OCME · Bilingual mathematics & derivations
Final Core80 Seeds and Five Rounds of Existing-Family Depth
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- Real derivatives at a point and finite difference quotients
- Point values and continuity of real functions
- Finite simple undirected graphs
- Function mapping
- Tends-to relation
- Absolute value
- Polynomial factorization over the rationals
- Univariate polynomials over the rationals
- Function composition
- Rational number
- Exact area of half-open rectangles and unit-cell unions
- Matrix-represented linear maps on the real plane
- Determinants of real 2×2 matrices
- Quadratic equations over the reals
- Differentiating polynomial sums, products, and compositions
- Nonempty finite trees: connectivity, cycles, and unique paths
- Function limits on punctured real domains
- Real eigenvalues and nonzero eigenvectors of 2×2 matrices
- Polynomial antiderivatives and the choice of constant
- Signed polynomial definite integrals on finite intervals
- Fundamental theorem of calculus: continuity and oriented endpoints
Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations.
This is a suggested learning sequence, not a universal logical dependency or a formal proof.
Real derivatives at a point and finite difference quotients
Existing canonical prerequisite for this pulse.
Point values and continuity of real functions
Existing canonical prerequisite for this pulse.
Finite simple undirected graphs
Existing canonical prerequisite for this pulse.
Function mapping
Existing canonical prerequisite for this pulse.
Tends-to relation
Existing canonical prerequisite for this pulse.
Absolute value
Existing canonical prerequisite for this pulse.
Polynomial factorization over the rationals
Existing canonical prerequisite for this pulse.
Univariate polynomials over the rationals
Existing canonical prerequisite for this pulse.
Function composition
Existing canonical prerequisite for this pulse.
Rational number
Existing canonical prerequisite for this pulse.
Exact area of half-open rectangles and unit-cell unions
Existing canonical prerequisite for this pulse.
Matrix-represented linear maps on the real plane
Existing canonical prerequisite for this pulse.
Determinants of real 2×2 matrices
Existing canonical prerequisite for this pulse.
Quadratic equations over the reals
Existing canonical prerequisite for this pulse.
Differentiating polynomial sums, products, and compositions
Deepen the full-real degree-at-most-three polynomial slice with second derivatives, product-value eligibility, ordered composition derivatives, and complete derivative coefficients without truncation.
Nonempty finite trees: connectivity, cycles, and unique paths
A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions.
Function limits on punctured real domains
Deepen the punctured model with a failing δ witness h=ε/|m|, center-versus-limit pairs, ordered one-sided limits, and quotient values only at x≠a.
Real eigenvalues and nonzero eigenvectors of 2×2 matrices
Deepen real 2×2 integer matrices with descending characteristic coefficients, exact discriminants, ordered kernel residuals, and a nonzero vector’s first mismatch; no diagonalization or complex-spectrum expansion.
Polynomial antiderivatives and the choice of constant
Deepen the rational-coefficient slice with a specified primitive coefficient, shared-primitive constant differences, anchor-selected constants, and candidate derivative coefficients.
Signed polynomial definite integrals on finite intervals
Deepen oriented moments with three-segment additivity, paired oddness/symmetric-integral output, signed finite-left-sum error, and endpoint recovery for a nonzero constant integrand.
Fundamental theorem of calculus: continuity and oriented endpoints
With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference.
Path source & status
path-pulse-foundations-20261009 · candidate