{"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-20261014","title":{"zh-Hant":"七輪基礎深化：角度、數線、方程、階乘與圓邊界","en":"Foundation Transforms: Angles, Number Lines, Equations, Factorials, and Circle Boundaries"},"summary":{"zh-Hant":"在七個既有知識物件及其前置上深化跨域題族；保持數學來源不變，以精確有限切片核對邊界。","en":"Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-variable-expression","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-euclidean-length","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-distance-formula","position":1,"requirement":"required","rationale":{"zh-Hant":"本密度脈衝所需的既有 Canonical 前置。","en":"Existing canonical prerequisite for this density pulse."}},{"id":"mko-angle","position":2,"requirement":"optional","rationale":{"zh-Hant":"0到180度非定向角的精確度數協議。拆分先核資格，互餘與互補分开，雙門檻殘差和三段累加只作明示算術；超180累加量不冒充本條目角，不由示意圖、射線長或定向旋轉推測量值。","en":"Use exact nondirected degree values in [0,180]: guard splits, distinguish complements/supplements, and compute two threshold defects and three-part budgets. Sums above 180 are not angles in this entry. Infer no measurements from sketches, ray lengths, or directed rotations."}},{"id":"mko-real-number","position":2,"requirement":"optional","rationale":{"zh-Hant":"只核有理點在R的嵌入切片：有序中點、有限唯一點列及閉包區間、含端點的閉區間交集與分離距離、四個點的有向偏移及距離。接觸是點區間而非空；不是Cauchy完備性、無理數分類或一般連續性證明。","en":"Check the rational-point embedding slice only: ordered midpoints, finite distinct points and least containing closed intervals, inclusive interval intersections/separation, and four directed offsets/distances. Touching is a point interval, not empty; the containing interval is not the point set’s topological closure. No Cauchy completeness, irrational classification, or general continuity proof."}},{"id":"mko-equation","position":2,"requirement":"optional","rationale":{"zh-Hant":"給定Q上的原式雙側值、有限候選與可逆條件。乘零能改變解集；含x-h分母時先排除x=h，交乘殘差零不恢復未定義點。同量平移核兩側值及殘差，不建立通用AST或一般方程求解完備性。","en":"Check original Q equation sides, finite candidates, and reversibility premises. Multiplication by zero may change solutions; with denominator x−h first exclude x=h, and cross-multiplication cannot restore it. Check both shifted sides and defects without a universal AST or general solving-completeness claim."}},{"id":"mko-factorial","position":2,"requirement":"optional","rationale":{"zh-Hant":"N從0起且0!=1，用BigInt核相鄰階乘精確比、含空乘積的窗口、候選自報的真值與遞迴缺陷、省去一次已存在因子後回乘。報告用有界正規十進整數字串，不以浮點或遞迴自洽冒充正確階乘；不聲稱排列、整除性或漸近普遍證明。","en":"Use N from 0, 0!=1, and exact BigInt products: adjacent ratios, empty-product windows, independently checked reports/base and recurrence defects, and omission of one existing factor with multiplication back. Bound canonical decimal reports; recurrence self-consistency and floating-point values are not factorial correctness. No general permutation, divisibility, or asymptotic proof."}},{"id":"mko-circle","position":2,"requirement":"optional","rationale":{"zh-Hant":"等單位正交歐氏框架與非負整數半徑。三點平方殘差分類但圓只取邊界；軸向兩個對徑端點、半徑非負尺度乘算、水平截線平方餘量與整數候選。r=0只有中心且軸向表示折疊，負餘量不取根，不聲稱一般圓API或相似定理的形式橋。","en":"Use one equal-unit orthogonal Euclidean frame and a nonnegative integer radius. Check three squared residual/classification sets but only boundary membership, axis antipodes, radius-parameter scaling, and fixed-horizontal-offset square remainders. r=0 is only the center with collapsed endpoints; negative remainders have no real root. No general circle-API or similarity-theorem formal bridge."}},{"id":"mko-linear-equation","position":3,"requirement":"optional","rationale":{"zh-Hant":"a非零才屬Q上一元一次方程；規一列、參數切片的退化常數等式、根周有理偏移回代與兩個一次方程的根／共同解。a=0可能全Q或無解但不是一次，有限核對不作一般線性系統或唯一性形式Evidence。","en":"Only a≠0 gives a one-variable linear equation over Q. Check normalized rows, parameter-dependent constant degeneracies, exact offsets around roots, and two rows’ roots/common solutions. a=0 can yield all Q or no solution but is not linear; no general linear-system or uniqueness formal Evidence."}},{"id":"mko-absolute-value","position":3,"requirement":"optional","rationale":{"zh-Hant":"只核有理嵌入的分段絕對值：負分母表示先正規化，三點距離與路徑差分開，閉半徑四點代入與有理目標的根／三候選核對。負目標無根，零根去重；有限切片非全部實數性質、三角不等式或一般不等式求解證明。","en":"Normalize rational representations before branch decisions; distinguish three-point distances from path defects, inclusive four-point radius tests, and rational-target roots/candidate audits. Negative targets have no roots and zero roots are deduplicated. Finite slices are not all real absolute-value properties, a universal triangle inequality, or general inequality solving."}}]},"title":"Foundation Transforms: Angles, Number Lines, Equations, Factorials, and Circle Boundaries"}