# Function Invertibility and Finite Probability Moments: Seven-Round Workflow Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. [← Return to results](/en/paths/) This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) - Stage 1 · required ## [Affine functions and degree-one functions](/en/mko/mko-linear-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-function) [Concept guides](/en/derivations/?q=mko-linear-function) - Stage 1 · required ## [Sample space](/en/mko/mko-sample-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sample-space) [Concept guides](/en/derivations/?q=mko-sample-space) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 2 · optional ## [Injectivity and distinct inputs](/en/mko/mko-injective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Injectivity requires distinct permitted inputs to have distinct outputs; it does not require coverage of the whole codomain. [Related questions](/en/explore/?mko=mko-injective) [Concept guides](/en/derivations/?q=mko-injective) - Stage 3 · optional ## [Surjectivity and codomain coverage](/en/mko/mko-surjective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Surjectivity requires a permitted preimage for every codomain element; an output may have several preimages. [Related questions](/en/explore/?mko=mko-surjective) [Concept guides](/en/derivations/?q=mko-surjective) - Stage 4 · optional ## [Bijectivity and unique preimages](/en/mko/mko-bijective/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) A bijection is both injective and surjective; equivalently, every codomain element has exactly one permitted preimage. [Related questions](/en/explore/?mko=mko-bijective) [Concept guides](/en/derivations/?q=mko-bijective) - Stage 5 · optional ## [Inverse functions of bijections](/en/mko/mko-inverse-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) For a bijection f:A→B, an inverse g:B→A restores values in both directions; exchanging a table is not an unconditional inverse construction. [Related questions](/en/explore/?mko=mko-inverse-function) [Concept guides](/en/derivations/?q=mko-inverse-function) - Stage 6 · optional ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) A random variable in a finite model is a total function from elementary outcomes to real numbers, distinct from an outcome or its probability weight. [Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable) - Stage 7 · optional ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Finite expectation is the weighted sum of random-variable values under a valid probability law; linearity does not require independence. [Related questions](/en/explore/?mko=mko-expectation) [Concept guides](/en/derivations/?q=mko-expectation) - Stage 8 · optional ## [Variance in finite probability models](/en/mko/mko-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261005%2F%3Ffrom%3D%252Fen%252Fpaths%252F) This entry defines variance as E[(X−E[X])²], distinct from sample variance with an n−1 denominator. Zero-weight outliers do not affect it. [Related questions](/en/explore/?mko=mko-variance) [Concept guides](/en/derivations/?q=mko-variance) Path source & statuspath-pulse-foundations-20261005 · candidate [Original paths JSON](/data/architecture/learning-paths.json)