{"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":{"page":"3"},"kind":"paths","total":59,"pages":3,"page":3,"entries":[{"id":"path-pulse-foundations-20261109","title":{"zh-Hant":"Wave6第三層：Stabilizer、Bolzano-Weierstrass、緊緻子集、全期望、二項式、Möbius與CFG","en":"Wave6 Third Layer: Stabilizers, Bolzano-Weierstrass, Compact Subsets, Total Expectation, Binomial Theorem, Mobius Function and CFGs"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group-action","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-subgroup-homomorphism","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-subsequence","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-limit","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-supremum-and-infimum","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-compact-topological-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-subspace-topology","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-conditional-expectation","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-expectation","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-event","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-permutation-combination","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-mathematical-induction","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-polynomial","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-prime-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-unique-prime-factorization","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-computational-model","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-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-finite-group-stabilizer","position":2,"requirement":"optional","rationale":{"zh-Hant":"指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。","en":"The stabilizer of a specified point consists of every group element fixing that point and forms a subgroup."}},{"id":"mko-bolzano-weierstrass-theorem-real","position":3,"requirement":"optional","rationale":{"zh-Hant":"每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。","en":"Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map."}},{"id":"mko-compact-subset","position":4,"requirement":"optional","rationale":{"zh-Hant":"K⊆X在subspace topology下compact，等價地每個relative-open cover有有限subcover。","en":"A subset K of X is compact in its subspace topology exactly when every relative-open cover has a finite subcover."}},{"id":"mko-law-of-total-expectation","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一有限機率空間上，完整partition各cell的conditional mean按cell probability加權後等於E[X]。","en":"On the same finite probability space, weighting every partition-cell conditional mean by its cell probability gives E[X]."}},{"id":"mko-binomial-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"交換乘法環境中，natural exponent n的二項式冪以binomial coefficients完整展開。","en":"With commutative multiplication, a binomial power of natural exponent n has a complete finite expansion with binomial coefficients."}},{"id":"mko-mobius-function","position":7,"requirement":"optional","rationale":{"zh-Hant":"Möbius函數在positive integers上由完整prime multiplicities決定：mu(1)=1、含平方質因數為0、squarefree取parity sign。","en":"On positive integers, complete prime multiplicities determine the Mobius function: mu(1)=1, a repeated prime gives0, and a squarefree integer has the parity sign."}},{"id":"mko-context-free-grammar","position":8,"requirement":"optional","rationale":{"zh-Hant":"CFG由有限且不交的nonterminals、terminals、start與單一nonterminal左側的finite productions生成語言。","en":"A CFG uses finite disjoint nonterminal and terminal alphabets, a start symbol and finite productions with a single nonterminal on each left side."}}]},{"id":"path-pulse-foundations-20261110","title":{"zh-Hant":"Wave6第四層：Orbit-Stabilizer、MVT、緊緻像、全變異數、遞迴、Möbius反演與PDA","en":"Wave6 Fourth Layer: Orbit-Stabilizer, MVT, Compact Images, Total Variance, Recurrences, Mobius Inversion and PDAs"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group-orbit","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-finite-group-stabilizer","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-cardinality-comparison","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","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-continuity","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-rate-of-change","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-compact-subset","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-topological-continuity","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-law-of-total-expectation","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-variance","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-conditional-expectation","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-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-mobius-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-natural-divisibility","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-counting-principle","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-computational-model","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-finite-automaton","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-context-free-grammar","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-orbit-stabilizer-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整有限群作用中，同一點的orbit大小乘以stabilizer大小等於群階。","en":"For a complete finite group action, the orbit size and stabilizer size of the same point multiply to the group order."}},{"id":"mko-mean-value-theorem","position":3,"requirement":"optional","rationale":{"zh-Hant":"若a<b，f在[a,b]連續且在(a,b)可微，則某個內點c的導數等於端點割線斜率。","en":"If a<b, f is continuous on [a,b] and differentiable on (a,b), some interior c has derivative equal to the endpoint secant slope."}},{"id":"mko-continuous-image-of-compact","position":4,"requirement":"optional","rationale":{"zh-Hant":"若K在X中compact且f:X→Y連續，則image f(K)在Y中compact。","en":"If K is compact in X and f:X→Y is continuous, then f(K) is compact in Y."}},{"id":"mko-law-of-total-variance","position":5,"requirement":"optional","rationale":{"zh-Hant":"同一有限law與完整partition生成的G下，總變異數等於cell內變異數期望加cell均值變異數。","en":"For the same finite law and partition-generated G, total variance is the expected within-cell variance plus the variance of the cell means."}},{"id":"mko-recurrence-relation","position":6,"requirement":"optional","rationale":{"zh-Hant":"order-r recurrence以明示initial values與total single-valued update rule逐項決定sequence。","en":"An order-r recurrence uses explicit initial values and a total single-valued state update to determine successive sequence terms."}},{"id":"mko-mobius-inversion-finite-divisors","position":7,"requirement":"optional","rationale":{"zh-Hant":"若F(n)是所有正因數d上的f(d)總和，則以Möbius權重可精確反演f(n)。","en":"If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n)."}},{"id":"mko-pushdown-automaton","position":8,"requirement":"optional","rationale":{"zh-Hant":"nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。","en":"A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state."}}]},{"id":"path-pulse-foundations-20261111","title":{"zh-Hant":"Wave6第五層：Lagrange、高階導數、Hausdorff、連續CDF、Catalan、二次剩餘與NFA","en":"Wave6 Fifth Layer: Lagrange, Higher Derivatives, Hausdorff Spaces, Continuous CDFs, Catalan Numbers, Quadratic Residues and NFAs"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group","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-subgroup-homomorphism","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-natural-divisibility","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-cardinality-comparison","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-quotient-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-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-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-mathematical-induction","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-topological-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-neighborhood","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-quantifier","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-cumulative-distribution-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-continuity","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-limit","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-sigma-algebra-measure-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-recurrence-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-binomial-theorem","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-counting-principle","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-modular-congruence","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-prime-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-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-finite-automaton","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-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-subset","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-lagrange-theorem-finite-groups","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群G的子群H以等大左陪集分割G，因此|G|=[G:H]|H|且|H|整除|G|。","en":"Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|."}},{"id":"mko-higher-order-derivative","position":3,"requirement":"optional","rationale":{"zh-Hant":"以f^(0)=f及f^(n+1)=(f^(n))'遞迴定義每個存在的有限階導數。","en":"Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'."}},{"id":"mko-hausdorff-space","position":4,"requirement":"optional","rationale":{"zh-Hant":"拓撲空間X若每對不同點都有不交open neighborhoods即為Hausdorff。","en":"A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods."}},{"id":"mko-continuous-real-distribution-function","position":5,"requirement":"optional","rationale":{"zh-Hant":"本節點以CDF本身連續定義continuous distribution；CDF仍須nondecreasing、right-continuous且兩端極限0與1。","en":"Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1."}},{"id":"mko-catalan-numbers","position":6,"requirement":"optional","rationale":{"zh-Hant":"Catalan數採C_0=1，並以convolution recurrence或兩個exact closed forms定義同一zero-based sequence。","en":"With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence."}},{"id":"mko-quadratic-residue","position":7,"requirement":"optional","rationale":{"zh-Hant":"本節點採odd prime p上的nonzero convention：a mod p非零且等於某個nonzero square時為quadratic residue。","en":"Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p."}},{"id":"mko-nondeterministic-finite-automaton","position":8,"requirement":"optional","rationale":{"zh-Hant":"epsilon-NFA以狀態、input symbol或epsilon對應可能後繼集合，存在一條full-input final path即接受。","en":"An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance."}}]},{"id":"path-pulse-foundations-20261112","title":{"zh-Hant":"Wave6第六層：Cauchy、Taylor、緊緻閉集、瑕積分、收斂半徑、Legendre與子集合構造","en":"Wave6 Sixth Layer: Cauchy, Taylor, Compact Closed Subsets, Improper Integrals, Convergence Radii, Legendre and Subset Construction"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-lagrange-theorem-finite-groups","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-prime-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-finite-group","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-finite-group-action","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-polynomial","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-factorial","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-higher-order-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-hausdorff-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-compact-subset","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-open-closed-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-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-function-limit","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-limit","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-convergence-of-series","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-infinite-series","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-supremum-and-infimum","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-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-quadratic-residue","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-nondeterministic-finite-automaton","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-finite-automaton","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-subset","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-cauchy-theorem-finite-groups","position":2,"requirement":"optional","rationale":{"zh-Hant":"若質數p整除有限群G的階，則G含有一個exact order為p的元素。","en":"If a prime p divides the order of a finite group G, G contains an element of exact order p."}},{"id":"mko-taylor-polynomial","position":3,"requirement":"optional","rationale":{"zh-Hant":"在中心a由f的零階到n階導數值依factorial縮放組成n階Taylor多項式。","en":"At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial."}},{"id":"mko-compact-subset-closed-in-hausdorff","position":4,"requirement":"optional","rationale":{"zh-Hant":"若K是Hausdorff空間X的緊緻子集，則K在X中為閉集。","en":"If K is compact in the Hausdorff space X, then K is closed in X."}},{"id":"mko-improper-integral","position":5,"requirement":"optional","rationale":{"zh-Hant":"無界區間或含奇點的積分以適當有限區間積分的one-sided limits定義，且每個必要分段須獨立收斂。","en":"Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently."}},{"id":"mko-power-series-radius-of-convergence","position":6,"requirement":"optional","rationale":{"zh-Hant":"冪級數在中心c周圍具有內部絕對收斂、外部發散的半徑R，邊界點須逐案判定。","en":"Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions."}},{"id":"mko-legendre-symbol","position":7,"requirement":"optional","rationale":{"zh-Hant":"對奇質數p，Legendre符號以0、1、-1分別編碼zero class、nonzero quadratic residue與nonresidue。","en":"For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively."}},{"id":"mko-subset-construction-nfa-to-dfa","position":8,"requirement":"optional","rationale":{"zh-Hant":"以epsilon-NFA狀態子集合為DFA狀態，start、transition與acceptance按epsilon closure定義即可保持語言。","en":"Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language."}}]},{"id":"path-pulse-foundations-20261113","title":{"zh-Hant":"Wave6第七層：正規子群、實指數、解析OGF、Euler判準、Turing判定、梯度與流網路","en":"Wave6 Seventh Layer: Normal Subgroups, Real Exponential, Analytic OGFs, Euler Criterion, Turing Decisions, Gradients and Flow Networks"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group","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-subgroup-homomorphism","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-quantifier","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-infinite-series","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-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-sequence-limit","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-series-radius-of-convergence","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-factorial","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-convergence-of-series","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-binomial-theorem","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-recurrence-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-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-euler-theorem-number-theory","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-residue","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-legendre-symbol","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-computational-model","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-decision-problem","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-finite-automaton","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-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-finite-directed-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-programming","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-finite-normal-subgroup","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。","en":"A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset."}},{"id":"mko-real-exponential-function","position":3,"requirement":"optional","rationale":{"zh-Hant":"實指數函數以對每個實數絕對收斂的冪級數E(x)=sum_{n≥0}x^n/n!定義，並滿足E(x+y)=E(x)E(y)與E(x)>0。","en":"Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0."}},{"id":"mko-analytic-ordinary-generating-function","position":4,"requirement":"optional","rationale":{"zh-Hant":"數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。","en":"An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain."}},{"id":"mko-euler-criterion-quadratic-residue","position":5,"requirement":"optional","rationale":{"zh-Hant":"若p為奇質數且p不整除a，則a^((p-1)/2)模p等於Legendre符號(a/p)的模p代表。","en":"For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p)."}},{"id":"mko-turing-machine-decision-model","position":6,"requirement":"optional","rationale":{"zh-Hant":"固定deterministic single-tape Turing機語義；decider必須對每個合法輸入停機並給正確accept/reject答案。","en":"Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer."}},{"id":"mko-gradient-vector","position":7,"requirement":"optional","rationale":{"zh-Hant":"在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。","en":"In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector."}},{"id":"mko-finite-flow-network","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限有向網路中的非負流須逐邊不超過capacity，並在source與sink以外每個頂點滿足流量守恆。","en":"A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink."}}]},{"id":"path-pulse-foundations-20261114","title":{"zh-Hant":"Wave6第八層：kernel、商群、共軛類、標準常態、MSE、LP標準形與頂點覆蓋","en":"Wave6 Eighth Layer: Kernels, Quotient Groups, Conjugacy, Standard Normal, MSE, LP Standard Form and Vertex Covers"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-subgroup-homomorphism","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-finite-normal-subgroup","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-quotient-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-finite-group","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-finite-group-action","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-finite-group-orbit","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-continuous-real-distribution-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-improper-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-real-exponential-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-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-expectation","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-variance","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-estimator-bias","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-estimator-variance","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-sampling-distribution","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-parametric-statistical-model","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-programming","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","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-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-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-subset","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-quantifier","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-finite-group-homomorphism-kernel","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。","en":"For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets."}},{"id":"mko-finite-quotient-group","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。","en":"Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives."}},{"id":"mko-finite-group-conjugacy-class","position":4,"requirement":"optional","rationale":{"zh-Hant":"有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。","en":"A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group."}},{"id":"mko-standard-normal-distribution","position":5,"requirement":"optional","rationale":{"zh-Hant":"標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。","en":"The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1."}},{"id":"mko-estimator-mean-squared-error","position":6,"requirement":"optional","rationale":{"zh-Hant":"scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。","en":"For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately."}},{"id":"mko-linear-program-standard-form","position":7,"requirement":"optional","rationale":{"zh-Hant":"本節點固定唯一primal convention：maximize c^T x，subject to Ax≤b與x≥0。","en":"This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0."}},{"id":"mko-vertex-cover","position":8,"requirement":"optional","rationale":{"zh-Hant":"有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。","en":"A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C."}}]},{"id":"path-pulse-foundations-20261115","title":{"zh-Hant":"Wave6第九層：第一同構、群中心、bias–variance、LP對偶、max-flow/min-cut、最小頂點覆蓋與many-one reduction","en":"Wave6 Ninth Layer: First Isomorphism, Centers, Bias-Variance, LP Duality, Max-Flow Min-Cut, Minimum Covers and Many-One Reduction"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-finite-group-homomorphism-kernel","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-finite-quotient-group","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-subgroup-homomorphism","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-bijective","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-finite-group","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-finite-group-conjugacy-class","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-finite-normal-subgroup","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-quantifier","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-estimator-mean-squared-error","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-estimator-bias","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-estimator-variance","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-program-standard-form","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-optimization-problem","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-finite-flow-network","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-programming","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-vertex-cover","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-cardinality-comparison","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-decision-problem","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-composition","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-turing-machine-decision-model","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-first-isomorphism-theorem-finite-groups","position":2,"requirement":"optional","rationale":{"zh-Hant":"有限群同態f:G→Q由kernel cosets誘導出G/ker f到im f的群同構，而非自動同構到整個codomain。","en":"A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain."}},{"id":"mko-finite-group-center","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限群G的中心Z(G)由與每個群元素交換的全部元素構成，且是G的正規子群。","en":"The center consists of all elements commuting with every group element and is a normal subgroup."}},{"id":"mko-bias-variance-decomposition","position":4,"requirement":"optional","rationale":{"zh-Hant":"逐參數theta，scalar estimator的MSE等於其variance加上bias平方。","en":"For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias."}},{"id":"mko-linear-program-duality","position":5,"requirement":"optional","rationale":{"zh-Hant":"對固定primal max c^Tx、Ax≤b、x≥0，其dual為min b^Ty、A^Ty≥c、y≥0；feasible pair滿足弱對偶。","en":"For primal maximize c^T*x with A*x≤b,x≥0, the dual minimizes b^T*y with A^T*y≥c,y≥0; feasible pairs satisfy weak duality."}},{"id":"mko-max-flow-min-cut-theorem","position":6,"requirement":"optional","rationale":{"zh-Hant":"有限容量網路的最大feasible flow value等於全部s–t cuts中的最小capacity。","en":"In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts."}},{"id":"mko-minimum-vertex-cover","position":7,"requirement":"optional","rationale":{"zh-Hant":"有限simple graph中，cardinality在全部vertex covers裡最小的候選是minimum vertex cover。","en":"A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph."}},{"id":"mko-many-one-reduction","position":8,"requirement":"optional","rationale":{"zh-Hant":"A≤mB表示存在total computable f，使每個x皆滿足x∈A恰當f(x)∈B。","en":"A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x."}}]},{"id":"path-pulse-foundations-20261116","title":{"zh-Hant":"Wave6第十層：contraction、IVT、Taylor remainder、finite MLE、sufficiency、toposort與multiplicative order","en":"Wave6 Tenth Layer: Contraction, IVT, Taylor Remainders, Finite MLE, Sufficiency, Topological Orders and Modular Order"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-orbit-fixed-periodic","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-complete-metric-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-metric-convergence","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-infinite-series","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-continuity","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-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-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-supremum-and-infimum","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-taylor-polynomial","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-higher-order-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-mean-value-theorem","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-parametric-statistical-model","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-statistical-estimator","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-optimization-problem","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-sampling-distribution","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-conditional-probability","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-finite-directed-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-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-mathematical-induction","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-modular-inverse","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-gcd-coprime","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-natural-divisibility","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-euler-theorem-number-theory","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-contraction-mapping-theorem","position":2,"requirement":"optional","rationale":{"zh-Hant":"非空完備度量空間上的self-map若以固定q<1壓縮所有距離，則有唯一fixed point且Picard orbit收斂。","en":"A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates."}},{"id":"mko-intermediate-value-theorem","position":3,"requirement":"optional","rationale":{"zh-Hant":"連續實函數在閉區間端點值之間取到每個中間值。","en":"A continuous real function on a closed interval attains each value between its endpoint values."}},{"id":"mko-taylor-remainder-theorem","position":4,"requirement":"optional","rationale":{"zh-Hant":"足夠高階可微函數的Taylor approximation error由Lagrange remainder與derivative bound控制。","en":"Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error."}},{"id":"mko-finite-maximum-likelihood-estimator","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限參數模型中，MLE選取使觀察likelihood最大的參數，tie需固定且與theta無關。","en":"A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter."}},{"id":"mko-finite-sufficient-statistic","position":6,"requirement":"optional","rationale":{"zh-Hant":"finite model中，給定statistic value後sample conditional law不依theta時，T對theta充分。","en":"A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass."}},{"id":"mko-topological-ordering-dag","position":7,"requirement":"optional","rationale":{"zh-Hant":"有限有向圖存在topological ordering恰當其無directed cycle。","en":"A finite directed graph has a topological ordering exactly when it has no directed cycle."}},{"id":"mko-multiplicative-order-mod-n","position":8,"requirement":"optional","rationale":{"zh-Hant":"gcd(a,n)=1時，ord_n(a)是a^k≡1 mod n的最小正整數k。","en":"For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n."}}]},{"id":"path-pulse-foundations-20261117","title":{"zh-Hant":"Wave6第十一層：series tests、uniform limits、product／quotient topology、path connectivity與components","en":"Wave6 Eleventh Layer: Series Tests, Uniform Limits, Product and Quotient Topologies, Paths and Components"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-infinite-series","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-sequence-limit","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-monotone-sequence-convergence-theorem","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-convergence-of-series","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-uniform-convergence-of-functions","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-continuity","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-supremum-and-infimum","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-topological-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-topological-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-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-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-topological-continuity","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-composition","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-quotient-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-surjective","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-connected-topological-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-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-intermediate-value-theorem","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-equivalence-class","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-subset","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-closure-and-interior","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-connected-component","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-subspace-topology","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-series-comparison-test","position":2,"requirement":"optional","rationale":{"zh-Hant":"非負項級數可由逐項或最終逐項上界與下界傳遞收斂或發散。","en":"Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series."}},{"id":"mko-ratio-test-series","position":3,"requirement":"optional","rationale":{"zh-Hant":"非零項絕對值相鄰比極限小於一時絕對收斂，大於一時發散，等於一時不判定。","en":"For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive."}},{"id":"mko-uniform-limit-continuity-theorem","position":4,"requirement":"optional","rationale":{"zh-Hant":"同一domain上的連續函數列若一致收斂至f，則極限函數f連續。","en":"Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain."}},{"id":"mko-finite-product-topology","position":5,"requirement":"optional","rationale":{"zh-Hant":"有限多個拓撲空間的Cartesian product由open rectangles生成乘積拓撲。","en":"Open rectangles generate the topology of a finite Cartesian product of topological spaces."}},{"id":"mko-quotient-topology","position":6,"requirement":"optional","rationale":{"zh-Hant":"滿射q:X→Y使U⊆Y恰在q逆像開時為開，從而賦予Y商拓撲。","en":"For a surjection q:X→Y, the quotient topology declares U open exactly when its full preimage under q is open."}},{"id":"mko-path-connected-space","position":7,"requirement":"optional","rationale":{"zh-Hant":"空間中任意兩點若可由單位區間的連續映射連接，則空間道路連通。","en":"A space is path connected when every pair of its points is joined by a continuous map from the unit interval."}},{"id":"mko-topological-connected-component","position":8,"requirement":"optional","rationale":{"zh-Hant":"指定點的連通分支是包含該點的極大連通子集；所有分支分割空間且各自閉。","en":"The component of a point is its unique inclusion-maximal connected subset; components partition the space and are closed."}}]},{"id":"path-pulse-foundations-20261118","title":{"zh-Hant":"Wave6第十二層：CLT、SLLN、confidence coverage、Wilson、p-adic valuation與regular languages","en":"Wave6 Twelfth Layer: CLT, SLLN, Confidence Coverage, Wilson, p-adic Valuations and Regular Languages"},"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 the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-iid-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-convergence-in-distribution","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-expectation","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-variance","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-standard-normal-distribution","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-sigma-algebra-measure-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-binomial-theorem","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-almost-sure-convergence","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-sampling-distribution","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-statistical-estimator","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-parametric-statistical-model","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-probability","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-prime-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-modular-congruence","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-factorial","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-modular-inverse","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-natural-divisibility","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-unique-prime-factorization","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-integer","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-regular-language","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-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-mathematical-induction","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-finite-automaton","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-proof","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-pigeonhole-principle","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-quantifier","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-central-limit-theorem-finite-variance-iid","position":2,"requirement":"optional","rationale":{"zh-Hant":"共同law且0<σ²<∞的無限IID序列，其正確中心化與sqrt(n)標準化和依分布收斂至標準常態。","en":"An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law."}},{"id":"mko-strong-law-large-numbers-finite-valued-iid","position":3,"requirement":"optional","rationale":{"zh-Hant":"有限值無限IID序列的樣本平均幾乎必然收斂至共同期望。","en":"The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation."}},{"id":"mko-finite-confidence-interval-coverage","position":4,"requirement":"optional","rationale":{"zh-Hant":"有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。","en":"Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter."}},{"id":"mko-wilson-theorem","position":5,"requirement":"optional","rationale":{"zh-Hant":"整數n>1為質數，恰當(n-1)!模n同餘於-1。","en":"An integer n>1 is prime exactly when (n−1)! is congruent to−1 modulo n."}},{"id":"mko-p-adic-valuation-integers","position":6,"requirement":"optional","rationale":{"zh-Hant":"對質數p與非零整數a，v_p(a)是p整除|a|的最高次數；本節固定v_p(0)=infinity。","en":"For prime p and nonzero integer a, v_p(a) is the largest exponent of p dividing |a|; this node fixes v_p(0)=infinity."}},{"id":"mko-regular-expression","position":7,"requirement":"optional","rationale":{"zh-Hant":"正規表示式由empty language、epsilon、alphabet symbols、union、concatenation與Kleene star遞迴建構語言。","en":"Regular expressions recursively build languages from the empty language, epsilon, alphabet symbols, union, concatenation and Kleene star."}},{"id":"mko-pumping-lemma-regular-languages","position":8,"requirement":"optional","rationale":{"zh-Hant":"每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。","en":"Every regular language has a positive pumping length such that each sufficiently long member word has a split that remains in the language for every pump count."}}]},{"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":"Learning paths"}