{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261024-r06-partial-differential-equation-0087", "pilot_number": "R0022-11637", "number": 11637, "family_id": "qf-mod24-pde-heat-polynomial-identity", "title": { "zh-Hant": "第 11637 題：線性PDE切片", "en": "Question 11637: A Symbolic Heat Residual Identity, Not a Point Sample" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261024-r06-partial-differential-equation-0087", "batch_id": "batch-20261013-advance-20261024-r06-partial-differential-equation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-partial-differential-equation" ], "title_zh": "線性PDE切片", "stem_zh": "明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-1,-1,24,3,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。", "answer": { "kind": "expression", "value": "([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], 是)", "explanation_zh": "依賴結構核對=([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], 是)；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；點殘差不證全域解；不宣稱一般解存在唯一或適定性。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "線性PDE切片" ] }, "variant": { "family_id": "qf-mod24-pde-heat-polynomial-identity", "seed": 202610246087, "parameters": { "coefficients": [ -1, -1, 24, 3, 0, 0 ], "diffusivity": 4, "domain": "Polynomial_linear_PDE_operator_slice", "operation": "pde-heat-polynomial-identity", "semantic_template": "qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-1,-1,24,3,0,0],\"diffusivity\":4}" } }, "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、Markov長期性、PDE適定性、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-20261024-r06-partial-differential-equation-0087", "batch_id": "batch-20261013-advance-20261024-r06-partial-differential-equation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "130760f82634fb7defd2577ac1173d73014eb78d57a0beec559cb820a877c12f", "family_id": "qf-mod24-pde-heat-polynomial-identity", "seed": 202610246087, "semantic_template": "qf-mod24-pde-heat-polynomial-identity/{\"coefficients\":[-1,-1,24,3,0,0],\"diffusivity\":4}" }, "fields": { "title": "A Symbolic Heat Residual Identity, Not a Point Sample", "stem": "Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-1, -1, 24, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy].", "answer": { "kind": "expression", "value": "([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], Yes)", "explanation": "A zero residual coefficient vector proves this specified polynomial satisfies the operator equation everywhere. A nonzero residual that vanishes at some sampled points is not an identity; neither case supplies a general solution theory. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], Yes)." }, "skills": [ "A Symbolic Heat Residual Identity, Not a Point Sample" ], "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": "明示線性PDE算子與係數convention。操作=pde-heat-polynomial-identity；資料={\"coefficients\":[-1,-1,24,3,0,0],\"diffusivity\":4}。請依 [coefficients,diffusivity,residual_coefficients,identity_solution] 欄序回答。", "en": "Use the explicit linear constant-coefficient PDE operator and polynomial candidate coefficients=[-1, -1, 24, 3, 0, 0] in order [1,x,y,x²,x*y,y²]. Fix the operator convention before evaluating residuals. A point residual proves only a point check; general solution, existence, uniqueness and well-posedness remain separate obligations. Here y is time, x is space and positive integer diffusivity=4. The equation is u_y=diffusivity*u_xx. Test the residual identically as a polynomial, not only at sampled points. Give (coefficients,diffusivity,residual coefficients in basis [1,x,y],all residual coefficients zero?). The residual coefficients are [c_y−2*diffusivity*c_xx,c_xy,2*c_yy]." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "original coefficients=[-1, -1, 24, 3, 0, 0];basis=[1,x,y,x^2,xy,y^2]", "after": "u(x,y)=(-1)+(-1)*x+(24)*y+(3)*x^2+(0)*x*y+(0)*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, 3)", "after": "u_xx constant = 6", "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", "3" ], "output": "6", "label": "u_xx constant" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "u_yy constant := mul(2, 0)", "after": "u_yy constant = 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 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", "0" ], "output": "0", "label": "u_yy constant" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "u coefficients=[-1, -1, 24, 3, 0, 0];u_x=c1+2*c3*x+c4*y;u_y=c2+c4*x+2*c5*y", "after": "u_xx=6;u_xy=0;u_yy=0", "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": "heat diffusivity times u_xx := mul(4, 6)", "after": "heat diffusivity times u_xx = 24", "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 heat diffusivity times u_xx，不以小數近似替代。", "en": "Recompute heat diffusivity times u_xx with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "6" ], "output": "24", "label": "heat diffusivity times u_xx" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "heat constant residual := sub(24, 24)", "after": "heat constant 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 heat constant residual，不以小數近似替代。", "en": "Recompute heat constant residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "24", "24" ], "output": "0", "label": "heat constant residual" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "u_y=(24)+(0)*x+(0)*y;diffusivity*u_xx=24", "after": "residual coefficients in [1,x,y]=[0, 0, 0];identically zero=是", "relation": "implies", "rule_id": "heat_residual_coefficient_identity_not_samples", "conditions": [], "explanation": { "zh-Hant": "y是時間，x是空間；把u_y−d*u_xx完整收集到[1,x,y]。單項式線性獨立，全部係數零當且僅當此多項式處處為零，非零係數即使某點抵消也不是恆等。", "en": "Here y is time and x is space.Collect the whole residual u_y−d*u_xx in[1,x,y].Linear independence of monomials makes all-zero coefficients equivalent to the polynomial identity.A nonzero coefficient cannot be repaired by cancellation at a sampled 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": [ [ -1, -1, 24, 3, 0, 0 ], 4, [ 0, 0, 0 ], true ] }, "conclusion": { "canonical_value": "([-1, -1, 24, 3, 0, 0], 4, [0, 0, 0], 是)", "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-20261024-r06-partial-differential-equation-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations/questions.jsonl", "batch_sha256": "3694f95d9344a162d98b9c0854b262ae4f31f1b65ec74f3c45f754d256e09dea", "original_line": 87, "original_line_sha256": "6dcc4c0644c78f0c1bd862b2da2b3a654f09e12a6771b91ad0ff559b1422f59e", "question_compact_sha256": "130760f82634fb7defd2577ac1173d73014eb78d57a0beec559cb820a877c12f", "english_row_compact_sha256": "6e201c176f15969c1cbec45a33bb2abad44b19c093138a648cd9b1a39e2ed601", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261024-r06-partial-differential-equation-foundations.en.jsonl" }, "mko_bindings": { "mko-partial-differential-equation": { "sha256": "b78ef1b1a613523b784d17e0507361cf263e495850e2b7aa880558f1294758b8", "english_sha256": "62e82ace9246a9a6a884ee7e0a1a332257e56eba985caac8dbc5c8d5efa7df75", "original_path": "public/data/mko/partial-differential-equation.json", "english_path": "public/data/i18n/editions/mko-partial-differential-equation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "844102be97423689e3a5e0c418b86b56773cf737eab263267081710eff80cbea", "original_candidate_sha256": "13f6022b7eba7da023f714a693b326dbd4f8817b22d9a72bff207d153040e258", "original_candidate_line_sha256": "13f6022b7eba7da023f714a693b326dbd4f8817b22d9a72bff207d153040e258" }, "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": 7, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "818baffd19ce4e89f7b85983ece5344665b07cfc663e34dd8a20db2256e178dc", "arithmetic_steps": 4 }, "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." } }