第 27412 題：整數p進賦值

原題與已知條件

prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={"mode":"equal_valuations_cancellation","p":5,"a":60}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-boundary；資料={"mode":"equal_valuations_cancellation","p":5,"a":60}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-p-adic-valuation-integers

逐步推導

source roles = [pval-boundary]

derived = [[mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim]]

Valuation boundary ORIGINAL a ORIGINAL division step1 := div(60, 5)

Valuation boundary ORIGINAL a ORIGINAL division step1 = 12

Valuation boundary ORIGINAL a dividing prime power1 := mul(1, 5)

Valuation boundary ORIGINAL a dividing prime power1 = 5

source roles = [Valuation boundary ORIGINAL a, 5, 60, 60, 1]

derived = [12, 5, 0, 是]

Valuation boundary ORIGINAL a terminal factorization product := mul(5, 12)

Valuation boundary ORIGINAL a terminal factorization product = 60

source roles = [Valuation boundary ORIGINAL a, 5, 60, 60, 1, 12, [5]]

derived = [60, 2, 是, 是]

Valuation ORIGINAL opposite cancellation sum := sum(60, -60)

Valuation ORIGINAL opposite cancellation sum = 0

source roles = [Valuation boundary ACTUAL cancellation, 5, 0]

derived = [infinity, 0, 0, []]

source roles = [5, 60, -60, 1]

derived = [1, infinity, 否]

逐步反向核對

• S009 · premise_and_result_audit — 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。
• S008 · premise_and_result_audit — 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。
• S007 · premise_and_result_audit — 以精確有理數重算 Valuation ORIGINAL opposite cancellation sum，不以小數近似替代。
• S006 · premise_and_result_audit — 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。
• S005 · premise_and_result_audit — 以精確有理數重算 Valuation boundary ORIGINAL a terminal factorization product，不以小數近似替代。
• S004 · premise_and_result_audit — 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。
• S003 · premise_and_result_audit — 以精確有理數重算 Valuation boundary ORIGINAL a dividing prime power1，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Valuation boundary ORIGINAL a ORIGINAL division step1，不以小數近似替代。
• S001 · premise_and_result_audit — 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。

結論

(equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, 否)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。