{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261107-r05-pigeonhole-principle-0043", "pilot_number": "R0039-20368", "number": 20368, "family_id": "qf-mod38-pigeon-generalized-load", "title": { "zh-Hant": "第 20368 題：鴿籠原理", "en": "Question 20368: The Largest Fibre Is at Least the Exact Ceiling Average" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261107-r05-pigeonhole-principle-0043", "batch_id": "batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-pigeonhole-principle" ], "title_zh": "鴿籠原理", "stem_zh": "完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,0,3,1,4,2,0,3,1],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([2, 2, 2, 2, 1], 2, 2, 是, generalized_pigeonhole_load_bound)", "explanation_zh": "Wave6根層核對=([2, 2, 2, 2, 1], 2, 2, 是, generalized_pigeonhole_load_bound)；依完整明示資料獨立重算；domain較大時須有兩個不同元素映到同一值；有限抽樣不足；domain不大於codomain不保證給定map單射；generalized負載界須明示ceiling。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "鴿籠原理" ] }, "variant": { "family_id": "qf-mod38-pigeon-generalized-load", "seed": 202611075043, "parameters": { "mapping": [ 2, 0, 3, 1, 4, 2, 0, 3, 1 ], "codomain_count": 5, "domain": "Complete_finite_total_map_pigeonhole", "operation": "pigeon-generalized-load", "semantic_template": "qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2,0,3,1]}" } }, "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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、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-20261107-r05-pigeonhole-principle-0043", "batch_id": "batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1cf71bbccead9eea283f9f7a4be526c49bf9df309939751de3db2e8334749ef1", "family_id": "qf-mod38-pigeon-generalized-load", "seed": 202611075043, "semantic_template": "qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2,0,3,1]}" }, "fields": { "title": "The Largest Fibre Is at Least the Exact Ceiling Average", "stem": "Use the complete total mapping=[2, 0, 3, 1, 4, 2, 0, 3, 1] from source indices Fin(9)={0,…,8} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).", "answer": { "kind": "expression", "value": "([2, 2, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)", "explanation": "List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)." }, "skills": [ "The Largest Fibre Is at Least the Exact Ceiling Average" ], "verification_warning": "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": "完整有限集合大小與total map。操作=pigeon-generalized-load；資料={\"mapping\":[2,0,3,1,4,2,0,3,1],\"codomain_count\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete total mapping=[2, 0, 3, 1, 4, 2, 0, 3, 1] from source indices Fin(9)={0,…,8} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=pigeon-generalized-load;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[loads, maximum_load, ceiling_lower_bound, bound_holds, certificate]", "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。欄位標籤不是額外數學證據，No保持原題的聲稱範圍。", "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the 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 source-indexed total map=[2, 0, 3, 1, 4, 2, 0, 3, 1];source Fin(9);ACTUAL target Fin(5)", "after": "every source has ONE in-range image;all target labels0..4 retained even if unused;empty source allowed", "relation": "implies", "rule_id": "pigeon_complete_total_map_not_sample_or_bin_relabeling", "conditions": [], "explanation": { "zh-Hant": "原source索引順序與target標籤保留，不以首次遇見次序重新編號桶。完整total map才可算全部fibre與碰撞；空domain也仍映入非空codomain。", "en": "Retain original source order and target labels rather than relabeling bins by first appearance.Complete total-map data support full fibres and collisions,including an empty source into a nonempty codomain." }, "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": "pigeon target0 complete fibre load := sum(1, 1)", "after": "pigeon target0 complete fibre load = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon target0 complete fibre load，不以小數近似替代。", "en": "Recompute pigeon target0 complete fibre load with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "pigeon target0 complete fibre load" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "ACTUAL original target=0;ALL source fibre indices=[1, 6]", "after": "complete load=2;unused targets remain with load0", "relation": "implies", "rule_id": "pigeon_every_target_load_null_bins_retained", "conditions": [], "explanation": { "zh-Hant": "逐全部codomain槽列出fibre，空桶也保0；不只計用過的桶而改變平均分母。", "en": "List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator." }, "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": "S005", "uses": [ "G001", "S004" ], "before": "pigeon target1 complete fibre load := sum(1, 1)", "after": "pigeon target1 complete fibre load = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon target1 complete fibre load，不以小數近似替代。", "en": "Recompute pigeon target1 complete fibre load with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "pigeon target1 complete fibre load" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "ACTUAL original target=1;ALL source fibre indices=[3, 8]", "after": "complete load=2;unused targets remain with load0", "relation": "implies", "rule_id": "pigeon_every_target_load_null_bins_retained", "conditions": [], "explanation": { "zh-Hant": "逐全部codomain槽列出fibre，空桶也保0；不只計用過的桶而改變平均分母。", "en": "List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator." }, "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": "S007", "uses": [ "G001", "S006" ], "before": "pigeon target2 complete fibre load := sum(1, 1)", "after": "pigeon target2 complete fibre load = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon target2 complete fibre load，不以小數近似替代。", "en": "Recompute pigeon target2 complete fibre load with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "pigeon target2 complete fibre load" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "ACTUAL original target=2;ALL source fibre indices=[0, 5]", "after": "complete load=2;unused targets remain with load0", "relation": "implies", "rule_id": "pigeon_every_target_load_null_bins_retained", "conditions": [], "explanation": { "zh-Hant": "逐全部codomain槽列出fibre，空桶也保0；不只計用過的桶而改變平均分母。", "en": "List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "pigeon target3 complete fibre load := sum(1, 1)", "after": "pigeon target3 complete fibre load = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon target3 complete fibre load，不以小數近似替代。", "en": "Recompute pigeon target3 complete fibre load with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "pigeon target3 complete fibre load" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "ACTUAL original target=3;ALL source fibre indices=[2, 7]", "after": "complete load=2;unused targets remain with load0", "relation": "implies", "rule_id": "pigeon_every_target_load_null_bins_retained", "conditions": [], "explanation": { "zh-Hant": "逐全部codomain槽列出fibre，空桶也保0；不只計用過的桶而改變平均分母。", "en": "List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "pigeon target4 complete fibre load := sum(1)", "after": "pigeon target4 complete fibre load = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon target4 complete fibre load，不以小數近似替代。", "en": "Recompute pigeon target4 complete fibre load with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1" ], "output": "1", "label": "pigeon target4 complete fibre load" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "ACTUAL original target=4;ALL source fibre indices=[4]", "after": "complete load=1;unused targets remain with load0", "relation": "implies", "rule_id": "pigeon_every_target_load_null_bins_retained", "conditions": [], "explanation": { "zh-Hant": "逐全部codomain槽列出fibre，空桶也保0；不只計用過的桶而改變平均分母。", "en": "List every codomain fibre,including load0 for unused targets.Do not count only occupied bins and thereby change the average's denominator." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "pigeon complete sum of all target loads := sum(2, 2, 2, 2, 1)", "after": "pigeon complete sum of all target loads = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon complete sum of all target loads，不以小數近似替代。", "en": "Recompute pigeon complete sum of all target loads with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "2", "2", "2", "1" ], "output": "9", "label": "pigeon complete sum of all target loads" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "pigeon exact average load := div(9, 5)", "after": "pigeon exact average load = 9/5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon exact average load，不以小數近似替代。", "en": "Recompute pigeon exact average load with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "9", "5" ], "output": "9/5", "label": "pigeon exact average load" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "pigeon exact ceiling lower bound := sum(1, 1)", "after": "pigeon exact ceiling lower bound = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 pigeon exact ceiling lower bound，不以小數近似替代。", "en": "Recompute pigeon exact ceiling lower bound with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "pigeon exact ceiling lower bound" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "ALL original-order loads=[2, 2, 2, 2, 1];sum=9;nonempty target count=5;exact average=9/5;division m=n*1+4", "after": "ceiling average=2;actual max load=2 attained at target0;maximum>=ceiling;not necessarily equality", "relation": "implies", "rule_id": "pigeon_generalized_ceiling_full_load_argument", "conditions": [], "explanation": { "zh-Hant": "若每桶都小於ceil(m/n)，整數負載都≤ceil−1，總和