R0043-22079 · q-20261014-advance-20261110-r03-continuous-image-of-compact-0079
compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={"left":-5,"right":5,"value":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={"left":-5,"right":5,"value":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
操作前
source operation=compactimg-reverse-implication-boundary;original snapshot and mathematics retained
操作後
explicit ordered source output fields=[left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary]
條件：無
原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL WHOLE real interval(-5,5);constant real image value=3
操作後
for integers j>=0,relative opens U_j=(L+(R-L)/(j+2),R) cover the interval;ANY finite subfamily is contained in U_max_j and misses a nonempty interval near L,so domain NOT compact
條件：無
任意x>L由Archimedean性選j令門檻低於x，故全族覆蓋；有限族有最大j，取L與其門檻間的點便漏掉。這不是有限取樣或僅靠非closed口號判非compact。
規則：compactimg_constant_open_interval_counterexample_whole_cover · 依賴：G001, S001
操作前
ORIGINAL noncompact domain=(-5,5);constant f(x)=3
操作後
preimage of ANY target open is either the whole domain or empty,so continuous;image singleton has a one-member subcover from any cover;reverse implication claim=否
條件：無
常數函數逆像的兩種情況都open；singleton任意cover中有一個open包含該點即可。故compact像不反推domain compact，也不暗推一般preimage compact或非Hausdorff下像closed。
規則：compactimg_constant_map_continuity_singleton_compact_no_converse · 依賴：G001, S002
S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。(-5, 5, 3, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)
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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 00f2bb0b527247501575cc57e00822ed304dac9499af1213fc608e66beae0080candidate record SHA-256: 96b09e6fbc19dc5ba085221b2fbb9e2e0559a3b7259313a78735f992ba97e8f0semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5eLean: NOT_RUN · human review: NOT_PERFORMED