{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-pulse-20261119-r06-0044", "pilot_number": "R0066-33894", "number": 33894, "family_id": "qf-mod50-graddeep-negative-direction", "title": { "zh-Hant": "第 33894 題：梯度向量", "en": "Question 33894: A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-pulse-20261119-r06-0044", "batch_id": "batch-20261015-advance-20261119-r06-gradient-vector-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-gradient-vector" ], "title_zh": "梯度向量", "stem_zh": "fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[19,-19]}；請依 exact_ordered_tuple_declared_by_interface 回答。", "answer": { "kind": "expression", "value": "([19, -19], [-19, 19], -722, 722, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)", "explanation_zh": "Wave6第十三層核對=([19, -19], [-19, 19], -722, 722, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "梯度向量" ] }, "variant": { "family_id": "qf-mod50-graddeep-negative-direction", "seed": 202611196044, "parameters": { "gradient": [ 19, -19 ], "domain": "Exact_gradient_vector_density", "operation": "graddeep-negative-direction", "semantic_template": "graddeep-negative-direction|{\"gradient\":[[19,1],[-19,1]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_contract_recomputation", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-pulse-20261119-r06-0044", "batch_id": "batch-20261015-advance-20261119-r06-gradient-vector-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b5eb3524a7d7d4c99ca8f43336dd35e81a0d39ef93f84911d0ad69ee9eb5f107", "family_id": "qf-mod50-graddeep-negative-direction", "seed": 202611196044, "semantic_template": "graddeep-negative-direction|{\"gradient\":[[19,1],[-19,1]]}" }, "fields": { "title": "A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product", "stem": "Use nonzero Euclidean gradient=[19, -19] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope).", "answer": { "kind": "expression", "value": "([19, -19], [-19, 19], -722, 722, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)", "explanation": "Set the unnormalized direction to minus the gradient and compute its inner product with the gradient as minus the squared norm. This is a first-order directional fact, not a guarantee that every positive finite step decreases the objective. The exact result, retaining the requested order, is ([19, -19], [-19, 19], -722, 722, Yes, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)." }, "skills": [ "A Nonzero Negative Gradient Has Strictly Negative Directional Inner Product" ], "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": "fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[19,-19]}；請依 exact_ordered_tuple_declared_by_interface 回答。", "en": "Use nonzero Euclidean gradient=[19, -19] at a differentiable point. Give (gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod50-graddeep-negative-direction, [domain, gradient, operation]]", "after": "derived = [graddeep-negative-direction, Exact_gradient_vector_density]", "relation": "implies", "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain", "conditions": [], "explanation": { "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。", "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [negative direction ORIGINAL gradient raw coordinate0, 100000000, 19, 1]", "after": "derived = [[19, 1], 19]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator := div(19, 1)", "after": "negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator = 19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "19", "1" ], "output": "19", "label": "negative direction ORIGINAL gradient raw coordinate0 ORIGINAL numerator over denominator" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [negative direction ORIGINAL gradient raw coordinate1, 100000000, -19, 1]", "after": "derived = [[-19, 1], -19]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator := div(-19, 1)", "after": "negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator = -19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-19", "1" ], "output": "-19", "label": "negative direction ORIGINAL gradient raw coordinate1 ORIGINAL numerator over denominator" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [negative direction ORIGINAL gradient, [19, -19]]", "after": "derived = [[19, -19]]", "relation": "implies", "rule_id": "R66_VECTOR_ALL_original_order_typed_nonempty", "conditions": [], "explanation": { "zh-Hant": "先驗完整raw vector的每個分量、非空與長度，保持原座標序；不能只驗將用的cell。", "en": "Validate EVERY raw vector component,nonemptiness and required length BEFORE use.Keep ORIGINAL coordinate order,including cells not ultimately selected." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Negative gradient coordinate0 SOURCE naive mul, 100000000, -19, 1]", "after": "derived = [[-19, 1], -19]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator := div(-19, 1)", "after": "Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator = -19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-19", "1" ], "output": "-19", "label": "Negative gradient coordinate0 SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Negative gradient coordinate0 ORIGINAL rational mul := mul(-1, 19)", "after": "Negative gradient coordinate0 ORIGINAL rational mul = -19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Negative gradient coordinate0 ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute Negative gradient coordinate0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "19" ], "output": "-19", "label": "Negative gradient coordinate0 ORIGINAL rational mul" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [Negative gradient coordinate1 SOURCE naive mul, 100000000, 19, 1]", "after": "derived = [[19, 1], 19]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator := div(19, 1)", "after": "Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator = 19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "19", "1" ], "output": "19", "label": "Negative gradient coordinate1 SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Negative gradient coordinate1 ORIGINAL rational mul := mul(-1, -19)", "after": "Negative gradient coordinate1 ORIGINAL rational mul = 19", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Negative gradient coordinate1 ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute Negative gradient coordinate1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "-19" ], "output": "19", "label": "Negative gradient coordinate1 ORIGINAL rational mul" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "source roles = [negative direction inner product product0 SOURCE naive mul, 100000000, -361, 1]", "after": "derived = [[-361, 1], -361]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator := div(-361, 1)", "after": "negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator = -361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-361", "1" ], "output": "-361", "label": "negative direction inner product product0 SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "negative direction inner product product0 ORIGINAL rational mul := mul(19, -19)", "after": "negative direction inner product product0 ORIGINAL rational mul = -361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product product0 ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute negative direction inner product product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "19", "-19" ], "output": "-361", "label": "negative direction inner product product0 ORIGINAL rational mul" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "source roles = [negative direction inner product prefix0 SOURCE naive sum, 100000000, -361, 1]", "after": "derived = [[-361, 1], -361]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(-361, 1)", "after": "negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator = -361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-361", "1" ], "output": "-361", "label": "negative direction inner product prefix0 SOURCE naive sum ORIGINAL numerator over denominator" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "negative direction inner product prefix0 ORIGINAL rational sum := sum(0, -361)", "after": "negative direction inner product prefix0 ORIGINAL rational sum = -361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product prefix0 ORIGINAL rational sum，不以小數近似替代。", "en": "Recompute negative direction inner product prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-361" ], "output": "-361", "label": "negative direction inner product prefix0 ORIGINAL rational sum" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "source roles = [negative direction inner product product1 SOURCE naive mul, 100000000, -361, 1]", "after": "derived = [[-361, 1], -361]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator := div(-361, 1)", "after": "negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator = -361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-361", "1" ], "output": "-361", "label": "negative direction inner product product1 SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "negative direction inner product product1 ORIGINAL rational mul := mul(-19, 19)", "after": "negative direction inner product product1 ORIGINAL rational mul = -361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product product1 ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute negative direction inner product product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-19", "19" ], "output": "-361", "label": "negative direction inner product product1 ORIGINAL rational mul" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "source roles = [negative direction inner product prefix1 SOURCE naive sum, 100000000, -722, 1]", "after": "derived = [[-722, 1], -722]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(-722, 1)", "after": "negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator = -722", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "-722", "1" ], "output": "-722", "label": "negative direction inner product prefix1 SOURCE naive sum ORIGINAL numerator over denominator" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "negative direction inner product prefix1 ORIGINAL rational sum := sum(-361, -361)", "after": "negative direction inner product prefix1 ORIGINAL rational sum = -722", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction inner product prefix1 ORIGINAL rational sum，不以小數近似替代。", "en": "Recompute negative direction inner product prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-361", "-361" ], "output": "-722", "label": "negative direction inner product prefix1 ORIGINAL rational sum" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "source roles = [negative direction inner product, [19, -19], [-19, 19]]", "after": "derived = [-722]", "relation": "implies", "rule_id": "R66_DOT_complete_ordered_terms_and_source_prefixes", "conditions": [], "explanation": { "zh-Hant": "逐對乘積再依來源次序從0累加，核所有pre-reduction中間界；不可只核最後值或換序跳過失敗的prefix。", "en": "Multiply EVERY aligned pair,then accumulate from0 in SOURCE order.Check every pre-reduction intermediate;the final value alone or a reordered sum cannot bypass a failing prefix." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "source roles = [negative direction norm squared product0 SOURCE naive mul, 100000000, 361, 1]", "after": "derived = [[361, 1], 361]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator := div(361, 1)", "after": "negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator = 361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "361", "1" ], "output": "361", "label": "negative direction norm squared product0 SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "negative direction norm squared product0 ORIGINAL rational mul := mul(19, 19)", "after": "negative direction norm squared product0 ORIGINAL rational mul = 361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared product0 ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute negative direction norm squared product0 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "19", "19" ], "output": "361", "label": "negative direction norm squared product0 ORIGINAL rational mul" } }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "source roles = [negative direction norm squared prefix0 SOURCE naive sum, 100000000, 361, 1]", "after": "derived = [[361, 1], 361]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator := div(361, 1)", "after": "negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator = 361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "361", "1" ], "output": "361", "label": "negative direction norm squared prefix0 SOURCE naive sum ORIGINAL numerator over denominator" } }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "negative direction norm squared prefix0 ORIGINAL rational sum := sum(0, 361)", "after": "negative direction norm squared prefix0 ORIGINAL rational sum = 361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared prefix0 ORIGINAL rational sum，不以小數近似替代。", "en": "Recompute negative direction norm squared prefix0 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "361" ], "output": "361", "label": "negative direction norm squared prefix0 ORIGINAL rational sum" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "source roles = [negative direction norm squared product1 SOURCE naive mul, 100000000, 361, 1]", "after": "derived = [[361, 1], 361]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator := div(361, 1)", "after": "negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator = 361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "361", "1" ], "output": "361", "label": "negative direction norm squared product1 SOURCE naive mul ORIGINAL numerator over denominator" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "negative direction norm squared product1 ORIGINAL rational mul := mul(-19, -19)", "after": "negative direction norm squared product1 ORIGINAL rational mul = 361", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared product1 ORIGINAL rational mul，不以小數近似替代。", "en": "Recompute negative direction norm squared product1 ORIGINAL rational mul with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-19", "-19" ], "output": "361", "label": "negative direction norm squared product1 ORIGINAL rational mul" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "source roles = [negative direction norm squared prefix1 SOURCE naive sum, 100000000, 722, 1]", "after": "derived = [[722, 1], 722]", "relation": "implies", "rule_id": "R66_SOURCE_PREreduction_components_then_sign_gcd", "conditions": [], "explanation": { "zh-Hant": "先驗來源傳入rational的原num/den範圍與非零den，之後才sign/gcd約分。MOD50界1億且naive運算不先cross-cancel；Wave4此均值界100萬。超界是來源切片不合，不是數學為假。", "en": "Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S036", "uses": [ "G001", "S035" ], "before": "negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator := div(722, 1)", "after": "negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator = 722", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。", "en": "Recompute negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "722", "1" ], "output": "722", "label": "negative direction norm squared prefix1 SOURCE naive sum ORIGINAL numerator over denominator" } }, { "id": "S037", "uses": [ "G001", "S036" ], "before": "negative direction norm squared prefix1 ORIGINAL rational sum := sum(361, 361)", "after": "negative direction norm squared prefix1 ORIGINAL rational sum = 722", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 negative direction norm squared prefix1 ORIGINAL rational sum，不以小數近似替代。", "en": "Recompute negative direction norm squared prefix1 ORIGINAL rational sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "361", "361" ], "output": "722", "label": "negative direction norm squared prefix1 ORIGINAL rational sum" } }, { "id": "S038", "uses": [ "G001", "S037" ], "before": "source roles = [negative direction norm squared, [19, -19], [19, -19]]", "after": "derived = [722]", "relation": "implies", "rule_id": "R66_DOT_complete_ordered_terms_and_source_prefixes", "conditions": [], "explanation": { "zh-Hant": "逐對乘積再依來源次序從0累加，核所有pre-reduction中間界；不可只核最後值或換序跳過失敗的prefix。", "en": "Multiply EVERY aligned pair,then accumulate from0 in SOURCE order.Check every pre-reduction intermediate;the final value alone or a reordered sum cannot bypass a failing prefix." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S039", "uses": [ "G001", "S038" ], "before": "source roles = [[19, -19]]", "after": "derived = [[-19, 19], -722, 722, 是]", "relation": "implies", "rule_id": "Gradient_NEGATIVE_direction_NONZERO_first_order_NOT_finite_step", "conditions": [], "explanation": { "zh-Hant": "非零gradient才有正norm²，方向是未正規化−gradient，內積=−norm²<0。可微前提給局部一階下降方向，不保每個正有限步長都降。", "en": "A NONZERO gradient has positive squared norm.The UNNORMALIZED direction−gradient gives inner product−norm²<0.Differentiability supports a first-order descent direction,not descent for every positive finite step." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null } ], "math_model": { "result": [ [ "19", "-19" ], [ "-19", "19" ], "-722", "722", true, "negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point" ] }, "conclusion": { "canonical_value": "([19, -19], [-19, 19], -722, 722, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)", "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-20261015-advance-20261119-r06-gradient-vector-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261119-r06-gradient-vector-foundations/questions.jsonl", "batch_sha256": "7aeac0f5d0efe49c24d6c45348a1a0f24e40ba206983593e56a4cf547072e1ff", "original_line": 44, "original_line_sha256": "78bcedab0cb9864ae5944a6b10e9cfacb03dd4732f176018c71181c3fa1e2ed8", "question_compact_sha256": "b5eb3524a7d7d4c99ca8f43336dd35e81a0d39ef93f84911d0ad69ee9eb5f107", "english_row_compact_sha256": "702bffcd042751a4cfa4801e88ed04237c3778e79cc6963d79b052d51ff1bdc3", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261119-r06-gradient-vector-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261119-r06-gradient-vector-foundations.en.jsonl" }, "mko_bindings": { "mko-gradient-vector": { "sha256": "953e582b90145e3769b9b22891783891e5dbfe6a99446edc1b06a277683029fa", "english_sha256": "6ba53da50d59e50b734006c70a0c1e5cb7e4986a4a57e6d0b1f27a8ae84bd363", "original_path": "public/data/mko/gradient-vector.json", "english_path": "public/data/i18n/editions/mko-gradient-vector.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "df58fcce25614251047f7c41d31ff93b5bfec19cd9bbcaa5fbb42a4daceb0310", "original_candidate_sha256": "2f4acee29d7ead73cf70ed451f5f44ca172b8ea31c28938bf96413257a7cb46a", "original_candidate_line_sha256": "2f4acee29d7ead73cf70ed451f5f44ca172b8ea31c28938bf96413257a7cb46a" }, "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-20261026-012-p67", "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": "99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267", "question_first_witness_sha256": "d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 39, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "c408ecdf789f8b151b49b38e8574fc587456525b83971766125928a0282ebc9b", "arithmetic_steps": 22 }, "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." } }