{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261030-r02-real-hilbert-space-0090", "pilot_number": "R0029-15290", "number": 15290, "family_id": "qf-mod30-hilbert-pythagorean", "title": { "zh-Hant": "第 15290 題：實Hilbert空間", "en": "Question 15290: Check Orthogonality before Using the Inner-Product Pythagorean Identity" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261030-r02-real-hilbert-space-0090", "batch_id": "batch-20261014-advance-20261030-r02-real-hilbert-space-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-real-hilbert-space" ], "title_zh": "實Hilbert空間", "stem_zh": "inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[16,2],[2,5]],\"x\":[1,2],\"y\":[2,-16]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。", "answer": { "kind": "expression", "value": "([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)", "explanation_zh": "Wave4第四層核對=([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；有限維命名定理提供完備全稱證書；不由樣本推完備且不推出compact separable。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "實Hilbert空間" ] }, "variant": { "family_id": "qf-mod30-hilbert-pythagorean", "seed": 202610302090, "parameters": { "matrix": [ [ 16, 2 ], [ 2, 5 ] ], "x": [ 1, 2 ], "y": [ 2, -16 ], "domain": "Named_finite_dimensional_real_Hilbert_space", "operation": "hilbert-pythagorean", "semantic_template": "qf-mod30-hilbert-pythagorean/{\"matrix\":[[16,2],[2,5]],\"x\":[1,2],\"y\":[2,-16]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261030-r02-real-hilbert-space-0090", "batch_id": "batch-20261014-advance-20261030-r02-real-hilbert-space-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "449e2ec26b4f20f92fdfec792d284031e2989a1e8ac12cd2735c4e9f4e09e5ac", "family_id": "qf-mod30-hilbert-pythagorean", "seed": 202610302090, "semantic_template": "qf-mod30-hilbert-pythagorean/{\"matrix\":[[16,2],[2,5]],\"x\":[1,2],\"y\":[2,-16]}" }, "fields": { "title": "Check Orthogonality before Using the Inner-Product Pythagorean Identity", "stem": "Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[2, -16]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity).", "answer": { "kind": "expression", "value": "([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)", "explanation": "Compute both quadratic expressions and the same-form inner product. If the inner product is zero, the last flag checks their equality; otherwise retain orthogonality_premise_false rather than treating a failed premise as the theorem being false. No completeness theorem is needed for this finite identity. The exact result, retaining the requested order, is ([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)." }, "skills": [ "Check Orthogonality before Using the Inner-Product Pythagorean 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": "inner product norm metric carrier completeness綁同一空間。操作=hilbert-pythagorean；資料={\"matrix\":[[16,2],[2,5]],\"x\":[1,2],\"y\":[2,-16]}。請依 [x,y,inner,lhs,rhs,result] 欄序回答。", "en": "Use the same real plane with symmetric positive-definite matrix G=[[16, 2], [2, 5]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, 2], y=[2, -16]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "Hilbert Gram diagonal product := mul(16, 5)", "after": "Hilbert Gram diagonal product = 80", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert Gram diagonal product，不以小數近似替代。", "en": "Recompute Hilbert Gram diagonal product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "5" ], "output": "80", "label": "Hilbert Gram diagonal product" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Hilbert Gram off-diagonal product := mul(2, 2)", "after": "Hilbert Gram off-diagonal product = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert Gram off-diagonal product，不以小數近似替代。", "en": "Recompute Hilbert Gram off-diagonal product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Hilbert Gram off-diagonal product" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Hilbert Gram determinant := sub(80, 4)", "after": "Hilbert Gram determinant = 76", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert Gram determinant，不以小數近似替代。", "en": "Recompute Hilbert Gram determinant with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "80", "4" ], "output": "76", "label": "Hilbert Gram determinant" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Hilbert completing-square slope := div(2, 16)", "after": "Hilbert completing-square slope = 1/8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert completing-square slope，不以小數近似替代。", "en": "Recompute Hilbert completing-square slope with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2", "16" ], "output": "1/8", "label": "Hilbert completing-square slope" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Hilbert completing-square positive residual := div(76, 16)", "after": "Hilbert completing-square positive residual = 19/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert completing-square positive residual，不以小數近似替代。", "en": "Recompute Hilbert completing-square positive residual with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "76", "16" ], "output": "19/4", "label": "Hilbert completing-square positive residual" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "complete real carrier R^2;Gram matrix=[[16, 2], [2, 5]];a=16>0;det=76>0;dimension declaration=2", "after": "x^T M x=16*(x0+(1/8)*x1)^2+(19/4)*x1^2;zero only at x=0", "relation": "implies", "rule_id": "hilbert_same_complete_matrix_positive_inner_product", "conditions": [], "explanation": { "zh-Hant": "對稱矩陣加正主子式，以配方給所有實向量的正定性，不是只測本題向量。配方的三角變換可逆，故此內積norm與R²座標norm等價；其Cauchy序列座標收斂，極限仍在同一R²。", "en": "Symmetry and positive principal minors give positivity for every real vector by completing the square,not just testing the displayed vectors.The triangular transformation is invertible,so this inner-product norm is equivalent to the coordinate norm on R².A Cauchy sequence has convergent coordinates and its limit remains in this same R²." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Hilbert cross inner product coefficient(0,0) times left coordinate := mul(16, 1)", "after": "Hilbert cross inner product coefficient(0,0) times left coordinate = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product coefficient(0,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert cross inner product coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "1" ], "output": "16", "label": "Hilbert cross inner product coefficient(0,0) times left coordinate" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Hilbert cross inner product term(0,0) := mul(16, 2)", "after": "Hilbert cross inner product term(0,0) = 32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product term(0,0)，不以小數近似替代。", "en": "Recompute Hilbert cross inner product term(0,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "2" ], "output": "32", "label": "Hilbert cross inner product term(0,0)" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Hilbert cross inner product coefficient(0,1) times left coordinate := mul(2, 1)", "after": "Hilbert cross inner product coefficient(0,1) times left coordinate = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product coefficient(0,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert cross inner product coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "Hilbert cross inner product coefficient(0,1) times left coordinate" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Hilbert cross inner product term(0,1) := mul(2, -16)", "after": "Hilbert cross inner product term(0,1) = -32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product term(0,1)，不以小數近似替代。", "en": "Recompute Hilbert cross inner product term(0,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-16" ], "output": "-32", "label": "Hilbert cross inner product term(0,1)" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Hilbert cross inner product coefficient(1,0) times left coordinate := mul(2, 2)", "after": "Hilbert cross inner product coefficient(1,0) times left coordinate = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product coefficient(1,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert cross inner product coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Hilbert cross inner product coefficient(1,0) times left coordinate" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Hilbert cross inner product term(1,0) := mul(4, 2)", "after": "Hilbert cross inner product term(1,0) = 8", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product term(1,0)，不以小數近似替代。", "en": "Recompute Hilbert cross inner product term(1,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "2" ], "output": "8", "label": "Hilbert cross inner product term(1,0)" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Hilbert cross inner product coefficient(1,1) times left coordinate := mul(5, 2)", "after": "Hilbert cross inner product coefficient(1,1) times left coordinate = 10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product coefficient(1,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert cross inner product coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "2" ], "output": "10", "label": "Hilbert cross inner product coefficient(1,1) times left coordinate" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Hilbert cross inner product term(1,1) := mul(10, -16)", "after": "Hilbert cross inner product term(1,1) = -160", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product term(1,1)，不以小數近似替代。", "en": "Recompute Hilbert cross inner product term(1,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "10", "-16" ], "output": "-160", "label": "Hilbert cross inner product term(1,1)" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "Hilbert cross inner product complete bilinear sum := sum(32, -32, 8, -160)", "after": "Hilbert cross inner product complete bilinear sum = -152", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross inner product complete bilinear sum，不以小數近似替代。", "en": "Recompute Hilbert cross inner product complete bilinear sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "32", "-32", "8", "-160" ], "output": "-152", "label": "Hilbert cross inner product complete bilinear sum" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Hilbert vector sum coordinate0 := sum(1, 2)", "after": "Hilbert vector sum coordinate0 = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert vector sum coordinate0，不以小數近似替代。", "en": "Recompute Hilbert vector sum coordinate0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "2" ], "output": "3", "label": "Hilbert vector sum coordinate0" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "Hilbert vector sum coordinate1 := sum(2, -16)", "after": "Hilbert vector sum coordinate1 = -14", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert vector sum coordinate1，不以小數近似替代。", "en": "Recompute Hilbert vector sum coordinate1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "-16" ], "output": "-14", "label": "Hilbert vector sum coordinate1" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Hilbert norm square of sum coefficient(0,0) times left coordinate := mul(16, 3)", "after": "Hilbert norm square of sum coefficient(0,0) times left coordinate = 48", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum coefficient(0,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "3" ], "output": "48", "label": "Hilbert norm square of sum coefficient(0,0) times left coordinate" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Hilbert norm square of sum term(0,0) := mul(48, 3)", "after": "Hilbert norm square of sum term(0,0) = 144", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum term(0,0)，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum term(0,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "48", "3" ], "output": "144", "label": "Hilbert norm square of sum term(0,0)" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Hilbert norm square of sum coefficient(0,1) times left coordinate := mul(2, 3)", "after": "Hilbert norm square of sum coefficient(0,1) times left coordinate = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum coefficient(0,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "3" ], "output": "6", "label": "Hilbert norm square of sum coefficient(0,1) times left coordinate" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "Hilbert norm square of sum term(0,1) := mul(6, -14)", "after": "Hilbert norm square of sum term(0,1) = -84", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum term(0,1)，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum term(0,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "6", "-14" ], "output": "-84", "label": "Hilbert norm square of sum term(0,1)" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Hilbert norm square of sum coefficient(1,0) times left coordinate := mul(2, -14)", "after": "Hilbert norm square of sum coefficient(1,0) times left coordinate = -28", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum coefficient(1,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-14" ], "output": "-28", "label": "Hilbert norm square of sum coefficient(1,0) times left coordinate" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Hilbert norm square of sum term(1,0) := mul(-28, 3)", "after": "Hilbert norm square of sum term(1,0) = -84", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum term(1,0)，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum term(1,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-28", "3" ], "output": "-84", "label": "Hilbert norm square of sum term(1,0)" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "Hilbert norm square of sum coefficient(1,1) times left coordinate := mul(5, -14)", "after": "Hilbert norm square of sum coefficient(1,1) times left coordinate = -70", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum coefficient(1,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "-14" ], "output": "-70", "label": "Hilbert norm square of sum coefficient(1,1) times left coordinate" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Hilbert norm square of sum term(1,1) := mul(-70, -14)", "after": "Hilbert norm square of sum term(1,1) = 980", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum term(1,1)，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum term(1,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-70", "-14" ], "output": "980", "label": "Hilbert norm square of sum term(1,1)" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "Hilbert norm square of sum complete bilinear sum := sum(144, -84, -84, 980)", "after": "Hilbert norm square of sum complete bilinear sum = 956", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert norm square of sum complete bilinear sum，不以小數近似替代。", "en": "Recompute Hilbert norm square of sum complete bilinear sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "144", "-84", "-84", "980" ], "output": "956", "label": "Hilbert norm square of sum complete bilinear sum" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "Hilbert first norm square coefficient(0,0) times left coordinate := mul(16, 1)", "after": "Hilbert first norm square coefficient(0,0) times left coordinate = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square coefficient(0,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert first norm square coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "1" ], "output": "16", "label": "Hilbert first norm square coefficient(0,0) times left coordinate" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "Hilbert first norm square term(0,0) := mul(16, 1)", "after": "Hilbert first norm square term(0,0) = 16", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square term(0,0)，不以小數近似替代。", "en": "Recompute Hilbert first norm square term(0,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "1" ], "output": "16", "label": "Hilbert first norm square term(0,0)" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "Hilbert first norm square coefficient(0,1) times left coordinate := mul(2, 1)", "after": "Hilbert first norm square coefficient(0,1) times left coordinate = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square coefficient(0,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert first norm square coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "1" ], "output": "2", "label": "Hilbert first norm square coefficient(0,1) times left coordinate" } }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "Hilbert first norm square term(0,1) := mul(2, 2)", "after": "Hilbert first norm square term(0,1) = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square term(0,1)，不以小數近似替代。", "en": "Recompute Hilbert first norm square term(0,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Hilbert first norm square term(0,1)" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "Hilbert first norm square coefficient(1,0) times left coordinate := mul(2, 2)", "after": "Hilbert first norm square coefficient(1,0) times left coordinate = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square coefficient(1,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert first norm square coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Hilbert first norm square coefficient(1,0) times left coordinate" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "Hilbert first norm square term(1,0) := mul(4, 1)", "after": "Hilbert first norm square term(1,0) = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square term(1,0)，不以小數近似替代。", "en": "Recompute Hilbert first norm square term(1,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "1" ], "output": "4", "label": "Hilbert first norm square term(1,0)" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "Hilbert first norm square coefficient(1,1) times left coordinate := mul(5, 2)", "after": "Hilbert first norm square coefficient(1,1) times left coordinate = 10", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square coefficient(1,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert first norm square coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "2" ], "output": "10", "label": "Hilbert first norm square coefficient(1,1) times left coordinate" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "Hilbert first norm square term(1,1) := mul(10, 2)", "after": "Hilbert first norm square term(1,1) = 20", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square term(1,1)，不以小數近似替代。", "en": "Recompute Hilbert first norm square term(1,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "10", "2" ], "output": "20", "label": "Hilbert first norm square term(1,1)" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "Hilbert first norm square complete bilinear sum := sum(16, 4, 4, 20)", "after": "Hilbert first norm square complete bilinear sum = 44", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert first norm square complete bilinear sum，不以小數近似替代。", "en": "Recompute Hilbert first norm square complete bilinear sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "16", "4", "4", "20" ], "output": "44", "label": "Hilbert first norm square complete bilinear sum" } }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "Hilbert second norm square coefficient(0,0) times left coordinate := mul(16, 2)", "after": "Hilbert second norm square coefficient(0,0) times left coordinate = 32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square coefficient(0,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert second norm square coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "16", "2" ], "output": "32", "label": "Hilbert second norm square coefficient(0,0) times left coordinate" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "Hilbert second norm square term(0,0) := mul(32, 2)", "after": "Hilbert second norm square term(0,0) = 64", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square term(0,0)，不以小數近似替代。", "en": "Recompute Hilbert second norm square term(0,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "32", "2" ], "output": "64", "label": "Hilbert second norm square term(0,0)" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "Hilbert second norm square coefficient(0,1) times left coordinate := mul(2, 2)", "after": "Hilbert second norm square coefficient(0,1) times left coordinate = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square coefficient(0,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert second norm square coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "Hilbert second norm square coefficient(0,1) times left coordinate" } }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "Hilbert second norm square term(0,1) := mul(4, -16)", "after": "Hilbert second norm square term(0,1) = -64", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square term(0,1)，不以小數近似替代。", "en": "Recompute Hilbert second norm square term(0,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "4", "-16" ], "output": "-64", "label": "Hilbert second norm square term(0,1)" } }, { "id": "S040", "uses": [ "G001", "S039" ], "before": "Hilbert second norm square coefficient(1,0) times left coordinate := mul(2, -16)", "after": "Hilbert second norm square coefficient(1,0) times left coordinate = -32", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square coefficient(1,0) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert second norm square coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-16" ], "output": "-32", "label": "Hilbert second norm square coefficient(1,0) times left coordinate" } }, { "id": "S041", "uses": [ "G001", "S040" ], "before": "Hilbert second norm square term(1,0) := mul(-32, 2)", "after": "Hilbert second norm square term(1,0) = -64", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square term(1,0)，不以小數近似替代。", "en": "Recompute Hilbert second norm square term(1,0) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-32", "2" ], "output": "-64", "label": "Hilbert second norm square term(1,0)" } }, { "id": "S042", "uses": [ "G001", "S041" ], "before": "Hilbert second norm square coefficient(1,1) times left coordinate := mul(5, -16)", "after": "Hilbert second norm square coefficient(1,1) times left coordinate = -80", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square coefficient(1,1) times left coordinate，不以小數近似替代。", "en": "Recompute Hilbert second norm square coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "5", "-16" ], "output": "-80", "label": "Hilbert second norm square coefficient(1,1) times left coordinate" } }, { "id": "S043", "uses": [ "G001", "S042" ], "before": "Hilbert second norm square term(1,1) := mul(-80, -16)", "after": "Hilbert second norm square term(1,1) = 1280", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square term(1,1)，不以小數近似替代。", "en": "Recompute Hilbert second norm square term(1,1) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-80", "-16" ], "output": "1280", "label": "Hilbert second norm square term(1,1)" } }, { "id": "S044", "uses": [ "G001", "S043" ], "before": "Hilbert second norm square complete bilinear sum := sum(64, -64, -64, 1280)", "after": "Hilbert second norm square complete bilinear sum = 1216", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert second norm square complete bilinear sum，不以小數近似替代。", "en": "Recompute Hilbert second norm square complete bilinear sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "64", "-64", "-64", "1280" ], "output": "1216", "label": "Hilbert second norm square complete bilinear sum" } }, { "id": "S045", "uses": [ "G001", "S044" ], "before": "Hilbert sum of separate norm squares := sum(44, 1216)", "after": "Hilbert sum of separate norm squares = 1260", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert sum of separate norm squares，不以小數近似替代。", "en": "Recompute Hilbert sum of separate norm squares with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "44", "1216" ], "output": "1260", "label": "Hilbert sum of separate norm squares" } }, { "id": "S046", "uses": [ "G001", "S045" ], "before": "Hilbert cross term twice inner product := mul(2, -152)", "after": "Hilbert cross term twice inner product = -304", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert cross term twice inner product，不以小數近似替代。", "en": "Recompute Hilbert cross term twice inner product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "-152" ], "output": "-304", "label": "Hilbert cross term twice inner product" } }, { "id": "S047", "uses": [ "G001", "S046" ], "before": "Hilbert expanded square including cross term := sum(1260, -304)", "after": "Hilbert expanded square including cross term = 956", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色與已核中間量精確計算；保留分母、嚴格條件、索引及正負號。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Hilbert expanded square including cross term，不以小數近似替代。", "en": "Recompute Hilbert expanded square including cross term with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1260", "-304" ], "output": "956", "label": "Hilbert expanded square including cross term" } }, { "id": "S048", "uses": [ "G001", "S047" ], "before": "same inner product=-152;norm square of sum=956;separate squares sum=1260;twice cross term=-304", "after": "bilinear expansion=956;orthogonality premise=否;Pythagorean conclusion status=orthogonality_premise_false", "relation": "implies", "rule_id": "hilbert_pythagorean_requires_orthogonality_not_counterexample", "conditions": [], "explanation": { "zh-Hant": "恆等式是||x+y||²=||x||²+2〈x,y〉+||y||²；只有內積為零才刪交叉項。非零時回報前提不符，不把它說成畢氏定理反例。", "en": "The universal identity includes2〈x,y〉.Remove the cross term only when the inner product is zero.A nonzero result reports a failed premise,not a counterexample to the Pythagorean 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 } ], "math_model": { "result": [ [ 1, 2 ], [ 2, -16 ], -152, 956, 1260, "orthogonality_premise_false" ] }, "conclusion": { "canonical_value": "([1, 2], [2, -16], -152, 956, 1260, orthogonality_premise_false)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261030-r02-real-hilbert-space-foundations", "batch_path": "public/data/questions/revisions/ocme-parameter-revision-20261017/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations/questions.jsonl", "batch_sha256": "4e3813f5a6d26ef66584109baa548d64ea1c9ec1d0158cd3f4cfa19fdb5990e1", "original_line": 90, "original_line_sha256": "6b5b4ff07f935501ee338478c099b085d1c3c4c38eab647bf71b64d1ecd33b4b", "question_compact_sha256": "449e2ec26b4f20f92fdfec792d284031e2989a1e8ac12cd2735c4e9f4e09e5ac", "english_row_compact_sha256": "6000b68c84f11459b057008727bd4e54ab72ab33a5ad511fe77652deca208623", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations.en.jsonl" }, "mko_bindings": { "mko-real-hilbert-space": { "sha256": "89b1eab58b853963348761bf140ffff91d2903053edbdf9baf157b32a5ac731e", "english_sha256": "3ac4455274e5bba35d48841ef7df59ae0ed9e3c9645b4035ebd5961f396f7052", "original_path": "public/data/mko/real-hilbert-space.json", "english_path": "public/data/i18n/editions/mko-real-hilbert-space.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "54de60dca082018923bf089f2b1eb0a6666cc401916115af792056c8bae935db", "original_candidate_sha256": "69204e49a887026f96a91693f06a69e29b5cb6f7ce4ebe33741c8a86c976c9e2", "original_candidate_line_sha256": "69204e49a887026f96a91693f06a69e29b5cb6f7ce4ebe33741c8a86c976c9e2" }, "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-20261026-007-p30", "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": "8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725", "question_first_witness_sha256": "c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 48, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "cd98638641ec29a654fa3d41e482bd89f4aaaef53ffe3e581e13985a401612e6", "arithmetic_steps": 46 }, "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." } }