{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0001","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4ce50d1705350d9800345b15f0a634b023cf247d3963e9d0ac813a43ae3e8f5","family_id":"qf-mod28-component-of-vertex","seed":202610287001,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"n\":7,\"vertex\":0}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0002","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ff8b44c407f3cdb57ce0a10a9cec0e1026868b589f4bd96f8ce5451dadcae31","family_id":"qf-mod28-component-of-vertex","seed":202610287002,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"n\":7,\"vertex\":1}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0003","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb4c75982f39ce644480d6caa49355ced26289f785fec680518ce4021e3fac0f","family_id":"qf-mod28-component-of-vertex","seed":202610287003,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"n\":7,\"vertex\":2}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0004","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4b15af91e06ff29f0f3de41c4a38de33ee849d38d0c713a59e7b8411e8ffe81","family_id":"qf-mod28-component-of-vertex","seed":202610287004,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"n\":7,\"vertex\":3}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0005","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8c5cfc3019bb021782f3a71ed599fe63ac25317a72a10d2b5b1f820d56e659a","family_id":"qf-mod28-component-of-vertex","seed":202610287005,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"n\":7,\"vertex\":4}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=4. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0006","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0b54bbfce559066c59185964b93ffc045fc66b64a6f7a24697c62205e2d3156","family_id":"qf-mod28-component-of-vertex","seed":202610287006,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"n\":7,\"vertex\":5}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=5. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 5])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 5])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0007","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3127c374873bd24e41b677b3d6e4860b9b34ea624424bea91252e9135ab4059","family_id":"qf-mod28-component-of-vertex","seed":202610287007,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"n\":7,\"vertex\":6}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=6. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0008","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8592cbacf3c275a909683b4b1241512818415bc9c03c0c86604001d5669ae031","family_id":"qf-mod28-component-of-vertex","seed":202610287008,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"n\":7,\"vertex\":0}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0009","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f353fda925eb66abed5645071ea74bca9ec99840e777f977f30e8cbdcc7b3c1b","family_id":"qf-mod28-component-of-vertex","seed":202610287009,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"n\":7,\"vertex\":1}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0010","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36c43b0637ab38aca156e5dc67a5da7a2cbdb26025efc484bab5a1dbd40d49fc","family_id":"qf-mod28-component-of-vertex","seed":202610287010,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"n\":7,\"vertex\":2}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0011","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51c55a394df8ba78558c20cb0018c0c1d85321bfd03cdc2528106b99556d7c7e","family_id":"qf-mod28-component-of-vertex","seed":202610287011,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"n\":7,\"vertex\":3}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0012","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee17d9436b990b7a86d055a69be8cfe7548f3f57fcf956a305952ead0bbdbc26","family_id":"qf-mod28-component-of-vertex","seed":202610287012,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"n\":7,\"vertex\":4}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=4. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0013","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d194732b678f853d46310e6865543a1111240bda728b63610b7d67b5d01309","family_id":"qf-mod28-component-of-vertex","seed":202610287013,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"n\":7,\"vertex\":5}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=5. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0014","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8763af4cc92f0b08df7e07a18b624cedce099a4a45688f05e3d22d927cb1eb6","family_id":"qf-mod28-component-of-vertex","seed":202610287014,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"n\":7,\"vertex\":6}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=6. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(6, [6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, [6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0015","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53a1b5e0a387c5e260c10856ae78d5fdb929ac2fcb3e7841bffe37713ecac8b1","family_id":"qf-mod28-component-of-vertex","seed":202610287015,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"n\":7,\"vertex\":0}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(0, [0, 1])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0016","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15e1752f0d0927486963a82a58af1c12de7205799a905509650f2e17ece11c0c","family_id":"qf-mod28-component-of-vertex","seed":202610287016,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"n\":7,\"vertex\":1}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0017","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a7d78120860f457dfd16c6f47e38d543e296bb7969f9a73ebc11e3b6b04fbcc","family_id":"qf-mod28-component-of-vertex","seed":202610287017,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"n\":7,\"vertex\":2}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0018","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"989d531bfb1c83858295bcfc2c31f5e01a1eb501f69de4bbfbee558bf9ec047f","family_id":"qf-mod28-component-of-vertex","seed":202610287018,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"n\":7,\"vertex\":3}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0019","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23a253849202fadd077c311c5cc55d0f2f7198cbec2f1202480f4d2d8c64df18","family_id":"qf-mod28-component-of-vertex","seed":202610287019,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"n\":7,\"vertex\":4}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=4. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0020","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4461bb935b65d758b918fcd49d7b5f32dbdcba628ffbc301491e31a11a41d17","family_id":"qf-mod28-component-of-vertex","seed":202610287020,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"n\":7,\"vertex\":5}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=5. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0021","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87ae4a94092a6f9522cbe6cd2fcad5ad69bacae8da4bedc46685b4057538fce4","family_id":"qf-mod28-component-of-vertex","seed":202610287021,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"n\":7,\"vertex\":6}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=6. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0022","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c28182b02f3f81c2e45deb0261a6748184d4f229387cc2981bbd3fed405f47ae","family_id":"qf-mod28-component-of-vertex","seed":202610287022,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"n\":7,\"vertex\":0}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=0. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0023","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bde6372618479b8e851418df52e5cda96c1f01032292a84bd77b4eb162b9329","family_id":"qf-mod28-component-of-vertex","seed":202610287023,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"n\":7,\"vertex\":1}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=1. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(1, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0024","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ebef9a6db61c1315f0a5dfefd68ee68279409bd90152394c68c7a095074139d","family_id":"qf-mod28-component-of-vertex","seed":202610287024,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"n\":7,\"vertex\":2}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=2. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(2, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0025","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1e7822f2fc3931942c58c3519fa33ce268d78911fe83bc1069cd4c8c23ecc42","family_id":"qf-mod28-component-of-vertex","seed":202610287025,"semantic_template":"qf-mod28-component-of-vertex/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"n\":7,\"vertex\":3}"},"fields":{"title":"Return the Entire Component, Not a Traversal Order","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Selected vertex=3. Give (vertex,component), with its vertices in increasing label order.","answer":{"kind":"expression","value":"(3, [0, 1, 2, 3, 4, 5, 6])","explanation":"Search the full reachable set from the selected vertex and sort its labels. Do not omit an isolated starting vertex. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, [0, 1, 2, 3, 4, 5, 6])."},"skills":["Return the Entire Component, Not a Traversal Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0026","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c37c37de5f263da92c38ebd99c1a909ec46509544d9d1300c81f73694055033","family_id":"qf-mod28-component-all","seed":202610287026,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0027","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4df1b39de00042763612e02e3a0e9c22eab270dd991c86321a1784b84a4594aa","family_id":"qf-mod28-component-all","seed":202610287027,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0028","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b26e2bb579f1729f6d44836be581ba0d5294382ff241ab3e6c73bf556b72eae8","family_id":"qf-mod28-component-all","seed":202610287028,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 6], [5]], 2)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], 2)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0029","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6757cda424edc780accaa8bafcba492e11d2160436543c9f02717b86e5e7f75b","family_id":"qf-mod28-component-all","seed":202610287029,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0030","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"561f2fef2315bb0c02edd22345343c1c438d157fd5218d5419dedbe41c296c86","family_id":"qf-mod28-component-all","seed":202610287030,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0031","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0b4c9017127c06075e5141e895f4dff469a4e6fcfa6f0f6085c36c170998d0e","family_id":"qf-mod28-component-all","seed":202610287031,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6]], 2)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], 2)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0032","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe0049ca38c3cc104d643708320ef796d23818f3156fbdb44d151da8431883be","family_id":"qf-mod28-component-all","seed":202610287032,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 6], [5]], 2)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], 2)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0033","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e53a886466dd63b79238fce8170e7ebdcb4f96900cdcb19b857d02ad0eb31f6","family_id":"qf-mod28-component-all","seed":202610287033,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0034","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88870346590dfd8858bf55c20fd4c572c0753b66ce6e2e5626d66d15e4e583ec","family_id":"qf-mod28-component-all","seed":202610287034,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0035","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"253d819e3d6746cd90d39dadcbbc2915855230e5157c433f982cac9864c922aa","family_id":"qf-mod28-component-all","seed":202610287035,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0036","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0cb1aad610b1110f566d150e46e3db7e5ebe2b2423229a44017569b1e004f2b","family_id":"qf-mod28-component-all","seed":202610287036,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0037","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b60577b32d486b5e86e09918460116c09d1e5286f36411fd8c7efa0b18b8dfd","family_id":"qf-mod28-component-all","seed":202610287037,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0038","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1346fa943d845430895e830076f60d89267c93aaf3637729f5535088d4b279b","family_id":"qf-mod28-component-all","seed":202610287038,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0039","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"494269fbeb6dd36ac567ff81b8c8fc61c03164e1d76a52870604a1944878f8ec","family_id":"qf-mod28-component-all","seed":202610287039,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6]], 2)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], 2)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0040","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03495d8c601424a9999a00144d5f6050e687477577e4e4ed8ae7c9b5c989f0ea","family_id":"qf-mod28-component-all","seed":202610287040,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5, 6]], 2)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5, 6]], 2)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0041","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6444691e05852e3c75c8fb28a72679f6ea742dcf6a375d9b83542750be4fdb9","family_id":"qf-mod28-component-all","seed":202610287041,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0042","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f9894d39b6ec79573e1c4591240e616164b850a2bb54d8a17b22786c60f0855","family_id":"qf-mod28-component-all","seed":202610287042,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0043","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78b6fd560c54a697df72267192d4aec0b0be59532e9e8cb14ef93a14bf0d3604","family_id":"qf-mod28-component-all","seed":202610287043,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0044","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd84fe84d9c2d07ccb4cf758d8469f93a07dc8ad3ff3c0a5bc904a1adf7101ea","family_id":"qf-mod28-component-all","seed":202610287044,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0045","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdaec5750864bd189708f7fe19dbaa3494e7279bb0102e09c36e7c2c9fee2417","family_id":"qf-mod28-component-all","seed":202610287045,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0046","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3801d9a824976d765b87d05706c28632a2564c30e588e13844b28a7fef7eb182","family_id":"qf-mod28-component-all","seed":202610287046,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0047","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c305d96b1f3b917d21c26989f8f887d27be6ed3e330331d73793b53ee80f9f0e","family_id":"qf-mod28-component-all","seed":202610287047,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0048","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8b4b62bcc7d389442331470b532603ff9923b93f2ecec628abd419cbf65203a","family_id":"qf-mod28-component-all","seed":202610287048,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0049","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c9c0da4223fcba6c9402d684caba89adac50b8050209146acd76e169839fff9","family_id":"qf-mod28-component-all","seed":202610287049,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0050","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d57a04363b64d6c84d193d1cf588e873c81e1d7b6da00616f66aa99e0d3c243","family_id":"qf-mod28-component-all","seed":202610287050,"semantic_template":"qf-mod28-component-all/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"n\":7}"},"fields":{"title":"Partition the Full Vertex Carrier into Reachability Classes","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,count). List each class increasingly and order classes by their least unseen vertex.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], 1)","explanation":"Start a complete reachability search at each least unvisited label. Include singleton classes of isolated vertices. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], 1)."},"skills":["Partition the Full Vertex Carrier into Reachability Classes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0051","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96543a1cce78bc7774267ecab16797fcb6fa9971de3514fc3e83f8779eb78fe0","family_id":"qf-mod28-component-same-audit","seed":202610287051,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"left\":0,\"n\":7,\"right\":1}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0052","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ac55f6727fc543fe630e626f0c06dee120c3ef745df289c663f6e6a4ec1c495","family_id":"qf-mod28-component-same-audit","seed":202610287052,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"left\":1,\"n\":7,\"right\":4}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0053","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"813b1d87bf750ee3b71d71011f12cd5d3ff61b9a8da426e93b48bf8d5c1bfd85","family_id":"qf-mod28-component-same-audit","seed":202610287053,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"left\":2,\"n\":7,\"right\":0}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0054","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ced6a3ea8c9cebe0922478f2fc064576de381bc8435f4376a6f8317f6b2906ad","family_id":"qf-mod28-component-same-audit","seed":202610287054,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"left\":3,\"n\":7,\"right\":3}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0055","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcd5c2a2ea7fb22596655a00bedb834d3afeb8061ec90a4e68a20ea0923c465c","family_id":"qf-mod28-component-same-audit","seed":202610287055,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"left\":4,\"n\":7,\"right\":6}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=4, right=6. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0056","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca59914e90c2bded9e408daaba54a7561246ff90f037a2694addcf30220f53e0","family_id":"qf-mod28-component-same-audit","seed":202610287056,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"left\":5,\"n\":7,\"right\":2}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=5, right=2. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(5, 2, [0, 1, 2, 3, 4, 5], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, 2, [0, 1, 2, 3, 4, 5], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0057","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6097188a0598404d68703ade8434422bfeb28cd14afe7f0b114e2f81fb2c6f0d","family_id":"qf-mod28-component-same-audit","seed":202610287057,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"left\":6,\"n\":7,\"right\":5}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(6, 5, [0, 1, 2, 3, 4, 6], No)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 5, [0, 1, 2, 3, 4, 6], No)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0058","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2349ed2968643183a84dd4d845e36c00832884cfc53b57ddd35211b919a3ec8","family_id":"qf-mod28-component-same-audit","seed":202610287058,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"left\":0,\"n\":7,\"right\":1}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0059","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2e54997f2addad88d8f5131861aa44645c6d2eb6e8e1af24f72d082c0991ef0","family_id":"qf-mod28-component-same-audit","seed":202610287059,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"left\":1,\"n\":7,\"right\":4}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0060","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2944a123cdef857fa0ddd028ab5edec4a8baa11c0aed8e4c134f45c76951d310","family_id":"qf-mod28-component-same-audit","seed":202610287060,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"left\":2,\"n\":7,\"right\":0}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0061","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"695579d1c97121dc277338618b4f09fbc5a19e238824d359be44144c2ecc5fc6","family_id":"qf-mod28-component-same-audit","seed":202610287061,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"left\":3,\"n\":7,\"right\":3}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0062","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeec877c16a3184ba4811610650132e4d8033286f26e314048fc0f77ee4e7038","family_id":"qf-mod28-component-same-audit","seed":202610287062,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"left\":4,\"n\":7,\"right\":6}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=4, right=6. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0063","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e115e59cde2d6eebaa17c2b535b59d4267fed408917702d27f401c49ea83a26","family_id":"qf-mod28-component-same-audit","seed":202610287063,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"left\":5,\"n\":7,\"right\":2}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=5, right=2. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(5, 2, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, 2, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0064","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"150a38ca953fa3c458c11199ed102e3c745c309e9e65307076ffb4d469338def","family_id":"qf-mod28-component-same-audit","seed":202610287064,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"left\":6,\"n\":7,\"right\":5}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(6, 5, [6], No)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 5, [6], No)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0065","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11aa48298bebc5f7eea23d04b33006fdd863ebeb4adafc5911abc3bb21138689","family_id":"qf-mod28-component-same-audit","seed":202610287065,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"left\":0,\"n\":7,\"right\":1}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(0, 1, [0, 1], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0066","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a841956c5280e794f645dc0d4519266d3c5be7cd76af1ff7c1c0c362af888ef","family_id":"qf-mod28-component-same-audit","seed":202610287066,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"left\":1,\"n\":7,\"right\":4}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0067","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03a2d421501247a0c9fd18bae7633db07178e95135afa5471aba8e523c97c043","family_id":"qf-mod28-component-same-audit","seed":202610287067,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"left\":2,\"n\":7,\"right\":0}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0068","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1836d47fdd3002b58d59f4d7accc9871203e22949580ada910686c9620a95e74","family_id":"qf-mod28-component-same-audit","seed":202610287068,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"left\":3,\"n\":7,\"right\":3}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0069","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"218416cae46de99e1658eadb9ef3fa0cb69b26703893400a2729204865c98981","family_id":"qf-mod28-component-same-audit","seed":202610287069,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"left\":4,\"n\":7,\"right\":6}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=4, right=6. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (4, 6, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0070","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e998c2fae48b0b6085948d1b5931e622e71d9a58d12c8e7e985188dde3d4848","family_id":"qf-mod28-component-same-audit","seed":202610287070,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"left\":5,\"n\":7,\"right\":2}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=5, right=2. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(5, 2, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (5, 2, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0071","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63ed5a659840f9054ba2f94fd201c872eb256d520fb00158fa56180c73915331","family_id":"qf-mod28-component-same-audit","seed":202610287071,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"left\":6,\"n\":7,\"right\":5}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=6, right=5. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(6, 5, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (6, 5, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0072","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf6082c13c872a5278ed41fffcca6c3ba047c8e1a35e5c7cd80b5721b7daf307","family_id":"qf-mod28-component-same-audit","seed":202610287072,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"left\":0,\"n\":7,\"right\":1}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=0, right=1. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (0, 1, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0073","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bdc178d99c88350514d216eaca2c9d86de239c43f9caed617a4dd40bac14801","family_id":"qf-mod28-component-same-audit","seed":202610287073,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"left\":1,\"n\":7,\"right\":4}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=1, right=4. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (1, 4, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0074","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30c309955da9deef82d274663cd29d7dc66a26c5e7466a9a83e612d0f0764152","family_id":"qf-mod28-component-same-audit","seed":202610287074,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"left\":2,\"n\":7,\"right\":0}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=2, right=0. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (2, 0, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0075","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e4f90a068a37eb677d9600a6d17915b195d2316d25ea35f201a092968434a3b","family_id":"qf-mod28-component-same-audit","seed":202610287075,"semantic_template":"qf-mod28-component-same-audit/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"left\":3,\"n\":7,\"right\":3}"},"fields":{"title":"Reflexive Connectivity Includes a Length-Zero Path","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. left=3, right=3. Give (left,right,left_component,same_component?).","answer":{"kind":"expression","value":"(3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)","explanation":"Compute the whole component of left and test membership of right. Equal endpoints are connected even in an edgeless graph. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is (3, 3, [0, 1, 2, 3, 4, 5, 6], Yes)."},"skills":["Reflexive Connectivity Includes a Length-Zero Path"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261028-r07-connected-component-0076","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4aff5035db807caaf390f5aa6d1b66a8660a45a839ef1ef66bc82ca6d6fef0b8","family_id":"qf-mod28-component-partition","seed":202610287076,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,4]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0077","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffd16f167f7e12125c4c00200ef79951e2ffc6ad116b02e937c428c222809eca","family_id":"qf-mod28-component-partition","seed":202610287077,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[2,4],[2,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [2, 4], [2, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0078","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1229e61d73c1de0410e6d491fb71ee9d1491ab618d5365db410f8316f5da1de","family_id":"qf-mod28-component-partition","seed":202610287078,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,4],[2,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 4], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0079","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61d877d8e7f498062f8733a563d9981b5d268cfa01470ad171f7e6bfd1fb7008","family_id":"qf-mod28-component-partition","seed":202610287079,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[2,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [2, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0080","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"893b1b51f56b08f0ffb6d43153c53c0835725df1d4ef4780dee7c5e3741ab559","family_id":"qf-mod28-component-partition","seed":202610287080,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,4],[0,6],[1,3],[1,5],[2,3],[2,4],[3,4]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 4], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0081","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8bb787b608f4e61f46e4bf0bf4469e430a84085c0679ea32e677d19d6ec229e","family_id":"qf-mod28-component-partition","seed":202610287081,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[2,4],[2,5],[3,4]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [2, 4], [2, 5], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0082","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c8d4b36b20b1cd491627af663afc864e1ba757291cde27cc8d6cfbfb7a4e4df","family_id":"qf-mod28-component-partition","seed":202610287082,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,4],[1,3],[2,3],[2,4],[2,6],[3,4]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [1, 3], [2, 3], [2, 4], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 6], [5]], Yes, Yes, [[5]], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0083","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e57f2efe7df8863422677d45fb4450558d07a49e53c4bd960256845e2f7443a5","family_id":"qf-mod28-component-partition","seed":202610287083,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 4], [2, 5], [2, 6], [3, 4]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0084","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25f3dd64cdf7c6614346d6ffa9d8d0f64290a3471fccb205e4d85deb69683e5b","family_id":"qf-mod28-component-partition","seed":202610287084,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,6],[1,2],[1,5],[1,6],[2,4],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 6], [1, 2], [1, 5], [1, 6], [2, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0085","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"878f19e035081ba1b4018c3d73592d1028771e1a35aefe11cf3a5167c780466c","family_id":"qf-mod28-component-partition","seed":202610287085,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[2,4],[2,5],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [2, 4], [2, 5], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0086","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6511d89424cbf4e285d35483c28dddd66969cedfc27f9e2aaca944c6b3394c10","family_id":"qf-mod28-component-partition","seed":202610287086,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,3],[1,2],[1,6],[2,4],[2,6],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [1, 2], [1, 6], [2, 4], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0087","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63607f8a13c95963ed13a81090c4b14e5cfbe7bbbfddf8cff87b284dbd841818","family_id":"qf-mod28-component-partition","seed":202610287087,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,3],[0,5],[0,6],[1,4],[1,5],[2,4],[2,5],[2,6],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 5], [0, 6], [1, 4], [1, 5], [2, 4], [2, 5], [2, 6], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0088","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"317b04743988bb5ca07d5e0ae820dadf021c09aca8d10a2f8f5fb3c707ab120d","family_id":"qf-mod28-component-partition","seed":202610287088,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,6],[1,5],[2,4],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 6], [1, 5], [2, 4], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0089","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"366ce1dd9de3d07b71e68c3d6b7647a2643ae410320e2cfc291252137698a9bd","family_id":"qf-mod28-component-partition","seed":202610287089,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,5],[1,4],[2,4],[2,5],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 5], [1, 4], [2, 4], [2, 5], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5], [6]], Yes, Yes, [[6]], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0090","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"087fceea2d82e8d235a5bd6006547b4ad77fa9b9fd2346b06f109d6e11bd13cf","family_id":"qf-mod28-component-partition","seed":202610287090,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[2,4],[2,6],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [2, 4], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0091","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba32eeb51031fce1d77ce07920084ed9701937f1030e832a18a8dff8e3a56a21","family_id":"qf-mod28-component-partition","seed":202610287091,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,5],[0,6],[1,2],[1,3],[1,5],[1,6],[2,3],[2,5],[2,6],[3,4],[3,5]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 5], [0, 6], [1, 2], [1, 3], [1, 5], [1, 6], [2, 3], [2, 5], [2, 6], [3, 4], [3, 5]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0092","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd5e79cdf584aa47967ba29bdced1836be39e55a478845814a7fef42fb6eb44d","family_id":"qf-mod28-component-partition","seed":202610287092,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,6],[2,3],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 6], [2, 3], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0093","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"593d346758b2e9ce3293a473ca04924cbf017e9d9f766d8b48f134f891f8ecf1","family_id":"qf-mod28-component-partition","seed":202610287093,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,3],[0,4],[1,2],[1,3],[1,6],[2,3],[2,5],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 6], [2, 3], [2, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0094","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce1e8220067f5f636e566a2cb99dc1263366be496f24f247c206ed8453c3fee7","family_id":"qf-mod28-component-partition","seed":202610287094,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,3],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[2,3],[2,6],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 3], [0, 4], [0, 5], [0, 6], [1, 3], [1, 4], [1, 5], [2, 3], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0095","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0941485a818841a5a7c34788adc3c45bdc40107d791fcfcff0e67875f59e8ba7","family_id":"qf-mod28-component-partition","seed":202610287095,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,3],[0,4],[0,6],[1,3],[1,5],[2,3],[2,5],[2,6],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 3], [0, 4], [0, 6], [1, 3], [1, 5], [2, 3], [2, 5], [2, 6], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0096","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47b84f748065b1bec4e20328d8e38ed76e0250a3073a850213da35924ef59595","family_id":"qf-mod28-component-partition","seed":202610287096,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,4],[0,5],[1,3],[1,4],[2,3],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 4], [0, 5], [1, 3], [1, 4], [2, 3], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0097","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52cc57f30d47fd220228c94a75ff02e709081e4051f74a0e986a2990af9c8f9a","family_id":"qf-mod28-component-partition","seed":202610287097,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,2],[0,4],[1,3],[2,3],[2,5],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 2], [0, 4], [1, 3], [2, 3], [2, 5], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0098","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"622a474ccaa93812379575490d907298bf0a64b3a9526a85ecfd0d65121520b8","family_id":"qf-mod28-component-partition","seed":202610287098,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,4],[0,5],[0,6],[1,2],[1,4],[1,5],[1,6],[2,6],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 4], [0, 5], [0, 6], [1, 2], [1, 4], [1, 5], [1, 6], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0099","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3fe0eef8bcb94e09924354976dae614d77b74550d12f0e2c42d1018552ed319","family_id":"qf-mod28-component-partition","seed":202610287099,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,4],[0,6],[1,2],[1,5],[1,6],[2,5],[2,6],[3,4],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 4], [0, 6], [1, 2], [1, 5], [1, 6], [2, 5], [2, 6], [3, 4], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261028-r07-connected-component-0100","batch_id":"batch-20261014-advance-20261028-r07-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"263985daf334cad70c79ba6669d36231d82843476e708d5be1709b1e2da9c445","family_id":"qf-mod28-component-partition","seed":202610287100,"semantic_template":"qf-mod28-component-partition/{\"edges\":[[0,1],[0,2],[0,3],[0,5],[1,2],[1,4],[1,6],[3,5],[3,6]],\"n\":7}"},"fields":{"title":"Coverage and Disjointness Must Include Isolated Vertices","stem":"Use the complete finite undirected loop-free simple graph on Fin(7)={0,…,6} with edge set=[[0, 1], [0, 2], [0, 3], [0, 5], [1, 2], [1, 4], [1, 6], [3, 5], [3, 6]]. Loops and duplicate undirected edges are forbidden. Connectivity is complete path reachability, not just one traversal fragment. Length-zero paths make every vertex reachable from itself, including isolated vertices. Do not switch to directed weak/strong, infinite or dynamic connectivity. Give (components,covers_vertices?,disjoint?,singleton_components,length_zero_paths_include_every_vertex). Use increasing labels within each class and least-unseen order across classes.","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)","explanation":"Check exact coverage of the full carrier and absence of overlaps, then retain all singleton components. A visitation list or edge-only carrier is not this partition. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], length_zero_paths_include_every_vertex)."},"skills":["Coverage and Disjointness Must Include Isolated Vertices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}