{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091", "pilot_number": "R0053-27416", "number": 27416, "family_id": "qf-mod49-pval-boundary", "title": { "zh-Hant": "第 27416 題：整數p進賦值", "en": "Question 27416: Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261118-r05-p-adic-valuation-integers-0091", "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-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":7,\"a\":112}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)", "explanation_zh": "Wave6第十二層核對=(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)；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": 2, "skills": [ "整數p進賦值" ] }, "variant": { "family_id": "qf-mod49-pval-boundary", "seed": 202611185091, "parameters": { "mode": "equal_valuations_cancellation", "p": 7, "a": 112, "domain": "Exact_integer_p_adic_valuation", "operation": "pval-boundary", "semantic_template": "pval-boundary|{a:112,mode:\"equal_valuations_cancellation\",p:7}" } }, "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-0091", "batch_id": "batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "3dac714ea461ba34a134b4bcd2be272c2c0e428810bdf15c7bab4a01395be526", "family_id": "qf-mod49-pval-boundary", "seed": 202611185091, "semantic_template": "pval-boundary|{a:112,mode:\"equal_valuations_cancellation\",p:7}" }, "fields": { "title": "Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions", "stem": "Use mode=equal_valuations_cancellation, base p=7, integer a=112. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).", "answer": { "kind": "expression", "value": "(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)", "explanation": "Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)." }, "skills": [ "Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions" ], "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-boundary；資料={\"mode\":\"equal_valuations_cancellation\",\"p\":7,\"a\":112}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=equal_valuations_cancellation, base p=7, integer a=112. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [pval-boundary]", "after": "derived = [[mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim]]", "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 boundary ORIGINAL a ORIGINAL division step1 := div(112, 7)", "after": "Valuation boundary ORIGINAL a ORIGINAL division step1 = 16", "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 boundary ORIGINAL a ORIGINAL division step1，不以小數近似替代。", "en": "Recompute Valuation boundary ORIGINAL a ORIGINAL division step1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "112", "7" ], "output": "16", "label": "Valuation boundary ORIGINAL a ORIGINAL division step1" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Valuation boundary ORIGINAL a dividing prime power1 := mul(1, 7)", "after": "Valuation boundary ORIGINAL a dividing prime power1 = 7", "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 boundary ORIGINAL a dividing prime power1，不以小數近似替代。", "en": "Recompute Valuation boundary ORIGINAL a dividing prime power1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "7" ], "output": "7", "label": "Valuation boundary ORIGINAL a dividing prime power1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [Valuation boundary ORIGINAL a, 7, 112, 112, 1]", "after": "derived = [16, 7, 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 boundary ORIGINAL a terminal factorization product := mul(7, 16)", "after": "Valuation boundary ORIGINAL a terminal factorization product = 112", "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 boundary ORIGINAL a terminal factorization product，不以小數近似替代。", "en": "Recompute Valuation boundary ORIGINAL a terminal factorization product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "7", "16" ], "output": "112", "label": "Valuation boundary ORIGINAL a terminal factorization product" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [Valuation boundary ORIGINAL a, 7, 112, 112, 1, 16, [7]]", "after": "derived = [112, 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": "S007", "uses": [ "G001", "S006" ], "before": "Valuation ORIGINAL opposite cancellation sum := sum(112, -112)", "after": "Valuation ORIGINAL opposite cancellation sum = 0", "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 opposite cancellation sum，不以小數近似替代。", "en": "Recompute Valuation ORIGINAL opposite cancellation sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "112", "-112" ], "output": "0", "label": "Valuation ORIGINAL opposite cancellation sum" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [Valuation boundary ACTUAL cancellation, 7, 0]", "after": "derived = [infinity, 0, 0, []]", "relation": "implies", "rule_id": "Valuation_zero_SINGLE_infinity_convention", "conditions": [], "explanation": { "zh-Hant": "零可被每個p冪整除，沒有最高有限指數；只用一個可序列化infinity sentinel，不當成數字0或未定義錯誤。", "en": "Zero is divisible by every power of p,so has no highest finite exponent.Use the single serializable infinity sentinel,not numeric0 or an undefined error." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [7, 112, -112, 1]", "after": "derived = [1, infinity, 否]", "relation": "implies", "rule_id": "Valuation_equal_values_cancellation_refutes_UNIVERSAL_equality", "conditions": [], "explanation": { "zh-Hant": "v_p(-a)=v_p(a)；相反數和0的賦值是infinity，比有限k嚴格大。因此兩值相等時不能保證sum valuation等於minimum。", "en": "v_p(-a)=v_p(a),while the opposite sum is0 with valuation infinity,strictly above finite k.Equal summand valuations therefore do not guarantee equality with their minimum." }, "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 } ], "math_model": { "result": [ "equal_valuations_cancellation", 7, 112, -112, 1, 1, "infinity", "strict_inequality_or_infinity_can_occur_when_valuations_equal", false ] }, "conclusion": { "canonical_value": "(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)", "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-20261118-r05-p-adic-valuation-integers-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations/questions.jsonl", "batch_sha256": "a2f6d43cd783e7dd6234ba65b2293735a102a99a2e2b7cb02a64e5f2431ede08", "original_line": 91, "original_line_sha256": "20051c4f5c83b2c47c053349cf4853f84e595ea64be72cb75cbc27aafd2d2ef5", "question_compact_sha256": "3dac714ea461ba34a134b4bcd2be272c2c0e428810bdf15c7bab4a01395be526", "english_row_compact_sha256": "d41e7d831fd078ab690b37ac4917df4dea6a1540e6ba010af5d178836acc65f8", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations.en.jsonl" }, "mko_bindings": { "mko-p-adic-valuation-integers": { "sha256": "9805aa95014a92e3ee82b127895171c719f8deb0548a932c13fb85c07d782a40", "english_sha256": "13f277e55f58c0bbb023895d24c591527ec1f4bd6f223d42e1a8f1070ec3facc", "original_path": "public/data/mko/p-adic-valuation-integers.json", "english_path": "public/data/i18n/editions/mko-p-adic-valuation-integers.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283", "original_candidate_sha256": "08c2e21529700e6c366b53e02d6717eaca5afe8a670329048706a63e2902f520", "original_candidate_line_sha256": "08c2e21529700e6c366b53e02d6717eaca5afe8a670329048706a63e2902f520" }, "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-20261026-010-p54", "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": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62", "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "27517b96194fe409d8924dc26eb2b02cac09fd2f14c0fb64972536982a2abf59", "arithmetic_steps": 4 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }