{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"paths","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":{"from":"/en/paths/"},"path":{"id":"path-pulse-foundations-20261006","title":{"zh-Hant":"實數順序、變化率與座標幾何、資料摘要：七輪脈衝","en":"Real Order, Average Rates, Coordinate Geometry, and Data Summaries: Seven-Round Workflow"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-quadratic-equation","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-ratio-proportion","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-graph","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-linear-function","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-coordinate-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-power-root","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-euclidean-length","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-order-relation","position":2,"requirement":"optional","rationale":{"zh-Hant":"在實數論域比較大小；精確有理數交叉乘積、有限極值及含端點區間是本輪切片。","en":"The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend."}},{"id":"mko-quadratic-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"實數全論域的 q(x)=a(x−h)²+k、a≠0；頂點、全域極值與對稱點的明示切片。","en":"A nonzero-coefficient quadratic on all of R has a vertex, an axis of symmetry, and a global minimum or maximum determined by the coefficient sign."}},{"id":"mko-rate-of-change","position":4,"requirement":"optional","rationale":{"zh-Hant":"平均變化率是輸出差除以非零輸入差，不是瞬時導數。","en":"Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative."}},{"id":"mko-distance-formula","position":5,"requirement":"optional","rationale":{"zh-Hant":"在同一等尺度正交座標系，用非負主平方根讀取二維歐氏距離。","en":"The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences."}},{"id":"mko-circle","position":6,"requirement":"optional","rationale":{"zh-Hant":"圓是到固定中心距離恰等於半徑的點集，不是含內部的圓盤。","en":"A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton."}},{"id":"mko-area","position":7,"requirement":"optional","rationale":{"zh-Hant":"先精確處理半開矩形及有限單位格聯集，含重疊、觸邊與空矩形；不冒充一般面積理論。","en":"This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases."}},{"id":"mko-descriptive-statistics","position":8,"requirement":"optional","rationale":{"zh-Hant":"保留重數的非空觀測列，精確計算均值、中位數、全距與所有最大頻率值。","en":"Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities."}}]},"title":"Real Order, Average Rates, Coordinate Geometry, and Data Summaries: Seven-Round Workflow"}