{"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-20261016","title":{"zh-Hant":"七輪線性與有限映射：向量、矩陣、方程及單滿射","en":"Seven Rounds of Linear and Finite-Mapping Depth"},"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-real-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-coordinate-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-function-mapping","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-linear-equation","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-quantifier","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-vector","position":2,"requirement":"optional","rationale":{"zh-Hant":"固定標準基底下的整數向量：兩純量加權、相反差、純量和兩側與候選分量殘差。零純量与零向量合法，逐分量不當內積或長度。","en":"Use integer vectors in a fixed standard basis: two weighted terms, opposite directional differences, two scalar-addition routes, and candidate component residuals. Zero scalars/vectors are legal; component arithmetic is not an inner product or length."}},{"id":"mko-matrix","position":2,"requirement":"optional","rationale":{"zh-Hant":"非空1至5行列整數矩陣：轉置再轉置、同尺寸加法的兩側轉置、三個1起算索引資格與候選轉置逐項殘差。形狀不合先拒絕求值，不當求逆或反序。","en":"Use complete nonempty 1–5-row/column integer matrices: transpose round trips, both transpose-of-sum routes, three 1-based index guards, and candidate-transpose residuals. Reject mismatched shapes before subtraction; transpose is neither inversion nor element reversal."}},{"id":"mko-injective","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表單射：列出全部碰撞、移去一個輸入後保留原標籤、附加輸出的陪域資格與限制到明示子集合。空源单射真，碰撞必有兩個不同輸入；重複表示不當多值函數。","en":"Validate complete finite total tables before enumerating every distinct-input collision, deleting an input with original labels retained, guarding appended outputs, or restricting to an explicit source subset. Empty sources are injective; a valid append may create nine labels and collisions still use the original numbering. Representation repetition is not a multivalued function."}},{"id":"mko-surjective","position":2,"requirement":"optional","rationale":{"zh-Hant":"完整Fin總表滿射：全纖維與漏項、更換陪域先容納全部輸出、刪輸入保原標籤後漏項、兩個候選目標的資格與首原像。空到空滿射真，空到非空不滿；值域不自動當陪域。","en":"Enumerate all declared codomain fibers/missing targets; a changed codomain must contain every existing output before evaluating its surjectivity. Preserve original source labels on deletion and distinguish typed empty fibers from outside-codomain candidates. Empty-to-empty is surjective, empty-to-nonempty is not; actual range is not automatically the declared codomain."}},{"id":"mko-linear-transformation","position":3,"requirement":"optional","rationale":{"zh-Hant":"明示全R²到R²列向量模型Mv：基底列重建、加法兩側、純量兩側與Mv+b的零點障礙。零矩陣可以線性，有限測值不替任意函數證明線性。","en":"Declare the full R²→R² column-vector matrix model Mv: reconstruct basis-image columns, compare additivity/scalar routes, and test the zero-point obstruction for Mv+b. The zero matrix is linear; finite values do not prove arbitrary functions linear."}},{"id":"mko-determinant","position":3,"requirement":"optional","rationale":{"zh-Hant":"R上2×2有號行列式：交换兩行的負號、t²縮放、通常行乘列的兩序乘積、絕對值候選誤判。實數域非零可逆不搬至整數環，負行列式不取絕對值。","en":"Retain signed real 2×2 determinants: swapped-row sign, t² matrix scaling, ordinary row-by-column products in both orders, and absolute-value candidate audits. Nonzero real determinants permit real-field invertibility, not a transfer to integer-ring invertibility; matching product determinants do not make the products equal."}},{"id":"mko-system-linear-equations","position":3,"requirement":"optional","rationale":{"zh-Hant":"R上整數係數的兩元兩式：有理候選逐式殘差、係數與增廣秩區分無解／唯一／無限多解、加倍列消元與第一列縮放。t=0不是可逆列操作，不能只看行列式零便稱無解。","en":"Use integer-coefficient two-variable/two-equation systems over R: substitute rational candidates into both rows, distinguish solution classes with coefficient/augmented ranks, add a multiple of one complete row to the other, and scale the first row. A zero determinant does not decide inconsistency; zero scaling is not invertible even if a redundant system’s solution set happens to remain unchanged."}}]},"title":"Seven Rounds of Linear and Finite-Mapping Depth"}