{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-discrete-dynamical-system"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-discrete-dynamical-system-01","title":{"zh-Hant":"離散時間動力系統：在七個狀態上迭代 T(x)=(3x+1) mod 7","en":"Discrete-time dynamical system: iterating T(x)=(3x+1) mod 7 on seven states"},"summary":{"zh-Hant":"狀態空間 X={0,1,2,3,4,5,6}，更新映射 T(x)=(3x+1) mod 7。由 x_0=0 依 x_(n+1)=T(x_n) 算出軌道，證明對所有自然數 n 有 x_(n+6)=x_n，據此求 x_2026；完整枚舉固定點；判定 A={0,1,2,4,5,6} 是正向不變子集而 B={0,1} 不是，並證明從 0 出發的軌道永遠不會到達固定點。","en":"State space X={0,1,2,3,4,5,6}, update map T(x)=(3x+1) mod 7. From x_0=0 compute the orbit given by x_(n+1)=T(x_n), prove x_(n+6)=x_n for every natural number n and use it to find x_2026; enumerate the fixed points completely; decide that A={0,1,2,4,5,6} is forward invariant while B={0,1} is not, and prove that the orbit from 0 never reaches the fixed point."},"mko_ids":["mko-discrete-dynamical-system"]}],"title":"Concept derivation guides"}