# First Modern Structures: Oscillators, Discounting, Complex Numbers, Vector Spaces, Finite Groups, Metrics and Distributions Complete seven traceable knowledge objects in dependency order on 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 ## [Ordinary differential equation](/en/mko/mko-ordinary-differential-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-ordinary-differential-equation) [Concept guides](/en/derivations/?q=mko-ordinary-differential-equation) - Stage 1 · required ## [Differentiating polynomial sums, products, and compositions](/en/mko/mko-derivative-rules/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-derivative-rules) [Concept guides](/en/derivations/?q=mko-derivative-rules) - Stage 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 1 · required ## [Compound-interest model](/en/mko/mko-compound-interest/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-compound-interest) [Concept guides](/en/derivations/?q=mko-compound-interest) - Stage 1 · required ## [Fraction](/en/mko/mko-fraction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-fraction) [Concept guides](/en/derivations/?q=mko-fraction) - Stage 1 · required ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-equation) [Concept guides](/en/derivations/?q=mko-equation) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Binary relations and set projections](/en/mko/mko-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-relation) [Concept guides](/en/derivations/?q=mko-relation) - Stage 1 · required ## [Mathematical proof](/en/mko/mko-proof/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-proof) [Concept guides](/en/derivations/?q=mko-proof) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 1 · required ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) - Stage 2 · optional ## [Linear harmonic-oscillator model](/en/mko/mko-harmonic-oscillator-model/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) The undamped linear harmonic oscillator is defined by m*x″+k*x=0 with m,k>0; this slice asserts no trigonometric closed-form solution. [Related questions](/en/explore/?mko=mko-harmonic-oscillator-model) [Concept guides](/en/derivations/?q=mko-harmonic-oscillator-model) - Stage 3 · optional ## [Present value and discounting](/en/mko/mko-present-value-discounting/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) For a fixed positive accumulation factor and period count, divide a future amount by the factor power to obtain present value. [Related questions](/en/explore/?mko=mko-present-value-discounting) [Concept guides](/en/derivations/?q=mko-present-value-discounting) - Stage 4 · optional ## [Complex number](/en/mko/mko-complex-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1. [Related questions](/en/explore/?mko=mko-complex-number) [Concept guides](/en/derivations/?q=mko-complex-number) - Stage 5 · optional ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) A real vector space has vector addition and real scalar multiplication, satisfying the additive abelian-group and compatible scalar-action axioms. [Related questions](/en/explore/?mko=mko-real-vector-space) [Concept guides](/en/derivations/?q=mko-real-vector-space) - Stage 6 · optional ## [Finite group](/en/mko/mko-finite-group/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) A finite group has a finite carrier and complete closed Cayley table, satisfying associativity, a unique identity and inverses for every element. [Related questions](/en/explore/?mko=mko-finite-group) [Concept guides](/en/derivations/?q=mko-finite-group) - Stage 7 · optional ## [Metric space](/en/mko/mko-metric-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) A metric space has a carrier set and a distance function that is nonnegative, zero exactly at equal points, symmetric and satisfies the triangle inequality. [Related questions](/en/explore/?mko=mko-metric-space) [Concept guides](/en/derivations/?q=mko-metric-space) - Stage 8 · optional ## [Finite discrete probability distribution](/en/mko/mko-finite-discrete-probability-distribution/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261022%2F) A finite discrete distribution assigns nonnegative probability masses to finitely many outcome values, with exact total mass one. [Related questions](/en/explore/?mko=mko-finite-discrete-probability-distribution) [Concept guides](/en/derivations/?q=mko-finite-discrete-probability-distribution) Path source & statuspath-pulse-foundations-20261022 · candidate [Original paths JSON](/data/architecture/learning-paths.json)