{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r01-boundary-value-problem-0027", "pilot_number": "R0022-11702", "number": 11702, "family_id": "qf-mod25-bvp-dirichlet-edge-identity", "title": { "zh-Hant": "第 11702 題：邊界值問題", "en": "Question 11702: A Whole-Edge Dirichlet Coefficient Identity" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r01-boundary-value-problem-0027", "batch_id": "batch-20261013-advance-20261025-r01-boundary-value-problem-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-boundary-value-problem" ], "title_zh": "邊界值問題", "stem_zh": "內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"bounds\":[-3,3,-2,4],\"edge\":\"right\",\"boundary_coefficients\":[4,-2,-1]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。", "answer": { "kind": "expression", "value": "([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], 是, declared_edge_only)", "explanation_zh": "收束波核對=([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], 是, declared_edge_only)；依完整明示資料重算；精確有限切片不自動升格為一般無限結論；符號恆等只證宣告的候選BVP；不宣稱存在唯一穩定正則或適定性。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "邊界值問題" ] }, "variant": { "family_id": "qf-mod25-bvp-dirichlet-edge-identity", "seed": 202610251027, "parameters": { "coefficients": [ -11, -1, -2, 2, 0, -1 ], "bounds": [ -3, 3, -2, 4 ], "edge": "right", "boundary_coefficients": [ 4, -2, -1 ], "domain": "Polynomial_rectangle_boundary_value_certificates", "operation": "bvp-dirichlet-edge-identity", "semantic_template": "qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[4,-2,-1],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"edge\":\"right\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、BVP適定性、全域最佳解、Markov長期性、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-13", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261013-advance-20261025-r01-boundary-value-problem-0027", "batch_id": "batch-20261013-advance-20261025-r01-boundary-value-problem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "f9a0004e7d594001aaf87e5988f50f46cd227b256172593edfe2215b2065bf5a", "family_id": "qf-mod25-bvp-dirichlet-edge-identity", "seed": 202610251027, "semantic_template": "qf-mod25-bvp-dirichlet-edge-identity/{\"boundary_coefficients\":[4,-2,-1],\"bounds\":[-3,3,-2,4],\"coefficients\":[-11,-1,-2,2,0,-1],\"edge\":\"right\"}" }, "fields": { "title": "A Whole-Edge Dirichlet Coefficient Identity", "stem": "Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-3, 3, -2, 4]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[4, -2, -1]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem.", "answer": { "kind": "expression", "value": "([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], Yes, declared_edge_only)", "explanation": "Substitute the fixed edge coordinate and compare all three coefficients in the remaining variable. This proves only the declared edge identity, not all four boundaries or the interior equation. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], Yes, declared_edge_only)." }, "skills": [ "A Whole-Edge Dirichlet Coefficient Identity" ], "verification_warning": "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": "內部PDE責任與矩形邊界資料分開。操作=bvp-dirichlet-edge-identity；資料={\"coefficients\":[-11,-1,-2,2,0,-1],\"bounds\":[-3,3,-2,4],\"edge\":\"right\",\"boundary_coefficients\":[4,-2,-1]}。請依 [bounds,edge,edge_variable,restriction_coefficients,boundary_coefficients,edge_identity,scope] 欄序回答。", "en": "Use the quadratic candidate coefficients=[-11, -1, -2, 2, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-3, 3, -2, 4]. Left/right fix x, leaving y; bottom/top fix y, leaving x. Boundary polynomials use ascending coefficients [constant,linear,quadratic]. Compare every coefficient for an edge identity, not a few sampled boundary values. Selected edge=right; supplied coefficients=[4, -2, -1]. Give (bounds,edge,remaining variable,restriction coefficients,supplied coefficients,coefficientwise identity?,declared_edge_only). Interior PDE and boundary data are separate obligations. Symbolic success covers only this candidate and declared BVP, not a general existence, uniqueness, stability, regularity or well-posedness theorem." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "original coefficients=[-11, -1, -2, 2, 0, -1];basis=[1,x,y,x^2,xy,y^2]", "after": "u(x,y)=(-11)+(-1)*x+(-2)*y+(2)*x^2+(0)*x*y+(-1)*y^2", "relation": "implies", "rule_id": "bivariate_scalar_polynomial_basis_roles", "conditions": [], "explanation": { "zh-Hant": "六個係數按固定單項式順序讀，不與單變量二次式的[a,b,c]降冪次序混用。", "en": "Read the six coefficients in their fixed monomial order,not the descending[a,b,c] order of a univariate quadratic." }, "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": "selected boundary fixed coordinate squared := mul(3, 3)", "after": "selected boundary fixed coordinate squared = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary fixed coordinate squared，不以小數近似替代。", "en": "Recompute selected boundary fixed coordinate squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "selected boundary fixed coordinate squared" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "selected boundary fixed linear term := mul(-1, 3)", "after": "selected boundary fixed linear term = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary fixed linear term，不以小數近似替代。", "en": "Recompute selected boundary fixed linear term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "3" ], "output": "-3", "label": "selected boundary fixed linear term" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "selected boundary fixed square term := mul(2, 9)", "after": "selected boundary fixed square term = 18", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary fixed square term，不以小數近似替代。", "en": "Recompute selected boundary fixed square term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "9" ], "output": "18", "label": "selected boundary fixed square term" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "selected boundary constant coefficient := sum(-11, -3, 18)", "after": "selected boundary constant coefficient = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary constant coefficient，不以小數近似替代。", "en": "Recompute selected boundary constant coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-11", "-3", "18" ], "output": "4", "label": "selected boundary constant coefficient" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "selected boundary cross contribution := mul(0, 3)", "after": "selected boundary cross contribution = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary cross contribution，不以小數近似替代。", "en": "Recompute selected boundary cross contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "3" ], "output": "0", "label": "selected boundary cross contribution" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "selected boundary linear coefficient := sum(-2, 0)", "after": "selected boundary linear coefficient = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary linear coefficient，不以小數近似替代。", "en": "Recompute selected boundary linear coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "0" ], "output": "-2", "label": "selected boundary linear coefficient" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "selected boundary:vary y;fix x=3;original coefficients=[-11, -1, -2, 2, 0, -1]", "after": "ascending y slice coefficients=[4, -2, -1]", "relation": "implies", "rule_id": "fixed_other_coordinate_before_derivative", "conditions": [], "explanation": { "zh-Hant": "先代固定的另一座標，常數、一次與二次項分開收集；沒有讓兩座標一起變動。", "en": "Substitute the fixed other coordinate first,collecting constant,linear and quadratic terms separately.The two coordinates do not vary together." }, "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": "selected boundary coefficient residual0 := sub(4, 4)", "after": "selected boundary coefficient residual0 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary coefficient residual0，不以小數近似替代。", "en": "Recompute selected boundary coefficient residual0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "4", "4" ], "output": "0", "label": "selected boundary coefficient residual0" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "selected boundary coefficient residual1 := sub(-2, -2)", "after": "selected boundary coefficient residual1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary coefficient residual1，不以小數近似替代。", "en": "Recompute selected boundary coefficient residual1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-2", "-2" ], "output": "0", "label": "selected boundary coefficient residual1" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "selected boundary coefficient residual2 := sub(-1, -1)", "after": "selected boundary coefficient residual2 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依本步指定的原始角色與已算中間量精確運算；保留正負號、分母及係數次序。", "en": "Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected boundary coefficient residual2，不以小數近似替代。", "en": "Recompute selected boundary coefficient residual2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "selected boundary coefficient residual2" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "selected boundary:edge=right;bounds=[-3, 3, -2, 4];remaining variable=y;restriction=[4, -2, -1];supplied=[4, -2, -1]", "after": "all coefficient residuals=[0, 0, 0];polynomial identity=是", "relation": "implies", "rule_id": "whole_edge_polynomial_coefficient_certificate", "conditions": [], "explanation": { "zh-Hant": "比較整條限制多項式的三個係數，不用有限點抽樣。此協定核的是形式多項式恆等；若邊退化為單點，係數不符只是否定這個較強證書，不能據此否定該單點值相等。", "en": "Compare all three coefficients of the restriction polynomial,not sampled boundary values.This protocol audits a polynomial identity.If an edge collapses to a point,a coefficient mismatch rejects this stronger certificate but does not by itself refute equality at that single point." }, "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": [ [ -3, 3, -2, 4 ], "right", "y", [ 4, -2, -1 ], [ 4, -2, -1 ], true, "declared_edge_only" ] }, "conclusion": { "canonical_value": "([-3, 3, -2, 4], right, y, [4, -2, -1], [4, -2, -1], 是, declared_edge_only)", "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-20261013-advance-20261025-r01-boundary-value-problem-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261025-r01-boundary-value-problem-foundations/questions.jsonl", "batch_sha256": "8bfdd372f77c3a1c3875ae0132ddd93f279405ca2410bdc7c6e0f092d30b65a4", "original_line": 27, "original_line_sha256": "5356b7866a9c59fb0c098145c35646e761c18a5361a9e581ccfa3838b9fc4ca3", "question_compact_sha256": "f9a0004e7d594001aaf87e5988f50f46cd227b256172593edfe2215b2065bf5a", "english_row_compact_sha256": "e02048f37745dece3603b177b1eb45663ede990e0b6774f03383f4eb00df2f0c", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261025-r01-boundary-value-problem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261025-r01-boundary-value-problem-foundations.en.jsonl" }, "mko_bindings": { "mko-boundary-value-problem": { "sha256": "4ffaeecac13219cb387fd43a669823ff3c2b402bfadf2e3d04f3938d83bf3f0d", "english_sha256": "a666e5292df9be234d0c6cad5ac693748b1c2a78e2b39dd8dc732b0eafe2e60b", "original_path": "public/data/mko/boundary-value-problem.json", "english_path": "public/data/i18n/editions/mko-boundary-value-problem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "844102be97423689e3a5e0c418b86b56773cf737eab263267081710eff80cbea", "original_candidate_sha256": "573d624afebc6f40358ea84638ad8c1744dc7ebbb94543252586712863996bbe", "original_candidate_line_sha256": "573d624afebc6f40358ea84638ad8c1744dc7ebbb94543252586712863996bbe" }, "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-20261026-006-p23", "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": "94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac", "question_first_witness_sha256": "0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 12, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "d0eff32ae3d55ba5f67d546b3b0accf2f90c2732c51584c762707207027746c6", "arithmetic_steps": 9 }, "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." } }