# Cross-Domain Depth: Algebra, Sets, Distances, Finite Probability, and Quantifiers 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. - Stage 1 · required ## [Variables and algebraic expressions](/en/mko/mko-variable-expression/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-variable-expression) [Concept guides](/en/derivations/?q=mko-variable-expression) - Stage 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number) - Stage 1 · required ## [Set membership](/en/mko/mko-set-membership/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-set-membership) [Concept guides](/en/derivations/?q=mko-set-membership) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 1 · required ## [Euclidean length](/en/mko/mko-euclidean-length/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-euclidean-length) [Concept guides](/en/derivations/?q=mko-euclidean-length) - Stage 1 · required ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) - Stage 1 · required ## [Real random variables on finite sample spaces](/en/mko/mko-random-variable/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-random-variable) [Concept guides](/en/derivations/?q=mko-random-variable) - Stage 1 · required ## [Proposition](/en/mko/mko-proposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-proposition) [Concept guides](/en/derivations/?q=mko-proposition) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 2 · optional ## [Univariate polynomials over the rationals](/en/mko/mko-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Use the integer-coefficient slice of Q[x] through input degree 3: complete normalized sum/product coefficients (products through degree 6), ordered evaluation at two points, and complete coefficients of p(x+c) by exact expansion. Equality at one point is not polynomial identity. [Related questions](/en/explore/?mko=mko-polynomial) [Concept guides](/en/derivations/?q=mko-polynomial) - Stage 2 · optional ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Normalize finite explicit integer sets; compute union/intersection cardinalities, symmetric differences, both relative De Morgan sets, and ordered three-part partitions. Require A,B⊆the declared U before complements; finite instances do not create general set-theory formal Evidence. [Related questions](/en/explore/?mko=mko-set-operations) [Concept guides](/en/derivations/?q=mko-set-operations) - Stage 2 · optional ## [Euclidean distance in an equal-unit orthonormal frame](/en/mko/mko-distance-formula/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Vectors are integer coordinate differences from a common origin in one equal-unit orthogonal real plane. Check ordered squares, candidate sign and square defect, indexed three-distance tie groups, and sums of two integer principal distances. Exclude nonorthogonal/unequal-scale and default product-metric interpretations; a two-segment sum is not another segment’s length. [Related questions](/en/explore/?mko=mko-distance-formula) [Concept guides](/en/derivations/?q=mko-distance-formula) - Stage 2 · optional ## [Expectation in finite models](/en/mko/mko-expectation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) On a shared normalized finite integer-valued model, compute wX contributions, E[X−Y], ordered (E[XY],E[X]E[Y]), and an integer value at an explicit null slot of positive mass from a known mean. No independence, sample-estimation, or general integrability claim; equal numerical expressions do not establish independence. [Related questions](/en/explore/?mko=mko-expectation) [Concept guides](/en/derivations/?q=mko-expectation) - Stage 2 · optional ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Enumerate finite explicit integer domains with order, equality, or nonzero-divisibility predicates: true/false partitions, (¬∀P,¬∃P), typed counterexample masks, and ordered nested existence/universality counts on U/V. Empty-domain universal is True and existential False. Do not exchange quantifiers or extrapolate to all integers or general first-order proofs. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 3 · optional ## [Conditional probability in finite models](/en/mko/mko-conditional-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Use explicit finite S, its full powerset, nonnegative integer masses, and positive D with total mass D. Require positive B mass, and positive A mass for both directions. Compute probability pairs, directional conditionals, every conditional singleton weight, and signed candidate defects. Nonempty zero-mass events cannot condition; no Bayes or independence proof is claimed. [Related questions](/en/explore/?mko=mko-conditional-probability) [Concept guides](/en/derivations/?q=mko-conditional-probability) - Stage 3 · optional ## [Variance in finite probability models](/en/mko/mko-variance/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261010%2F) Compute w(X−μ)², exact (E[X²],μ²), variance after forming the shared-model difference, and the first positive-support mismatch relative to the first positive-weight value. Preserve zero-weight outliers; do not use n−1 sample variance, split Var(X−Y) into two variances, or claim complete covariance theory. [Related questions](/en/explore/?mko=mko-variance) [Concept guides](/en/derivations/?q=mko-variance) Path source & statuspath-pulse-foundations-20261010 · candidate [Original paths JSON](/data/architecture/learning-paths.json)