{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261118-r05-p-adic-valuation-integers-0073", "pilot_number": "R0053-27498", "number": 27498, "family_id": "qf-mod49-pval-sum-ultrametric", "title": { "zh-Hant": "第 27498 題：整數p進賦值", "en": "Question 27498: Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261118-r05-p-adic-valuation-integers-0073", "batch_id": "batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-p-adic-valuation-integers" ], "title_zh": "整數p進賦值", "stem_zh": "prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":11,\"a\":2783,\"b\":2904}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(11, 2783, 2904, 2, 2, 5687, 2, 2, 是, 否, not_guaranteed)", "explanation_zh": "Wave6第十二層核對=(11, 2783, 2904, 2, 2, 5687, 2, 2, 是, 否, not_guaranteed)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "整數p進賦值" ] }, "variant": { "family_id": "qf-mod49-pval-sum-ultrametric", "seed": 202611185073, "parameters": { "p": 11, "a": 2783, "b": 2904, "domain": "Exact_integer_p_adic_valuation", "operation": "pval-sum-ultrametric", "semantic_template": "pval-sum-ultrametric|{a:2783,b:2904,p:11}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "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-advance-20261118-r05-p-adic-valuation-integers-0073", "batch_id": "batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3217e2d9348a6c2584c39a9539e55492f62f4dd86d5d190d0a35e0f7d390229c", "family_id": "qf-mod49-pval-sum-ultrametric", "seed": 202611185073, "semantic_template": "pval-sum-ultrametric|{a:2783,b:2904,p:11}" }, "fields": { "title": "Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations", "stem": "Use prime p=11, nonzero summands a=2783, b=2904; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).", "answer": { "kind": "expression", "value": "(11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed)", "explanation": "Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed)." }, "skills": [ "Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations" ], "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": "prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-sum-ultrametric；資料={\"p\":11,\"a\":2783,\"b\":2904}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use prime p=11, nonzero summands a=2783, b=2904; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [pval-sum-ultrametric]", "after": "derived = [[p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。", "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Valuation ORIGINAL a ORIGINAL division step1 := div(2783, 11)", "after": "Valuation ORIGINAL a ORIGINAL division step1 = 253", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a ORIGINAL division step1，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL a ORIGINAL division step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2783", "11" ], "output": "253", "label": "Valuation ORIGINAL a ORIGINAL division step1" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Valuation ORIGINAL a dividing prime power1 := mul(1, 11)", "after": "Valuation ORIGINAL a dividing prime power1 = 11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a dividing prime power1，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL a dividing prime power1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "11" ], "output": "11", "label": "Valuation ORIGINAL a dividing prime power1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Valuation ORIGINAL a, 11, 2783, 2783, 1]", "after": "derived = [253, 11, 0, 是]", "relation": "implies", "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant", "conditions": [], "explanation": { "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。", "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Valuation ORIGINAL a ORIGINAL division step2 := div(253, 11)", "after": "Valuation ORIGINAL a ORIGINAL division step2 = 23", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a ORIGINAL division step2，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL a ORIGINAL division step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "253", "11" ], "output": "23", "label": "Valuation ORIGINAL a ORIGINAL division step2" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Valuation ORIGINAL a dividing prime power2 := mul(11, 11)", "after": "Valuation ORIGINAL a dividing prime power2 = 121", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a dividing prime power2，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL a dividing prime power2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "11" ], "output": "121", "label": "Valuation ORIGINAL a dividing prime power2" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [Valuation ORIGINAL a, 11, 2783, 253, 2]", "after": "derived = [23, 121, 0, 是]", "relation": "implies", "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant", "conditions": [], "explanation": { "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。", "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "Valuation ORIGINAL a terminal factorization product := mul(121, 23)", "after": "Valuation ORIGINAL a terminal factorization product = 2783", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a terminal factorization product，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL a terminal factorization product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "121", "23" ], "output": "2783", "label": "Valuation ORIGINAL a terminal factorization product" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [Valuation ORIGINAL a, 11, 2783, 2783, 2, 23, [11, 121]]", "after": "derived = [2783, 1, 是, 是]", "relation": "implies", "rule_id": "Valuation_terminal_NONdivisible_unit_highest_power", "conditions": [], "explanation": { "zh-Hant": "停止時p不整除unit，所以p^(k+1)不整除|a|，k就是最高指數。負號不改整除性，factorization回到|a|，不是把負a等於正product。", "en": "At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Valuation ORIGINAL b ORIGINAL division step1 := div(2904, 11)", "after": "Valuation ORIGINAL b ORIGINAL division step1 = 264", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b ORIGINAL division step1，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL b ORIGINAL division step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "2904", "11" ], "output": "264", "label": "Valuation ORIGINAL b ORIGINAL division step1" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Valuation ORIGINAL b dividing prime power1 := mul(1, 11)", "after": "Valuation ORIGINAL b dividing prime power1 = 11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b dividing prime power1，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL b dividing prime power1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "11" ], "output": "11", "label": "Valuation ORIGINAL b dividing prime power1" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "source roles = [Valuation ORIGINAL b, 11, 2904, 2904, 1]", "after": "derived = [264, 11, 0, 是]", "relation": "implies", "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant", "conditions": [], "explanation": { "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。", "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "Valuation ORIGINAL b ORIGINAL division step2 := div(264, 11)", "after": "Valuation ORIGINAL b ORIGINAL division step2 = 24", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b ORIGINAL division step2，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL b ORIGINAL division step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "264", "11" ], "output": "24", "label": "Valuation ORIGINAL b ORIGINAL division step2" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "Valuation ORIGINAL b dividing prime power2 := mul(11, 11)", "after": "Valuation ORIGINAL b dividing prime power2 = 121", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b dividing prime power2，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL b dividing prime power2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "11" ], "output": "121", "label": "Valuation ORIGINAL b dividing prime power2" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "source roles = [Valuation ORIGINAL b, 11, 2904, 264, 2]", "after": "derived = [24, 121, 0, 是]", "relation": "implies", "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant", "conditions": [], "explanation": { "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。", "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "Valuation ORIGINAL b terminal factorization product := mul(121, 24)", "after": "Valuation ORIGINAL b terminal factorization product = 2904", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b terminal factorization product，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL b terminal factorization product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "121", "24" ], "output": "2904", "label": "Valuation ORIGINAL b terminal factorization product" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "source roles = [Valuation ORIGINAL b, 11, 2904, 2904, 2, 24, [11, 121]]", "after": "derived = [2904, 2, 是, 是]", "relation": "implies", "rule_id": "Valuation_terminal_NONdivisible_unit_highest_power", "conditions": [], "explanation": { "zh-Hant": "停止時p不整除unit，所以p^(k+1)不整除|a|，k就是最高指數。負號不改整除性，factorization回到|a|，不是把負a等於正product。", "en": "At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "Valuation ORIGINAL a plus b := sum(2783, 2904)", "after": "Valuation ORIGINAL a plus b = 5687", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a plus b，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL a plus b with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2783", "2904" ], "output": "5687", "label": "Valuation ORIGINAL a plus b" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "Valuation ACTUAL sum ORIGINAL division step1 := div(5687, 11)", "after": "Valuation ACTUAL sum ORIGINAL division step1 = 517", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ACTUAL sum ORIGINAL division step1，不以小數近似替代。", "en": "Recompute Valuation ACTUAL sum ORIGINAL division step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "5687", "11" ], "output": "517", "label": "Valuation ACTUAL sum ORIGINAL division step1" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "Valuation ACTUAL sum dividing prime power1 := mul(1, 11)", "after": "Valuation ACTUAL sum dividing prime power1 = 11", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ACTUAL sum dividing prime power1，不以小數近似替代。", "en": "Recompute Valuation ACTUAL sum dividing prime power1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "11" ], "output": "11", "label": "Valuation ACTUAL sum dividing prime power1" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "source roles = [Valuation ACTUAL sum, 11, 5687, 5687, 1]", "after": "derived = [517, 11, 0, 是]", "relation": "implies", "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant", "conditions": [], "explanation": { "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。", "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "Valuation ACTUAL sum ORIGINAL division step2 := div(517, 11)", "after": "Valuation ACTUAL sum ORIGINAL division step2 = 47", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ACTUAL sum ORIGINAL division step2，不以小數近似替代。", "en": "Recompute Valuation ACTUAL sum ORIGINAL division step2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "517", "11" ], "output": "47", "label": "Valuation ACTUAL sum ORIGINAL division step2" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "Valuation ACTUAL sum dividing prime power2 := mul(11, 11)", "after": "Valuation ACTUAL sum dividing prime power2 = 121", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ACTUAL sum dividing prime power2，不以小數近似替代。", "en": "Recompute Valuation ACTUAL sum dividing prime power2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "11", "11" ], "output": "121", "label": "Valuation ACTUAL sum dividing prime power2" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "source roles = [Valuation ACTUAL sum, 11, 5687, 517, 2]", "after": "derived = [47, 121, 0, 是]", "relation": "implies", "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant", "conditions": [], "explanation": { "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。", "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "Valuation ACTUAL sum terminal factorization product := mul(121, 47)", "after": "Valuation ACTUAL sum terminal factorization product = 5687", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。", "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Valuation ACTUAL sum terminal factorization product，不以小數近似替代。", "en": "Recompute Valuation ACTUAL sum terminal factorization product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "121", "47" ], "output": "5687", "label": "Valuation ACTUAL sum terminal factorization product" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "source roles = [Valuation ACTUAL sum, 11, 5687, 5687, 2, 47, [11, 121]]", "after": "derived = [5687, 3, 是, 是]", "relation": "implies", "rule_id": "Valuation_terminal_NONdivisible_unit_highest_power", "conditions": [], "explanation": { "zh-Hant": "停止時p不整除unit，所以p^(k+1)不整除|a|，k就是最高指數。負號不改整除性，factorization回到|a|，不是把負a等於正product。", "en": "At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "source roles = [11, 2783, 2904, 2, 2, 2]", "after": "derived = [5687, 2, 是, 否, not_guaranteed]", "relation": "implies", "rule_id": "Valuation_GENERAL_common_power_ultrametric_AND_unequal_case", "conditions": [], "explanation": { "zh-Hant": "提取共同p^min後剩整數和，故至少有min個p因子。若r