# Final Core80 Seeds and Five Rounds of Existing-Family Depth 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. - Stage 1 · required ## [Real derivatives at a point and finite difference quotients](/en/mko/mko-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-derivative) [Concept guides](/en/derivations/?q=mko-derivative) - Stage 1 · required ## [Point values and continuity of real functions](/en/mko/mko-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-continuity) [Concept guides](/en/derivations/?q=mko-continuity) - Stage 1 · required ## [Finite simple undirected graphs](/en/mko/mko-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-graph) [Concept guides](/en/derivations/?q=mko-graph) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Tends-to relation](/en/mko/mko-tends-to-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-tends-to-relation) [Concept guides](/en/derivations/?q=mko-tends-to-relation) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) - Stage 1 · required ## [Polynomial factorization over the rationals](/en/mko/mko-factorization/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-factorization) [Concept guides](/en/derivations/?q=mko-factorization) - Stage 1 · required ## [Univariate polynomials over the rationals](/en/mko/mko-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-polynomial) [Concept guides](/en/derivations/?q=mko-polynomial) - Stage 1 · required ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number) - Stage 1 · required ## [Exact area of half-open rectangles and unit-cell unions](/en/mko/mko-area/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-area) [Concept guides](/en/derivations/?q=mko-area) - Stage 1 · required ## [Matrix-represented linear maps on the real plane](/en/mko/mko-linear-transformation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-linear-transformation) [Concept guides](/en/derivations/?q=mko-linear-transformation) - Stage 1 · required ## [Determinants of real 2×2 matrices](/en/mko/mko-determinant/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-determinant) [Concept guides](/en/derivations/?q=mko-determinant) - Stage 1 · required ## [Quadratic equations over the reals](/en/mko/mko-quadratic-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Existing canonical prerequisite for this pulse. [Related questions](/en/explore/?mko=mko-quadratic-equation) [Concept guides](/en/derivations/?q=mko-quadratic-equation) - Stage 2 · optional ## [Differentiating polynomial sums, products, and compositions](/en/mko/mko-derivative-rules/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) 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. [Related questions](/en/explore/?mko=mko-derivative-rules) [Concept guides](/en/derivations/?q=mko-derivative-rules) - Stage 2 · optional ## [Nonempty finite trees: connectivity, cycles, and unique paths](/en/mko/mko-tree/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions. [Related questions](/en/explore/?mko=mko-tree) [Concept guides](/en/derivations/?q=mko-tree) - Stage 2 · optional ## [Function limits on punctured real domains](/en/mko/mko-function-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) 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. [Related questions](/en/explore/?mko=mko-function-limit) [Concept guides](/en/derivations/?q=mko-function-limit) - Stage 2 · optional ## [Real eigenvalues and nonzero eigenvectors of 2×2 matrices](/en/mko/mko-eigenvalue/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) 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. [Related questions](/en/explore/?mko=mko-eigenvalue) [Concept guides](/en/derivations/?q=mko-eigenvalue) - Stage 3 · optional ## [Polynomial antiderivatives and the choice of constant](/en/mko/mko-antiderivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) Deepen the rational-coefficient slice with a specified primitive coefficient, shared-primitive constant differences, anchor-selected constants, and candidate derivative coefficients. [Related questions](/en/explore/?mko=mko-antiderivative) [Concept guides](/en/derivations/?q=mko-antiderivative) - Stage 4 · optional ## [Signed polynomial definite integrals on finite intervals](/en/mko/mko-definite-integral/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) 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. [Related questions](/en/explore/?mko=mko-definite-integral) [Concept guides](/en/derivations/?q=mko-definite-integral) - Stage 5 · optional ## [Fundamental theorem of calculus: continuity and oriented endpoints](/en/mko/mko-fundamental-theorem-calculus/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261009%2F) With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference. [Related questions](/en/explore/?mko=mko-fundamental-theorem-calculus) [Concept guides](/en/derivations/?q=mko-fundamental-theorem-calculus) Path source & statuspath-pulse-foundations-20261009 · candidate [Original paths JSON](/data/architecture/learning-paths.json)