OCME · Bilingual mathematics & derivations
Seven Rounds of Linear and Finite-Mapping Depth
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- Real number
- Arithmetic operations
- Coordinate plane
- Function mapping
- Linear equation in one variable
- Quantifiers and explicit domains
- Plane vectors in a fixed basis
- Finite matrices: dimensions, indexing, and transpose
- Injectivity and distinct inputs
- Surjectivity and codomain coverage
- Matrix-represented linear maps on the real plane
- Determinants of real 2×2 matrices
- Linear systems in two variables and degenerate constraints
Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices.
This is a suggested learning sequence, not a universal logical dependency or a formal proof.
Real number
Existing canonical prerequisite for this density pulse.
Arithmetic operations
Existing canonical prerequisite for this density pulse.
Coordinate plane
Existing canonical prerequisite for this density pulse.
Function mapping
Existing canonical prerequisite for this density pulse.
Linear equation in one variable
Existing canonical prerequisite for this density pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for this density pulse.
Plane vectors in a fixed basis
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.
Finite matrices: dimensions, indexing, and transpose
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.
Injectivity and distinct inputs
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.
Surjectivity and codomain coverage
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.
Matrix-represented linear maps on the real plane
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.
Determinants of real 2×2 matrices
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.
Linear systems in two variables and degenerate constraints
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.
Path source & status
path-pulse-foundations-20261016 · candidate