4922cc857a0744736ad735251895fbc59","family_id":"qf-mod37-clique-exact-extrema","seed":202611062064,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [1, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0065","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af34adb16097426d674e3d15a9c3e3e93116ceaeb1f595fd6525a711248e523c","family_id":"qf-mod37-clique-exact-extrema","seed":202611062065,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0066","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc0b8f6b4cb4da1ee0394b00374d231b95235a98a733b4b7d8c1d1bd6ebcf51","family_id":"qf-mod37-clique-exact-extrema","seed":202611062066,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0067","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea3ad9adf338a7491d4a4c2d5afb778ecbf32dae33e01f7f7f8b96b00dd3ad2a","family_id":"qf-mod37-clique-exact-extrema","seed":202611062067,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0068","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"258e05eb020c59232da4e81c6ca2947b90cbd8c66f4810fb82fb5e6c4f01cae5","family_id":"qf-mod37-clique-exact-extrema","seed":202611062068,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0069","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"937ec7193c02cd21ffd0a1d750c9aee352f65eeb4a724b34b126717436068572","family_id":"qf-mod37-clique-exact-extrema","seed":202611062069,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0070","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a882e0bb50425cb02ba08148fb7cf1761e39f950614d4a11afadf8044fe1f02","family_id":"qf-mod37-clique-exact-extrema","seed":202611062070,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0071","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"102a8313613c6d6b5c430d5e7dacb121610bae15e2e781da3b439ae3059f44fe","family_id":"qf-mod37-clique-exact-extrema","seed":202611062071,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0072","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf842f035235b0c602a725fbcc2ee3d3cb4dd696baf0290022dbf21d693f181d","family_id":"qf-mod37-clique-exact-extrema","seed":202611062072,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0073","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41e3ff2017992c9475492f9764f1c4ed7dd0f93ab91b4e80f4f5de0e5448df88","family_id":"qf-mod37-clique-exact-extrema","seed":202611062073,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [0, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0074","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f547bf9afa65f15749579f49f30979612c2432b7d6a0870393b48023b2bcd80c","family_id":"qf-mod37-clique-exact-extrema","seed":202611062074,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0075","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05d47c6abc359d3094d0124025b943861f1ced5929c782b1ca9cff9f2945f241","family_id":"qf-mod37-clique-exact-extrema","seed":202611062075,"semantic_template":"qf-mod37-clique-exact-extrema/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"n\":6}"},"fields":{"title":"Exact Extremal Sizes Need the Complete Vertex-Subset Search","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Give (maximum_clique, clique_number, maximum_independent_set, independence_number, all_subsets_checked, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)","explanation":"Inspect all vertex subsets. Return a maximum clique and independent set, preserving the first maximum witness in increasing vertex-bitmask order, not re-sorting tied witnesses lexicographically. This bounded exhaustive certificate is not a large-graph efficiency claim. The exact result, retaining the requested order, is ([0, 1, 2], 3, [2, 3, 5], 3, Yes, exhaustive_all_vertex_subsets)."},"skills":["Exact Extremal Sizes Need the Complete Vertex-Subset Search"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0076","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dd0c1d0dcbe03b0993693d078fb7ea3672d6813a20fbcd252681df50d67394a","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062076,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0077","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f5834d102503f1191d4683d617738e8a69101e03c8f3b68ee53d7e5260b21b0","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062077,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0078","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0832bcf071906532c7d09262fe08bed77b7c3d77669628fd9b2a3361277878a2","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062078,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0079","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecdbd97116fa0c8cf56ab79a31417aa5d24471d4c7b9db852e4ee5bffb9e5b42","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062079,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0080","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"005d1c6e9d4d5a2ec2107bf81a7df6ff6c7b4c22e881548d09c30ea4fb01a742","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062080,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0081","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d975b8d99d4278f263a1777d5286e00492da8d05b92b68d110a638fb8fc5c699","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062081,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0082","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53b2bc585c9050ceec30344dc643cf5d48f3a40be13d46f4bacdd0241ed1e55c","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062082,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0083","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"827a3ac2cb09343480ade820131f5cef385d572758eba3b0dd37a39e7ff3e098","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062083,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0084","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26958d1cb02e97e78b911503f3fdeed2ec923f6ac42e267de6648fa629a39175","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062084,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0085","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89f23782d87def97c32986b2dd622884b897fb2a96ade813946e247aa87eecb3","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062085,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0086","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"904638fb8e09e18a038a6aba549a0b48b253b9bcc77ad40ae4ed6f6bf4a47202","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062086,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0087","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec90f6f7417b50fd743461b6289766f6c4cecf83c00486f84fb4c0e1fd4425d8","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062087,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0088","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40b810ab478ca144567230b275a45c07786139d04be63ad8cab8ac4e9ee20411","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062088,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0089","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a352f598bf7823aca12909c6a89f46b4bf3b821e645c056bf9c7a9ba5e93c7e","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062089,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0090","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"053444e62834380452c0e7cbd9f50f1168539aa72753470f003880af10d35407","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062090,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0091","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d6339eab19bc851b3a345c1c176d3bb3ae4827f3d64aa80dd3a133a0e4613e1","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062091,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,3],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 3], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0092","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"498afc931ff4f23af2ce70fe7dfe567d26166b1782d18c69495b56927f5f03ef","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062092,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0093","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab751a2a995c15779261c496a13cdec70e0cba7f61f5843d4177dd209bcc6e67","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062093,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0094","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7f192f1ee0decdf5bd815f2b54d4f11814bb3a10eca90f4126750b8a4e93a25","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062094,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0095","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e29dcc90f7aa6163764f3a67903d6cef83f8d88dfe9b7fd6d94e5fa9beb58218","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062095,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0096","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0886db0c976960f71f6c9ccde0db91e2afb05772b62f3c45d97384577f8b2ad2","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062096,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0097","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96d2dd30677017e9df2955720b4cfb9344e9f8116c8644f9eb6cfc1d3fcc4f10","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062097,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0098","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a0df1d12c85d69b5256b83d588dd776bf2413a6d72cf7e52c55d02ec73f4a23","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062098,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[1,2],[1,3],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0099","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc10def29f0b664376ffde2eb4e3e803a297c65af37be57eb7c45b6987f14685","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062099,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,3],[1,2],[1,3],[1,5],[2,4],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [1, 5], [2, 4], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r02-clique-and-independent-set-0100","batch_id":"batch-20261014-advance-20261106-r02-clique-and-independent-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07f144d39ed213cc4c140afcf4c7564a7617a07bc22ab2833acd87a307ca9d8f","family_id":"qf-mod37-clique-maximal-vs-maximum","seed":202611062100,"semantic_template":"qf-mod37-clique-maximal-vs-maximum/{\"edges\":[[0,1],[0,2],[0,5],[1,2],[1,5],[3,4]],\"kind\":\"clique\",\"n\":6,\"vertices\":[3,4]}"},"fields":{"title":"Validity, Inclusion-Maximality and Global Maximum Are Separate","stem":"Use the complete finite loop-free simple undirected graph on Fin(6)={0,…,5} with canonical edges=[[0, 1], [0, 2], [0, 5], [1, 2], [1, 5], [3, 4]]. All graph vertices and edges are supplied, and exact searches are limited to n≤9. Candidate vertices=[3, 4], with distinct vertex indices. Check every pair, not a sample; an invalid candidate's size is not a certified lower bound. Requested kind=clique. Give (kind, vertices, valid, maximal, maximum_size, is_maximum, boundary).","answer":{"kind":"expression","value":"(clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)","explanation":"A valid candidate is maximal if no outside vertex can be added while preserving the property. It is maximum only if valid and its size equals the independently computed global maximum. An invalid candidate cannot become maximum by matching the maximum size. A clique gives only a chromatic lower bound in general. The exact result, retaining the requested order, is (clique, [3, 4], Yes, Yes, 3, No, maximal_not_maximum_and_clique_only_lower_bounds_chromatic_number)."},"skills":["Validity, Inclusion-Maximality and Global Maximum Are Separate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}