{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261025-r01-boundary-value-problem-0067", "pilot_number": "R0022-11767", "number": 11767, "family_id": "qf-mod25-bvp-joint-candidate-audit", "title": { "zh-Hant": "第 11767 題：邊界值問題", "en": "Question 11767: Close Interior and All Four Boundary Obligations Separately" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261025-r01-boundary-value-problem-0067", "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-joint-candidate-audit；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"bounds\":[-18,18,-17,19],\"source\":0,\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。", "answer": { "kind": "expression", "value": "([4, -1, -1, 1, 0, -1], 0, 0, 是, [[left, y, [346, -1, -1], [346, -1, -1], 是], [right, y, [310, -1, -1], [310, -1, -1], 是], [bottom, x, [-268, -1, 1], [-268, -1, 1], 是], [top, x, [-376, -1, 1], [-376, -1, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_only)", "explanation_zh": "收束波核對=([4, -1, -1, 1, 0, -1], 0, 0, 是, [[left, y, [346, -1, -1], [346, -1, -1], 是], [right, y, [310, -1, -1], [310, -1, -1], 是], [bottom, x, [-268, -1, 1], [-268, -1, 1], 是], [top, x, [-376, -1, 1], [-376, -1, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_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": 2, "skills": [ "邊界值問題" ] }, "variant": { "family_id": "qf-mod25-bvp-joint-candidate-audit", "seed": 202610251067, "parameters": { "coefficients": [ 4, -1, -1, 1, 0, -1 ], "bounds": [ -18, 18, -17, 19 ], "source": 0, "boundary_data": [ [ "left", [ 346, -1, -1 ] ], [ "right", [ 310, -1, -1 ] ], [ "bottom", [ -268, -1, 1 ] ], [ "top", [ -376, -1, 1 ] ] ], "domain": "Polynomial_rectangle_boundary_value_certificates", "operation": "bvp-joint-candidate-audit", "semantic_template": "qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"source\":0}" } }, "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-0067", "batch_id": "batch-20261013-advance-20261025-r01-boundary-value-problem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5059d11892d6ec083a591c551ccb59c5113f9bf2cbb38cb357e940647a95a615", "family_id": "qf-mod25-bvp-joint-candidate-audit", "seed": 202610251067, "semantic_template": "qf-mod25-bvp-joint-candidate-audit/{\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]],\"bounds\":[-18,18,-17,19],\"coefficients\":[4,-1,-1,1,0,-1],\"source\":0}" }, "fields": { "title": "Close Interior and All Four Boundary Obligations Separately", "stem": "Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19]. 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. Interior source=0; supplied [edge,coefficients] records=[[left, [346, -1, -1]], [right, [310, -1, -1]], [bottom, [-268, -1, 1]], [top, [-376, -1, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_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": "([4, -1, -1, 1, 0, -1], 0, 0, Yes, [[left, y, [346, -1, -1], [346, -1, -1], Yes], [right, y, [310, -1, -1], [310, -1, -1], Yes], [bottom, x, [-268, -1, 1], [-268, -1, 1], Yes], [top, x, [-376, -1, 1], [-376, -1, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)", "explanation": "Recompute the interior residual and every boundary restriction independently before taking their conjunction. This verifies a candidate for the declared problem without claiming general well-posedness. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], 0, 0, Yes, [[left, y, [346, -1, -1], [346, -1, -1], Yes], [right, y, [310, -1, -1], [310, -1, -1], Yes], [bottom, x, [-268, -1, 1], [-268, -1, 1], Yes], [top, x, [-376, -1, 1], [-376, -1, 1], Yes]], Yes, Yes, candidate_satisfies_declared_BVP_only)." }, "skills": [ "Close Interior and All Four Boundary Obligations Separately" ], "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-joint-candidate-audit；資料={\"coefficients\":[4,-1,-1,1,0,-1],\"bounds\":[-18,18,-17,19],\"source\":0,\"boundary_data\":[[\"left\",[346,-1,-1]],[\"right\",[310,-1,-1]],[\"bottom\",[-268,-1,1]],[\"top\",[-376,-1,1]]]}。請依 [coefficients,laplacian,source,interior_identity,boundary_rows,boundary_identity,candidate_satisfies_declared_BVP,scope] 欄序回答。", "en": "Use the quadratic candidate coefficients=[4, -1, -1, 1, 0, -1] in fixed order [1,x,y,x²,x*y,y²]. Rectangle bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19]. 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. Interior source=0; supplied [edge,coefficients] records=[[left, [346, -1, -1]], [right, [310, -1, -1]], [bottom, [-268, -1, 1]], [top, [-376, -1, 1]]] must include all four distinct edges. Preserve their supplied order. Give (coefficients,Laplacian,source,interior identity?,every [edge,remaining variable,restriction,supplied,identity?] row,all boundary identities?,both obligation groups?,candidate_satisfies_declared_BVP_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=[4, -1, -1, 1, 0, -1];basis=[1,x,y,x^2,xy,y^2]", "after": "u(x,y)=(4)+(-1)*x+(-1)*y+(1)*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": "u_xx constant := mul(2, 1)", "after": "u_xx constant = 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 u_xx constant，不以小數近似替代。", "en": "Recompute u_xx constant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "u_xx constant" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "u_yy constant := mul(2, -1)", "after": "u_yy constant = -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 u_yy constant，不以小數近似替代。", "en": "Recompute u_yy constant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-1" ], "output": "-2", "label": "u_yy constant" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "u coefficients=[4, -1, -1, 1, 0, -1];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y", "after": "u_xx=2;u_xy=0;u_yy=-2", "relation": "implies", "rule_id": "polynomial_second_partials_explicit_monomials", "conditions": [], "explanation": { "zh-Hant": "以固定另一座標的切片規則逐項微分：x的一次項再微分成常數，xy交叉項只貢獻c4；這是多項式恆等式，不是由一般混偏導交換假設。", "en": "Differentiate each monomial by fixed-coordinate slice rules.The linear x term differentiates to a constant,and xy contributes c4 to u_xy.These are polynomial identities,not an assumed general mixed-partial exchange theorem." }, "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": "interior Laplacian := sum(2, -2)", "after": "interior Laplacian = 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 interior Laplacian，不以小數近似替代。", "en": "Recompute interior Laplacian with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "-2" ], "output": "0", "label": "interior Laplacian" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "interior constant signed residual := sub(0, 0)", "after": "interior constant signed residual = 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 interior constant signed residual，不以小數近似替代。", "en": "Recompute interior constant signed residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "0", "0" ], "output": "0", "label": "interior constant signed residual" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "u coefficients=[4, -1, -1, 1, 0, -1];constant Laplacian=0;source=0", "after": "constant residual=0;interior polynomial identity=是", "relation": "implies", "rule_id": "interior_identity_separate_boundary_obligation", "conditions": [], "explanation": { "zh-Hant": "u_xx+u_yy在此候選為常數，常數殘差零才是宣告的符號恆等；這一步沒有檢查任何邊界，也不是一般Poisson解存在唯一性證明。", "en": "For this candidate u_xx+u_yy is constant.A zero constant residual is the declared symbolic identity.No boundary data have been checked here,nor is this a general Poisson existence/uniqueness proof." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "ordered bounds=[-18, 18, -17, 19];supplied edge order=[left, right, bottom, top]", "after": "all four distinct edges present;preserve original record order", "relation": "implies", "rule_id": "four_boundary_records_complete_not_sorted_away", "conditions": [], "explanation": { "zh-Hant": "四邊各出現一次，不以同一邊的重複紀錄補數；輸出維持原資料配對次序。退化矩形仍按宣告的多項式證書協定，不捏造非空開內部。", "en": "Each of the four edges appears once;duplicates do not substitute for missing edges.Preserve supplied record order and pairing.Degenerate rectangles still use the declared polynomial-certificate protocol;do not invent a nonempty open interior." }, "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": "boundary record0 fixed coordinate squared := mul(-18, -18)", "after": "boundary record0 fixed coordinate squared = 324", "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 boundary record0 fixed coordinate squared，不以小數近似替代。", "en": "Recompute boundary record0 fixed coordinate squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-18", "-18" ], "output": "324", "label": "boundary record0 fixed coordinate squared" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "boundary record0 fixed linear term := mul(-1, -18)", "after": "boundary record0 fixed linear 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 boundary record0 fixed linear term，不以小數近似替代。", "en": "Recompute boundary record0 fixed linear term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-18" ], "output": "18", "label": "boundary record0 fixed linear term" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "boundary record0 fixed square term := mul(1, 324)", "after": "boundary record0 fixed square term = 324", "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 boundary record0 fixed square term，不以小數近似替代。", "en": "Recompute boundary record0 fixed square term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "324" ], "output": "324", "label": "boundary record0 fixed square term" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "boundary record0 constant coefficient := sum(4, 18, 324)", "after": "boundary record0 constant coefficient = 346", "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 boundary record0 constant coefficient，不以小數近似替代。", "en": "Recompute boundary record0 constant coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "18", "324" ], "output": "346", "label": "boundary record0 constant coefficient" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "boundary record0 cross contribution := mul(0, -18)", "after": "boundary record0 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 boundary record0 cross contribution，不以小數近似替代。", "en": "Recompute boundary record0 cross contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-18" ], "output": "0", "label": "boundary record0 cross contribution" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "boundary record0 linear coefficient := sum(-1, 0)", "after": "boundary record0 linear coefficient = -1", "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 boundary record0 linear coefficient，不以小數近似替代。", "en": "Recompute boundary record0 linear coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "0" ], "output": "-1", "label": "boundary record0 linear coefficient" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "boundary record0:vary y;fix x=-18;original coefficients=[4, -1, -1, 1, 0, -1]", "after": "ascending y slice coefficients=[346, -1, -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": "S016", "uses": [ "G001", "S015" ], "before": "boundary record0 coefficient residual0 := sub(346, 346)", "after": "boundary record0 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 boundary record0 coefficient residual0，不以小數近似替代。", "en": "Recompute boundary record0 coefficient residual0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "346", "346" ], "output": "0", "label": "boundary record0 coefficient residual0" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "boundary record0 coefficient residual1 := sub(-1, -1)", "after": "boundary record0 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 boundary record0 coefficient residual1，不以小數近似替代。", "en": "Recompute boundary record0 coefficient residual1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "boundary record0 coefficient residual1" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "boundary record0 coefficient residual2 := sub(-1, -1)", "after": "boundary record0 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 boundary record0 coefficient residual2，不以小數近似替代。", "en": "Recompute boundary record0 coefficient residual2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "boundary record0 coefficient residual2" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "boundary record0:edge=left;bounds=[-18, 18, -17, 19];remaining variable=y;restriction=[346, -1, -1];supplied=[346, -1, -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 }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "boundary record1 fixed coordinate squared := mul(18, 18)", "after": "boundary record1 fixed coordinate squared = 324", "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 boundary record1 fixed coordinate squared，不以小數近似替代。", "en": "Recompute boundary record1 fixed coordinate squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "18", "18" ], "output": "324", "label": "boundary record1 fixed coordinate squared" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "boundary record1 fixed linear term := mul(-1, 18)", "after": "boundary record1 fixed linear 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 boundary record1 fixed linear term，不以小數近似替代。", "en": "Recompute boundary record1 fixed linear term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "18" ], "output": "-18", "label": "boundary record1 fixed linear term" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "boundary record1 fixed square term := mul(1, 324)", "after": "boundary record1 fixed square term = 324", "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 boundary record1 fixed square term，不以小數近似替代。", "en": "Recompute boundary record1 fixed square term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "324" ], "output": "324", "label": "boundary record1 fixed square term" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "boundary record1 constant coefficient := sum(4, -18, 324)", "after": "boundary record1 constant coefficient = 310", "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 boundary record1 constant coefficient，不以小數近似替代。", "en": "Recompute boundary record1 constant coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "-18", "324" ], "output": "310", "label": "boundary record1 constant coefficient" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "boundary record1 cross contribution := mul(0, 18)", "after": "boundary record1 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 boundary record1 cross contribution，不以小數近似替代。", "en": "Recompute boundary record1 cross contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "18" ], "output": "0", "label": "boundary record1 cross contribution" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "boundary record1 linear coefficient := sum(-1, 0)", "after": "boundary record1 linear coefficient = -1", "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 boundary record1 linear coefficient，不以小數近似替代。", "en": "Recompute boundary record1 linear coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "0" ], "output": "-1", "label": "boundary record1 linear coefficient" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "boundary record1:vary y;fix x=18;original coefficients=[4, -1, -1, 1, 0, -1]", "after": "ascending y slice coefficients=[310, -1, -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": "S027", "uses": [ "G001", "S026" ], "before": "boundary record1 coefficient residual0 := sub(310, 310)", "after": "boundary record1 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 boundary record1 coefficient residual0，不以小數近似替代。", "en": "Recompute boundary record1 coefficient residual0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "310", "310" ], "output": "0", "label": "boundary record1 coefficient residual0" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "boundary record1 coefficient residual1 := sub(-1, -1)", "after": "boundary record1 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 boundary record1 coefficient residual1，不以小數近似替代。", "en": "Recompute boundary record1 coefficient residual1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "boundary record1 coefficient residual1" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "boundary record1 coefficient residual2 := sub(-1, -1)", "after": "boundary record1 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 boundary record1 coefficient residual2，不以小數近似替代。", "en": "Recompute boundary record1 coefficient residual2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "boundary record1 coefficient residual2" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "boundary record1:edge=right;bounds=[-18, 18, -17, 19];remaining variable=y;restriction=[310, -1, -1];supplied=[310, -1, -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 }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "boundary record2 fixed coordinate squared := mul(-17, -17)", "after": "boundary record2 fixed coordinate squared = 289", "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 boundary record2 fixed coordinate squared，不以小數近似替代。", "en": "Recompute boundary record2 fixed coordinate squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-17", "-17" ], "output": "289", "label": "boundary record2 fixed coordinate squared" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "boundary record2 fixed linear term := mul(-1, -17)", "after": "boundary record2 fixed linear term = 17", "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 boundary record2 fixed linear term，不以小數近似替代。", "en": "Recompute boundary record2 fixed linear term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-17" ], "output": "17", "label": "boundary record2 fixed linear term" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "boundary record2 fixed square term := mul(-1, 289)", "after": "boundary record2 fixed square term = -289", "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 boundary record2 fixed square term，不以小數近似替代。", "en": "Recompute boundary record2 fixed square term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "289" ], "output": "-289", "label": "boundary record2 fixed square term" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "boundary record2 constant coefficient := sum(4, 17, -289)", "after": "boundary record2 constant coefficient = -268", "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 boundary record2 constant coefficient，不以小數近似替代。", "en": "Recompute boundary record2 constant coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "17", "-289" ], "output": "-268", "label": "boundary record2 constant coefficient" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "boundary record2 cross contribution := mul(0, -17)", "after": "boundary record2 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 boundary record2 cross contribution，不以小數近似替代。", "en": "Recompute boundary record2 cross contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "-17" ], "output": "0", "label": "boundary record2 cross contribution" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "boundary record2 linear coefficient := sum(-1, 0)", "after": "boundary record2 linear coefficient = -1", "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 boundary record2 linear coefficient，不以小數近似替代。", "en": "Recompute boundary record2 linear coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "0" ], "output": "-1", "label": "boundary record2 linear coefficient" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "boundary record2:vary x;fix y=-17;original coefficients=[4, -1, -1, 1, 0, -1]", "after": "ascending x slice coefficients=[-268, -1, 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": "S038", "uses": [ "G001", "S037" ], "before": "boundary record2 coefficient residual0 := sub(-268, -268)", "after": "boundary record2 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 boundary record2 coefficient residual0，不以小數近似替代。", "en": "Recompute boundary record2 coefficient residual0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-268", "-268" ], "output": "0", "label": "boundary record2 coefficient residual0" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "boundary record2 coefficient residual1 := sub(-1, -1)", "after": "boundary record2 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 boundary record2 coefficient residual1，不以小數近似替代。", "en": "Recompute boundary record2 coefficient residual1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "boundary record2 coefficient residual1" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "boundary record2 coefficient residual2 := sub(1, 1)", "after": "boundary record2 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 boundary record2 coefficient residual2，不以小數近似替代。", "en": "Recompute boundary record2 coefficient residual2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "1" ], "output": "0", "label": "boundary record2 coefficient residual2" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "boundary record2:edge=bottom;bounds=[-18, 18, -17, 19];remaining variable=x;restriction=[-268, -1, 1];supplied=[-268, -1, 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 }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "boundary record3 fixed coordinate squared := mul(19, 19)", "after": "boundary record3 fixed coordinate squared = 361", "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 boundary record3 fixed coordinate squared，不以小數近似替代。", "en": "Recompute boundary record3 fixed coordinate squared with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "19", "19" ], "output": "361", "label": "boundary record3 fixed coordinate squared" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "boundary record3 fixed linear term := mul(-1, 19)", "after": "boundary record3 fixed linear term = -19", "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 boundary record3 fixed linear term，不以小數近似替代。", "en": "Recompute boundary record3 fixed linear term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "19" ], "output": "-19", "label": "boundary record3 fixed linear term" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "boundary record3 fixed square term := mul(-1, 361)", "after": "boundary record3 fixed square term = -361", "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 boundary record3 fixed square term，不以小數近似替代。", "en": "Recompute boundary record3 fixed square term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "361" ], "output": "-361", "label": "boundary record3 fixed square term" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "boundary record3 constant coefficient := sum(4, -19, -361)", "after": "boundary record3 constant coefficient = -376", "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 boundary record3 constant coefficient，不以小數近似替代。", "en": "Recompute boundary record3 constant coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "-19", "-361" ], "output": "-376", "label": "boundary record3 constant coefficient" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "boundary record3 cross contribution := mul(0, 19)", "after": "boundary record3 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 boundary record3 cross contribution，不以小數近似替代。", "en": "Recompute boundary record3 cross contribution with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "19" ], "output": "0", "label": "boundary record3 cross contribution" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "boundary record3 linear coefficient := sum(-1, 0)", "after": "boundary record3 linear coefficient = -1", "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 boundary record3 linear coefficient，不以小數近似替代。", "en": "Recompute boundary record3 linear coefficient with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-1", "0" ], "output": "-1", "label": "boundary record3 linear coefficient" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "boundary record3:vary x;fix y=19;original coefficients=[4, -1, -1, 1, 0, -1]", "after": "ascending x slice coefficients=[-376, -1, 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": "S049", "uses": [ "G001", "S048" ], "before": "boundary record3 coefficient residual0 := sub(-376, -376)", "after": "boundary record3 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 boundary record3 coefficient residual0，不以小數近似替代。", "en": "Recompute boundary record3 coefficient residual0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-376", "-376" ], "output": "0", "label": "boundary record3 coefficient residual0" } }, { "id": "S050", "uses": [ "G001", "S049" ], "before": "boundary record3 coefficient residual1 := sub(-1, -1)", "after": "boundary record3 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 boundary record3 coefficient residual1，不以小數近似替代。", "en": "Recompute boundary record3 coefficient residual1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-1", "-1" ], "output": "0", "label": "boundary record3 coefficient residual1" } }, { "id": "S051", "uses": [ "G001", "S050" ], "before": "boundary record3 coefficient residual2 := sub(1, 1)", "after": "boundary record3 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 boundary record3 coefficient residual2，不以小數近似替代。", "en": "Recompute boundary record3 coefficient residual2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "1" ], "output": "0", "label": "boundary record3 coefficient residual2" } }, { "id": "S052", "uses": [ "G001", "S051" ], "before": "boundary record3:edge=top;bounds=[-18, 18, -17, 19];remaining variable=x;restriction=[-376, -1, 1];supplied=[-376, -1, 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 }, { "id": "S053", "uses": [ "G001", "S052" ], "before": "interior identity=是;boundary row flags=[是, 是, 是, 是]", "after": "all boundary identities=是;joint certificate=是;state=both_pass", "relation": "implies", "rule_id": "BVP_two_obligation_groups_conjunction", "conditions": [], "explanation": { "zh-Hant": "內部與四邊證書各自算完再合取，一組成功不能補救另一組。這只驗宣告候選，不自動給一般邊界值問題的存在、唯一、穩定、正則或適定性。", "en": "Finish interior and all boundary certificates separately before conjunction.One passing group does not repair another.This audits the declared candidate,not general BVP existence,uniqueness,stability,regularity or well-posedness." }, "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": [ [ 4, -1, -1, 1, 0, -1 ], 0, 0, true, [ [ "left", "y", [ 346, -1, -1 ], [ 346, -1, -1 ], true ], [ "right", "y", [ 310, -1, -1 ], [ 310, -1, -1 ], true ], [ "bottom", "x", [ -268, -1, 1 ], [ -268, -1, 1 ], true ], [ "top", "x", [ -376, -1, 1 ], [ -376, -1, 1 ], true ] ], true, true, "candidate_satisfies_declared_BVP_only" ] }, "conclusion": { "canonical_value": "([4, -1, -1, 1, 0, -1], 0, 0, 是, [[left, y, [346, -1, -1], [346, -1, -1], 是], [right, y, [310, -1, -1], [310, -1, -1], 是], [bottom, x, [-268, -1, 1], [-268, -1, 1], 是], [top, x, [-376, -1, 1], [-376, -1, 1], 是]], 是, 是, candidate_satisfies_declared_BVP_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": 67, "original_line_sha256": "09a5f697372a6c030bb634b9ed1d58e2db4d098fd4471d18a15bc224ae80f77b", "question_compact_sha256": "5059d11892d6ec083a591c551ccb59c5113f9bf2cbb38cb357e940647a95a615", "english_row_compact_sha256": "1a42f8228afd5b838d3c8a7fea8d14d7864d314c0c0c17ceb095b6b0d726e081", "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": "cde06b84730e1c165ee1161fbd664d4fb6b267e510056b422e8647362b44b4e2", "original_candidate_line_sha256": "cde06b84730e1c165ee1161fbd664d4fb6b267e510056b422e8647362b44b4e2" }, "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": 53, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "8bdc03ef8597c5a56e1fa6650065c6268eceb083e57dea85e74936bfd59d9caf", "arithmetic_steps": 40 }, "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." } }