{"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":{},"path":{"id":"path-pulse-foundations-20261119","title":{"zh-Hant":"Wave6第十三層閉合：Newton、梯形積分、梯度下降、Lagrange與三輪核心密度","en":"Wave6 Thirteenth Layer: Newton, Trapezoids, Gradient Descent, Lagrange Conditions and Numerical-Optimization Depth"},"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 four new traceable knowledge objects and deepen three existing objects in dependency order on the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-derivative","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-algorithm-correctness","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-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-sequence","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-derivative-rules","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-definite-integral","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-order-relation","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-real-vector-space","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-system-linear-equations","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-matrix-rank","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-independence-basis","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-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-absolute-value","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-partial-derivative","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-vector","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-multivariable-real-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-set","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-numerical-approximation-error","position":2,"requirement":"optional","rationale":{"zh-Hant":"在既有誤差MKO上加厚affine equation的residual／forward error、誤差傳播、conditioning witness與tolerance邊界。","en":"Extend the existing error object with affine residual versus forward error, propagation, conditioning witnesses and tolerance boundaries."}},{"id":"mko-gradient-vector","position":2,"requirement":"optional","rationale":{"zh-Hant":"在既有standard Euclidean coordinate語義上加厚gradient線性、負梯度內積、symmetric quadratic差分與metric邊界。","en":"Extend fixed standard Euclidean semantics with gradient linearity, negative-gradient inner products, symmetric-quadratic differences and metric boundaries."}},{"id":"mko-optimization-problem","position":2,"requirement":"optional","rationale":{"zh-Hant":"在完整有限可行域與鄰接關係上分開local/global最小、strict improvement、lower-bound certificate及空域邊界。","en":"Separate finite neighborhood-local and global minima, strict improvement, attained lower bounds and empty-domain boundaries on the complete feasible structure."}},{"id":"mko-newton-method","position":3,"requirement":"optional","rationale":{"zh-Hant":"在導數非零的目前點，以切線零點更新x至x-f(x)/f'(x)；收斂是另需局部假設的責任。","en":"At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses."}},{"id":"mko-trapezoidal-quadrature","position":3,"requirement":"optional","rationale":{"zh-Hant":"在嚴格遞增partition上逐段以端點平均乘各自寬度，得到定積分的複合梯形近似。","en":"On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral."}},{"id":"mko-gradient-descent","position":3,"requirement":"optional","rationale":{"zh-Hant":"在固定Euclidean coordinates中以正步長沿負梯度更新；下降與收斂仍需目標及步長假設。","en":"In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses."}},{"id":"mko-lagrange-multiplier-condition","position":3,"requirement":"optional","rationale":{"zh-Hant":"正則等式約束局部極值點的目標梯度，必為constraint gradients的線性組合。","en":"At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients."}}]},"title":"Wave6 Thirteenth Layer: Newton, Trapezoids, Gradient Descent, Lagrange Conditions and Numerical-Optimization Depth"}