# Foundation Transforms: Angles, Number Lines, Equations, Factorials, and Circle Boundaries Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices. [← Return to results](/en/paths/) This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Points, lines, and planes](/en/mko/mko-point-line-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-point-line-plane) [Concept guides](/en/derivations/?q=mko-point-line-plane) - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Variables and algebraic expressions](/en/mko/mko-variable-expression/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Euclidean length](/en/mko/mko-euclidean-length/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Euclidean distance in an equal-unit orthonormal frame](/en/mko/mko-distance-formula/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for this density pulse. [Related questions](/en/explore/?mko=mko-distance-formula) [Concept guides](/en/derivations/?q=mko-distance-formula) - Stage 2 · optional ## [Angle](/en/mko/mko-angle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Use exact nondirected degree values in [0,180]: guard splits, distinguish complements/supplements, and compute two threshold defects and three-part budgets. Sums above 180 are not angles in this entry. Infer no measurements from sketches, ray lengths, or directed rotations. [Related questions](/en/explore/?mko=mko-angle) [Concept guides](/en/derivations/?q=mko-angle) - Stage 2 · optional ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Check the rational-point embedding slice only: ordered midpoints, finite distinct points and least containing closed intervals, inclusive interval intersections/separation, and four directed offsets/distances. Touching is a point interval, not empty; the containing interval is not the point set’s topological closure. No Cauchy completeness, irrational classification, or general continuity proof. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 2 · optional ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Check original Q equation sides, finite candidates, and reversibility premises. Multiplication by zero may change solutions; with denominator x−h first exclude x=h, and cross-multiplication cannot restore it. Check both shifted sides and defects without a universal AST or general solving-completeness claim. [Related questions](/en/explore/?mko=mko-equation) [Concept guides](/en/derivations/?q=mko-equation) - Stage 2 · optional ## [Factorial](/en/mko/mko-factorial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Use N from 0, 0!=1, and exact BigInt products: adjacent ratios, empty-product windows, independently checked reports/base and recurrence defects, and omission of one existing factor with multiplication back. Bound canonical decimal reports; recurrence self-consistency and floating-point values are not factorial correctness. No general permutation, divisibility, or asymptotic proof. [Related questions](/en/explore/?mko=mko-factorial) [Concept guides](/en/derivations/?q=mko-factorial) - Stage 2 · optional ## [Circle boundaries and zero-radius degeneracy](/en/mko/mko-circle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Use one equal-unit orthogonal Euclidean frame and a nonnegative integer radius. Check three squared residual/classification sets but only boundary membership, axis antipodes, radius-parameter scaling, and fixed-horizontal-offset square remainders. r=0 is only the center with collapsed endpoints; negative remainders have no real root. No general circle-API or similarity-theorem formal bridge. [Related questions](/en/explore/?mko=mko-circle) [Concept guides](/en/derivations/?q=mko-circle) - Stage 3 · optional ## [Linear equation in one variable](/en/mko/mko-linear-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Only a≠0 gives a one-variable linear equation over Q. Check normalized rows, parameter-dependent constant degeneracies, exact offsets around roots, and two rows’ roots/common solutions. a=0 can yield all Q or no solution but is not linear; no general linear-system or uniqueness formal Evidence. [Related questions](/en/explore/?mko=mko-linear-equation) [Concept guides](/en/derivations/?q=mko-linear-equation) - Stage 3 · optional ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261014%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Normalize rational representations before branch decisions; distinguish three-point distances from path defects, inclusive four-point radius tests, and rational-target roots/candidate audits. Negative targets have no roots and zero roots are deduplicated. Finite slices are not all real absolute-value properties, a universal triangle inequality, or general inequality solving. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) Path source & statuspath-pulse-foundations-20261014 · candidate [Original paths JSON](/data/architecture/learning-paths.json)