{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261108-r07-finite-directed-graph-0010", "pilot_number": "R0040-20885", "number": 20885, "family_id": "qf-mod39-digraph-qualification", "title": { "zh-Hant": "第 20885 題：有限有向圖", "en": "Question 20885: A Directed Edge Is an Ordered Pair under the Declared Simple Convention" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261108-r07-finite-directed-graph-0010", "batch_id": "batch-20261014-advance-20261108-r07-finite-directed-graph-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-directed-graph" ], "title_zh": "有限有向圖", "stem_zh": "完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)", "explanation_zh": "Wave6第二層核對=(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)；依完整明示資料獨立重算；(u,v)與(v,u)不同，平行同向edge不重複且loop在主模型拒絕；out-degree與in-degree各自總和等於edge count；不沿用undirected degree或connectivity。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "有限有向圖" ] }, "variant": { "family_id": "qf-mod39-digraph-qualification", "seed": 202611087010, "parameters": { "n": 5, "edges": [ [ 0, 1 ], [ 0, 2 ], [ 1, 2 ], [ 2, 3 ], [ 3, 0 ], [ 3, 4 ], [ 4, 0 ] ], "domain": "Complete_finite_loopless_simple_digraph", "operation": "digraph-qualification", "semantic_template": "qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261108-r07-finite-directed-graph-0010", "batch_id": "batch-20261014-advance-20261108-r07-finite-directed-graph-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "7feb7656aa635ad86af7da7cf4e6e620d72e7cc815494b2fd2436ab2fe0c2d80", "family_id": "qf-mod39-digraph-qualification", "seed": 202611087010, "semantic_template": "qf-mod39-digraph-qualification/{\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]],\"n\":5}" }, "fields": { "title": "A Directed Edge Is an Ordered Pair under the Declared Simple Convention", "stem": "Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention).", "answer": { "kind": "expression", "value": "(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)", "explanation": "Validate every endpoint, canonical edge order, uniqueness and absence of loops. Opposite oriented edges may both be present and remain distinct; no undirected connectivity or degree convention is inferred. The exact result, retaining the requested order, is (5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], Yes, Yes, ordered_pairs_and_no_parallel_duplicate_edges)." }, "skills": [ "A Directed Edge Is an Ordered Pair under the Declared Simple Convention" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "完整有限ordered-pair edge set與loop-free simple digraph convention。操作=digraph-qualification；資料={\"n\":5,\"edges\":[[0,1],[0,2],[1,2],[2,3],[3,0],[3,4],[4,0]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete directed vertex carrier Fin(5)={0,…,4} and canonical ordered-pair edges=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]. (u,v) and (v,u) are different edges; duplicate same-direction edges are forbidden. The main model is loop-free simple with n≥1; only the explicit loop-boundary audit temporarily accepts loops to demonstrate rejection of that main convention. Give (vertex_count, edges, loop_free, simple_directed_set_qualified, convention)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=digraph-qualification;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[vertex_count, edges, loop_free, simple_directed_set_qualified, convention]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。", "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "ORIGINAL complete directed carrier Fin(5);ORIGINAL lexicographically canonical ORDERED arcs=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]]", "after": "all endpoints in range and same-direction arcs unique;actual loops=[];main loop-free directed-set convention qualified", "relation": "implies", "rule_id": "digraph_ordered_pairs_not_undirected_endpoint_sorting", "conditions": [], "explanation": { "zh-Hant": "只核整張arc列表的from/to排序，不把各arc端點排序。(u,v)與(v,u)不同且可同時存在；main禁止loop，boundary暫收loop用來比對慣例，不作一般digraph不存在的判決。", "en": "Check outer from/to order without sorting endpoints inside each arc.(u,v) and(v,u)are distinct and may coexist.The main convention forbids loops;the boundary temporarily accepts them to audit that convention,not to deny other digraph models." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "complete ordered-arc data passed this main loop-free simple-directed convention", "after": "qualification does NOT assert acyclicity,undirectedness,connectivity or any degree convention beyond this signature", "relation": "implies", "rule_id": "digraph_signature_qualification_not_unrequested_graph_properties", "conditions": [], "explanation": { "zh-Hant": "即使有兩方向arc或有向cycle，仍可是合格loop-free simple digraph。這個True不升格額外graph性質。", "en": "Opposite arcs or a directed cycle may occur in a qualified loop-free simple digraph.This true flag does not promote additional graph properties." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 5, [ [ 0, 1 ], [ 0, 2 ], [ 1, 2 ], [ 2, 3 ], [ 3, 0 ], [ 3, 4 ], [ 4, 0 ] ], true, true, "ordered_pairs_and_no_parallel_duplicate_edges" ] }, "conclusion": { "canonical_value": "(5, [[0, 1], [0, 2], [1, 2], [2, 3], [3, 0], [3, 4], [4, 0]], 是, 是, ordered_pairs_and_no_parallel_duplicate_edges)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261108-r07-finite-directed-graph-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations/questions.jsonl", "batch_sha256": "b76dd0131fe7faeb6a32b6d736c48955db141a0c42218e36d7e584617fde48c5", "original_line": 10, "original_line_sha256": "849b3a20d4147082f93ca60ab5cd08c9b63b62ffa359345317556c0aa96cee15", "question_compact_sha256": "7feb7656aa635ad86af7da7cf4e6e620d72e7cc815494b2fd2436ab2fe0c2d80", "english_row_compact_sha256": "17fcc35052dda12e54363cf9ed8031f1658ca09513336e7847594a9396bc44ac", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r07-finite-directed-graph-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-directed-graph": { "sha256": "09311b0ec711e3f02a16b1c332e69c9f718fe2a288f2005683d393695a41161b", "english_sha256": "1a9289ec7bd86c2ec776a919d938ee1c37455b7d8c5ec5c81e2293906e548c98", "original_path": "public/data/mko/finite-directed-graph.json", "english_path": "public/data/i18n/editions/mko-finite-directed-graph.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "f1ed3239c9185524aa4f0839ab4cfbe8f7d74200b283ea09e850a792f0178a3b", "original_candidate_sha256": "63555ce8ea83abab0888855613344e423f0c2b5a52798a282b4cb13af2f19989", "original_candidate_line_sha256": "63555ce8ea83abab0888855613344e423f0c2b5a52798a282b4cb13af2f19989" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-008-p41", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d", "question_first_witness_sha256": "e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "6f484b00db9a69c59afbcc9b0c75193178bbbaa557870982cb07221a2107bc29", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }